Open Access

Adaptive Rate Sampling and Filtering Based on Level Crossing Sampling

EURASIP Journal on Advances in Signal Processing20092009:971656

DOI: 10.1155/2009/971656

Received: 11 August 2008

Accepted: 14 April 2009

Published: 1 June 2009

Abstract

The recent sophistications in areas of mobile systems and sensor networks demand more and more processing resources. In order to maintain the system autonomy, energy saving is becoming one of the most difficult industrial challenges, in mobile computing. Most of efforts to achieve this goal are focused on improving the embedded systems design and the battery technology, but very few studies target to exploit the input signal time-varying nature. This paper aims to achieve power efficiency by intelligently adapting the processing activity to the input signal local characteristics. It is done by completely rethinking the processing chain, by adopting a non conventional sampling scheme and adaptive rate filtering. The proposed approach, based on the LCSS (Level Crossing Sampling Scheme) presents two filtering techniques, able to adapt their sampling rate and filter order by online analyzing the input signal variations. Indeed, the principle is to intelligently exploit the signal local characteristics—which is usually never considered—to filter only the relevant signal parts, by employing the relevant order filters. This idea leads towards a drastic gain in the computational efficiency and hence in the processing power when compared to the classical techniques.

1. Introduction

This work is part of a large project aimed to enhance the signal processing chain implemented in the mobile systems. The motivation is to reduce their size, cost, processing noise, electromagnetic emission and especially power consumption, as they are most often powered by batteries. This can be achieved by intelligently reorganizing their associated signal processing theory, and architecture. The idea is to combine event driven signal processing with asynchronous circuit design, in order to reduce the system processing activity and energy cost.

Almost all natural signals like speech, seismic, and biomedical are time varying in nature. Moreover, the man made signals like Doppler, Amplitude Shift Keying (ASK), and Frequency Shift Keying (FSK), also lay in the same category. The spectral contents of these signals vary with time, which is a direct consequence of the signal generation process [1]

The classical systems are based on the Nyquist signal processing architectures. These systems do not exploit the signal variations. Indeed, they sample the signal at a fixed rate without taking into account the intrinsic signal nature. Moreover they are highly constrained due to the Shannon theory especially in the case of low activity sporadic signals like electrocardiogram, phonocardiogram, seismic, and so forth. It causes to capture, and to process a large number of samples without any relevant information, a useless increase of the system activity, and its power consumption.

The power efficiency can be enhanced by intelligently adapting the system processing load according to the signal local variations. In this end, a signal driven sampling scheme, which is based on "level-crossing" is employed. The Level Crossing Sampling Scheme (LCSS) [2] adapts the sampling rate by following the local characteristics of the input signal [3, 4]. Hence, it drastically reduces the activity of the post-processing chain, because it only captures the relevant information [5, 6]. In this context, LCSS Based Analog to Digital Converters (LCADCs) have been developed [79]. Algorithms for processing [6, 1012], and analysis [3, 5, 13, 14] of the nonuniformly spaced out in time-sampled data, obtained with the LCSS have also been developed.

Filtering is a basic operation, almost required in every signal processing chain. Therefore, this paper focuses on the development of efficient Finite Impulse Response (FIR) filtering techniques. The idea is to pilot the system processing activity by the input signal variations. By following this idea, an efficient solution is proposed by intelligently combining the features of both nonuniform and uniform signal processing tools, which promise a drastic computational gain of the proposed techniques compared to the classical one.

Section 2 briefly reviews the nonuniform signal processing tools employed in the proposed approach. Complete functionality of the proposed filtering techniques is described in Section 3. Section 4 demonstrates the appealing features of the proposed techniques with the help of an illustrative example. The computational complexities of both proposed techniques are deduced and compared, among and to the classical case in Section 5. Section 6 discusses the processing error. In Section 7, the proposed techniques performance is evaluated for a speech signal. Section 8 finally concludes the article.

2. Nonuniform Signal Processing Tools

2.1. LCSS (Level Crossing Sampling Scheme)

The LCSS belongs to the signal-dependent sampling schemes like zero-crossing sampling [15], Lebesgue sampling [16], and reference signal crossing sampling [17]. The concept of LCSS is not new and has been known at least since 1950s [18]. It is also known as an event-based sampling [19, 20]. In recent years, there have been considerable interests in the LCSS, in a broad spectrum of technology and applications. In [2124], authors have employed it for monitoring and control systems. It has also been suggested in literature for compression [2], random processes [25], and band-limited Gaussian random processes [26].

The LCSS is a natural choice for sampling the time-varying signals. It lets the signal to dictate the sampling process [4]. The nonuniformity in the sampling process represents the signal local variations [3]. In the case of LCSS, a sample is captured only when the input analog signal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq1_HTML.gif crosses one of the predefined thresholds. The samples are not uniformly spaced in time because they depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq2_HTML.gif variations as it is clear from Figure 1.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig1_HTML.jpg
Figure 1

Level-crossing sampling scheme.

Let a set of levels which span the analog signal amplitude range be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq3_HTML.gif . These levels are equally spaced by a quantum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq4_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq5_HTML.gif crosses one of these predefined levels, a sample is taken [2]. This sample is the couple https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq6_HTML.gif of an amplitude https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq7_HTML.gif and a time https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq8_HTML.gif . However https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq9_HTML.gif is clearly equal to one of the levels and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq10_HTML.gif can be computed by employing
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ1_HTML.gif
(1)

In (1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq11_HTML.gif is the current sampling instant, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq12_HTML.gif is the previous one, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq13_HTML.gif is the time elapsed between the current and the previous sampling instants.

2.2. LCADC (LCSS-Based Analog to Digital Converter)

Classically, during an ideal A/D conversion process the sampling instants are exactly known, where as samples amplitudes are quantized at the ADC resolution [27], which is defined by the ADC number of bits. This error is characterized by the Signal to Noise Ratio (SNR) [27], which can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ2_HTML.gif
(2)

Here, M is the ADC number of bits. It follows that the SNR of an ideal ADC depends only on M and it can be improved by 6.02 dB for each increment in M.

The A/D conversion process, which occurs in the LCADCs [79], is dual in nature. Ideally in this case, samples amplitudes are exactly known since they are exactly equal to one of the predefined levels, while the sampling instants are quantized at the timer resolution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq14_HTML.gif . According to [7, 8], the SNR in this case is given by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ3_HTML.gif
(3)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq16_HTML.gif are the powers of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq17_HTML.gif and of its derivative, respectively. It shows that in this case, the SNR does not depend on M any more, but on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq18_HTML.gif characteristics and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq19_HTML.gif . An improvement of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq20_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq21_HTML.gif in the SNR can be achieved by simply halving https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq22_HTML.gif .

The choice of M is however crucial. It should be taken large enough to ensure a proper reconstruction of the signal. This problem has been addressed in [2831]. In particular, in [31], it is shown that a band-limited signal can be ideally reconstructed from nonuniformly spaced samples if the average number of samples satisfies the Nyquist criterion. In the case of LCADCs, the average sampling frequency depends on M and the signal characteristics [79]. Thus, for a given application an appropriate M should be chosen in order to respect the reconstruction criterion [31].

In [79], authors have shown advantages of the LCADCs over the classical ones. The major advantages are the reduced activity, the power saving, the reduced electromagnetic emission, and the processing noise reduction. Inspiring from these interesting features, the Asynchronous Analog to Digital Converter (AADC) [7] is employed to digitize https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq23_HTML.gif in the studied case. The characteristics of the filtering techniques described in the sequel are highly determined by the characteristics of the nonuniformly sampled signal produced by the AADC. We have already defined the AADC amplitude range https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq24_HTML.gif , the number of bits M and the quantum q. They are linked by the following relation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ4_HTML.gif
(4)
This quantum together with the AADC processing delay for one sample https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq25_HTML.gif yields the upper limit on the input signal slope, which can be captured properly:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ5_HTML.gif
(5)
In order to respect the reconstruction criterion [31] and the tracking condition [7], a band pass filter with pass-band https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq26_HTML.gif is employed at the AADC input. This together with a given M induces the AADC maximum and minimum sampling frequencies [6, 11], defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ6_HTML.gif
(6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ7_HTML.gif
(7)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq28_HTML.gif are the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq29_HTML.gif bandwidth and fundamental frequencies, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq31_HTML.gif are the AADC maximum and minimum sampling frequencies, respectively.

2.3. ASA (Activity Selection Algorithm)

The nonuniformly sampled signal obtained with the AADC can be used for further nonuniform digital processing [3, 10, 13]. However in the studied case, the nonuniformity of the sampling process, which yields information on the signal local features, is employed to select only the relevant signal parts. Furthermore, the characteristics of each signal selected part are analyzed and are employed later on to adapt the proposed system parameters accordingly. This selection and local-features extraction process is named as the ASA.

For activity selection, the ASA exploits the information laying in the level-crossing sampled signal nonuniformity [5]. This selection process corresponds to an adaptive length rectangular windowing. It defines a series of selected windows within the whole signal length. The ability of activity selection is extremely important to reduce the proposed system processing activity and consequently its power consumption. Indeed, in the proposed case, no processing is performed during idle signal parts, which is one of the reasons of the achieved computational gain compared to the classical case. The ASA is defined as follow:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ8_HTML.gif
(8)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq32_HTML.gif is clear from (1). https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq33_HTML.gif is the fundamental period of the bandlimited signal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq35_HTML.gif detect parts of the nonuniformly sampled signal with activity. If the measured time delay https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq36_HTML.gif is greater than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq37_HTML.gif is considered to be idle. The condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq38_HTML.gif is chosen to ensure the Nyquist sampling criterion for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq39_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq40_HTML.gif is the reference window length. Its choice depends on the input signal characteristics and the system resources. The upper bound on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq41_HTML.gif is posed by the maximum number of samples that the system can treat at once. Whereas the lower bound on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq42_HTML.gif is posed by the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq43_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq44_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq45_HTML.gif , which should be respected in order to achieve a proper spectral representation [5].

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq46_HTML.gif represents the length in seconds of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq47_HTML.gif selected window https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq48_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq49_HTML.gif poses the upper bound on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq50_HTML.gif represents the number of nonuniform samples laying in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq51_HTML.gif , which lies on the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq52_HTML.gif active part of the nonuniformly sampled signal. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq54_HTML.gif both belong to the set of natural numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq55_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq56_HTML.gif signal activity can be longer than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq57_HTML.gif . In this case, it will be splitted into more than one selected windows.

The above-described loop repeats for each selected window, which occurs during the observation length of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq58_HTML.gif . Every time before starting the next loop, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq59_HTML.gif is incremented and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq61_HTML.gif are initialized to zero.

The maximum number of samples https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq62_HTML.gif , which can take place within a chosen https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq63_HTML.gif can be calculated by employing
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ9_HTML.gif
(9)

The ASA displays interesting features, which are not available in the classical case. It only selects the active parts of the nonuniformly sampled signal. Moreover, it correlates the length of the selected window with the input signal activity, laying in it. In addition, it also provides an efficient reduction of the phenomenon of spectral leakage in the case of transient signals. The leakage reduction is achieved by avoiding the signal truncation problem with a simple and an efficient algorithm, instead of employing a smoothening (cosine) window function, which is used in the classical schemes [5]. These abilities make the ASA extremely effective in reducing the overall system processing activity, especially in the case of low activity sporadic signals [5, 6, 11, 12, 14].

3. Proposed Adaptive Rate Filtering

3.1. General Principle

Two techniques are described to filter the selected signal obtained at the ASA output. The signal processing chain common to both filtering techniques is shown in Figure 2.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig2_HTML.jpg
Figure 2

Signal processing chain common to both filtering techniques.

The activity selection and the local features extraction are the bases of the proposed techniques. They make to achieve the adaptive rate sampling (only relevant samples to process) along with the adaptive rate filtering (only relevant operations to deliver a filtered sample). Such an achievement assures a drastic computational gain of the proposed filtering techniques compared to the classical one. The steps of realizing these ideas are detailed in the following subsections.

3.1.1. Adaptive Rate Sampling

The AADC sampling frequency is correlated to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq64_HTML.gif local variations [6, 11, 12, 14]. It follows that the local sampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq65_HTML.gif can be specific for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq66_HTML.gif . According to [5] https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq67_HTML.gif can be calculated by employing
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ10_HTML.gif
(10)

The upper and the lower bounds on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq68_HTML.gif are posed by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq70_HTML.gif , respectively. In order to perform a classical filtering algorithm, the selected signal laying in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq71_HTML.gif is uniformly resampled before proceeding to the filtering stage (cf. Figure 2). Characteristics of the selected signal part laying in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq72_HTML.gif are employed to choose its resampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq73_HTML.gif . Once the resampling is done, there are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq74_HTML.gif samples in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq75_HTML.gif . Choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq76_HTML.gif is crucial and this procedure is detailed in the following subsection.

3.1.2. Adaptive Rate Filtering

It is known that for fixed design parameters (cut-off frequency, transition-band width, pass-band, and stop-band ripples) the FIR filter order varies as a function of the operational sampling frequency. For high sampling frequency, the order is high and vice versa. In the classical case, the sampling frequency and filter order both remains unique regardless of the input signal variations, so they have to be chosen for the worst case. This time invariant nature of the classical filtering causes a useless increase of the computational load. This drawback has been resolved up to a certain extent by employing the multirate filtering techniques [3234].

The proposed filtering techniques of this paper are the intelligent alternatives to the multirate filtering techniques. They achieve computational efficiency by adapting the sampling frequency and the filter order according to the input signal local variations. Both techniques have some common features, which are described in the following.

In both cases, a reference FIR filter is offline designed for a reference sampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq77_HTML.gif . Its impulse response is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq78_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq79_HTML.gif is indexing the reference filter coefficients. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq80_HTML.gif is chosen in order to satisfy the Nyquist sampling criterion for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq81_HTML.gif , namely https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq82_HTML.gif .

During online computation, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq83_HTML.gif and the local sampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq84_HTML.gif of window https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq85_HTML.gif are used to define the local resampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq86_HTML.gif and a decimation factor https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq87_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq88_HTML.gif is employed to uniformly resample the selected signal laying in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq89_HTML.gif , where as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq90_HTML.gif is employed to decimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq91_HTML.gif for filtering https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq92_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq93_HTML.gif can be specific depending upon https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq94_HTML.gif [11, 12]. For proper online filtering, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq96_HTML.gif should match. The approaches of keeping https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq98_HTML.gif coherent are explained below.

In the case, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq100_HTML.gif is chosen and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq101_HTML.gif remains unchanged. This case is treated similarly by both proposed techniques. This choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq102_HTML.gif makes to resample https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq103_HTML.gif closer to the Nyquist rate, so avoiding unnecessary interpolations during the data resampling process. It thus further improves the proposed technique computational efficiency. This case is included in the description (see flowcharts in Figures 3 and 4) of the following two filtering techniques.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig3_HTML.jpg
Figure 3

Flowchart of the ARD.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig4_HTML.jpg
Figure 4

Flowchart of the ARR.

In the opposite case, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq105_HTML.gif is chosen and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq106_HTML.gif is online decimated in order to reduce https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq107_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq108_HTML.gif . In this case, the reference filter order is reduced for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq109_HTML.gif , which reduces the number of operations to deliver a filtered sample [6, 11]. Hence, it improves the proposed techniques computational efficiency. In this case, it appears that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq110_HTML.gif may be lower than the Nyquist frequency of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq111_HTML.gif and so it can cause aliasing. According to [6, 11], if the local signal amplitude is of the order of the maximal range https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq112_HTML.gif then for a suitable choice of M (application-dependent) the signal crosses enough consecutive thresholds. Thus, it is locally oversampled with respect to its local bandwidth and so there is no aliasing problem. This statement is further illustrated with the results summarized in Table 3.

In order to decimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq113_HTML.gif the decimation factor https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq114_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq115_HTML.gif is online calculated by employing
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq116_HTML.gif can be specific for each selected window depending upon https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq117_HTML.gif . For an integral https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq118_HTML.gif both techniques decimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq119_HTML.gif in a similar way. Thus, a test on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq120_HTML.gif is made by computing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq121_HTML.gif and verifying if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq122_HTML.gif . Here, floor operation delivers only the integral part of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq123_HTML.gif . If the answer is yes, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq124_HTML.gif is decimated with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq125_HTML.gif , the process is clear from
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ12_HTML.gif
(12)

Equation (12) shows that the decimated filter impulse response for the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq126_HTML.gif selected window https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq127_HTML.gif is obtained by picking every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq128_HTML.gif coefficient from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq129_HTML.gif . Here, j is indexing the decimated filter coefficients. If the order of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq130_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq131_HTML.gif , then the order of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq132_HTML.gif is given as: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq133_HTML.gif .

A simple decimation causes a reduction of the decimated filter energy compared to the reference one. It will lead to an attenuated version of the filtered signal. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq134_HTML.gif is a good approximate of the ratio between the energy of the reference filter and that of the decimated one. Thus, this effect of decimation is compensated by scaling https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq135_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq136_HTML.gif . The process is clear from

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ13_HTML.gif
(13)

The two techniques mainly differ in the way of decimating https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq137_HTML.gif for a fractional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq138_HTML.gif . The process is explained in the following Sections.

3.2. ARD (Activity Reduction by Filter Decimation)

In the ARD technique, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq139_HTML.gif is decimated by employing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq140_HTML.gif . It calls for an adjustment of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq141_HTML.gif which is achieved as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq142_HTML.gif . As in this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq143_HTML.gif , so it makes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq144_HTML.gif . For the ARD https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq145_HTML.gif scaling is performed with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq146_HTML.gif . The complete procedure of obtaining https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq148_HTML.gif for the ARD is described in Figure 3.

3.3. ARR (Activity Reduction by Filter Resampling)

In the ARR technique, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq149_HTML.gif is employed to decimated https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq150_HTML.gif . In this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq151_HTML.gif is given as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq152_HTML.gif , so it remains equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq153_HTML.gif . The process of matching https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq155_HTML.gif requires a fractional decimation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq156_HTML.gif , which is achieved by resampling https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq157_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq158_HTML.gif . Again NNRI is employed for the purpose of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq159_HTML.gif resampling. For the ARR https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq160_HTML.gif scaling is performed with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq161_HTML.gif . The complete procedure of obtaining https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq163_HTML.gif for the ARR is described in Figure 4.

4. Illustrative Example

In order to illustrate the ARD and the ARR filtering techniques, an input signal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq164_HTML.gif shown on the left part of Figure 5 is employed. Its total duration is 20 seconds and it consists of three active parts. Summary of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq165_HTML.gif activities is given in Table 1.
Table 1

Summary of the input signal active parts.

Active part

Signal components

Length (s)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq166_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq167_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq168_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq169_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq170_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq172_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq173_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq174_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig5_HTML.jpg
Figure 5

The input signal (left) and the selected signal obtained with the ASA (right).

Table 1 shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq175_HTML.gif is band limited between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq176_HTML.gif Hz and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq177_HTML.gif  kHz. In this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq178_HTML.gif is digitized by employing a 3-bit resolution AADC. Thus, for given ENOB the corresponding minimum and maximum sampling frequencies are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq179_HTML.gif  Hz and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq180_HTML.gif  kHz. The AADC amplitude range https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq181_HTML.gif  v is chosen, which results into a quantum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq182_HTML.gif  v.

Each activity contains a low- and a high-frequency component (cf. Table 1). In order to filter out the high-frequency parts from each activity, a low pass reference FIR filter is implemented by employing the standard Parks-McClellan algorithm. The reference filter parameters are summarized in Table 2.
Table 2

Summary of the reference filter parameters.

Cut-off frequency (Hz)

Transition band (Hz)

Pass-band ripples (dB)

Stop-band ripples (dB)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq183_HTML.gif (Hz)

P

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq184_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq185_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq186_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq187_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq188_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq189_HTML.gif

Table 3

Summary of the selected windows parameters.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq190_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq191_HTML.gif (Sec.)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq192_HTML.gif (Samples)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq193_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq194_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq195_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq196_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq197_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq198_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq199_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq200_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq201_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq202_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq203_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq204_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq205_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq206_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq207_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq208_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq209_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq210_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq211_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq212_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq213_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq214_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq215_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq216_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq217_HTML.gif

For this example the reference window length https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq218_HTML.gif second is chosen. It satisfies the boundary conditions discussed in Section 2.3. The given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq219_HTML.gif delivers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq220_HTML.gif samples in this case (cf. Equation (9). The ASA delivers three selected windows for the whole https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq221_HTML.gif span of 20 seconds, which are shown on the right part of Figure 5. The selected windows parameters are displayed in Table 3.

Table 3 shows that the first window is an example of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq222_HTML.gif case, so it is tackled similarly by both techniques. In the other windows, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq223_HTML.gif is valid, so the online https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq224_HTML.gif decimation is employed. As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq226_HTML.gif , calculated by employing Equation (11) are fractional ones, so this case is tackled in a different way by the ARD and the ARR.

Values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq229_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq230_HTML.gif are calculated for the ARD, and the ARR by employing the methods shown in Figures 3 and 4, respectively. The obtained results are summarized in Tables 4 and 5.
Table 4

Values of Frs i , Nr i , D i , and P i for each selected window in the ARD.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq231_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq232_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq233_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq234_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq235_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq236_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq237_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq238_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq239_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq240_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq241_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq242_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq243_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq244_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq245_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq246_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq247_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq248_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq249_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq250_HTML.gif

Table 5

Values of Frs i , Nr i , d i and P i for each selected window in the ARR.

i

Frs i (Hz)

Nr i

d i

P i

1

2500

1250

1

127

2

1083

1083

2.3

54

3

464

464

5.4

24

Tables 3, 4, and 5 jointly exhibit the interesting features of the proposed filtering techniques, which are achieved by an intelligent combination of the nonuniform, and the uniform signal processing tools (cf. Figure 2). https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq251_HTML.gif represents the sampling frequency adaptation by following the local variations of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq252_HTML.gif shows that the relevant signal parts are locally over-sampled in time with respect to their local bandwidths [6, 11]. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq253_HTML.gif shows the adaptation of the resampling frequency for each selected window. It further adds to the computational gain of the proposed techniques by avoiding the unnecessary interpolations during the resampling process. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq254_HTML.gif shows how the adjustment of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq255_HTML.gif avoids the processing of unnecessary samples during the post filtering process. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq256_HTML.gif represents how the adaptation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq257_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq258_HTML.gif avoids the unnecessary operations to deliver the filtered signal. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq259_HTML.gif exhibits the dynamic feature of ASA, which is to correlate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq260_HTML.gif with the signal activity laying in it [5].

These results have to be compared with what is done in the corresponding classical case. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq261_HTML.gif is chosen as the sampling frequency, then the total https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq262_HTML.gif span is sampled at 2500 Hz. It makes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq263_HTML.gif samples to process with the 127th-order FIR filter. On the other hand, in both proposed techniques the total number of resampled data points is much lower, 3000 and 2794 for the ARD and the ARR, respectively. Moreover, the local filter orders in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq264_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq265_HTML.gif are also lower than 127. It promises the computational efficiency of the proposed techniques compared to the classical one. A detailed complexity comparison is made in the following Section.

5. Computational Complexity

In the classical case, with a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq266_HTML.gif order filter, it is well known that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq267_HTML.gif multiplications and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq268_HTML.gif additions are required to compute each filtered sample. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq269_HTML.gif is the number of samples then the total computational complexity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq270_HTML.gif can be calculated by employing

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ14_HTML.gif
(14)

In the adaptive techniques presented here, the adaptation process requires extra operations for each selected window. The computational complexities of both techniques, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq272_HTML.gif are deduces as follow.

The following steps are common to both the ARD and the ARR techniques. The choice of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq273_HTML.gif is a common operation for both proposed techniques. It requires one comparison between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq275_HTML.gif . The data resampling operation is also required in both techniques before filtering. In the studied case, the resampling process is performed by employing the Nearest Neighbour Resampling Interpolation (NNRI). The NNRI is chosen because of its simplicity, as it employs only one nonuniform observation for each resampled one. Moreover, it provides an unbiased estimate of the original signal variance. Due to this reason, it is also known as a robust interpolation method [35, 36]. The detailed reasons of inclination toward NNRI are discussed in [5, 35, 36]. The NNRI is performed as follow.

For each interpolation instant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq276_HTML.gif , the interval of nonuniform samples https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq277_HTML.gif , within which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq278_HTML.gif lies is determined. Then the distance of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq279_HTML.gif to each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq280_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq281_HTML.gif is computed and a comparison among the computed distances is performed to decide the smaller among them. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq282_HTML.gif , the complexity of the first step is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq283_HTML.gif comparisons and the complexity of the second step is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq284_HTML.gif additions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq285_HTML.gif comparisons. Hence, the NNRI total complexity for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq286_HTML.gif becomes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq287_HTML.gif comparisons and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq288_HTML.gif additions.

In the case, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq289_HTML.gif , the decimation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq290_HTML.gif is performed in both techniques. In order to do so, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq291_HTML.gif is computed by performing a division between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq293_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq294_HTML.gif is calculated by employing a floor operation on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq295_HTML.gif . A comparison is made between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq297_HTML.gif . In the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq298_HTML.gif , the process of obtaining https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq299_HTML.gif is similar for both techniques (cf. Figures 3 and 4). In this case, the decimator simply picks every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq300_HTML.gif th coefficient from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq301_HTML.gif . It has a negligible complexity compared to the operations like addition and multiplication. This is the reason why its complexity is not taken into account during the complexity evaluation process. In both techniques, the decimated filter impulse response is scaled, it requires https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq302_HTML.gif multiplications. The fractional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq303_HTML.gif is tackled in a different way by each filtering technique and is detailed in the following subsections.

5.1. Complexity of the ARD Technique

Even if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq304_HTML.gif is fractional in the case of ARD technique, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq305_HTML.gif decimation is performed by employing D i . Frs i is modified in order to keep it coherent with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq306_HTML.gif and it requires one division (cf. Figure 3). Finally, a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq307_HTML.gif -order filter performs https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq308_HTML.gif multiplications and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq309_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq310_HTML.gif additions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq311_HTML.gif . The combine computational complexity for the ARD technique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq312_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ15_HTML.gif
(15)

5.2. Complexity of the ARR Technique

In the case of ARR technique, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq313_HTML.gif is employed as the decimation factor. The fractional decimation is achieved by resampling https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq314_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq315_HTML.gif . The resampling is performed by employing the NNRI, which performs https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq316_HTML.gif comparisons and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq317_HTML.gif additions to deliver h j i . The remaining operation cost between the ARD and the ARR is common. The combine computational complexity for the ARR technique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq318_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ16_HTML.gif
(16)

In Equations (15) and (16), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq319_HTML.gif represents the selected windows index. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq320_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq321_HTML.gif are the multiplying factors. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq322_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq323_HTML.gif for the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq324_HTML.gif and it is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq325_HTML.gif otherwise. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq326_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq327_HTML.gif for the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq328_HTML.gif and it is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq329_HTML.gif otherwise.

5.3. Complexity Comparison of the ARD and the ARR with the Classical Filtering

From (14), (15), and (16), it is clear that there are uncommon operations between the classical and the proposed adaptive rate filtering techniques. In order to make them approximately comparable, it is assumed that a comparison has the same processing cost as that of an addition and a division or a floor has the same processing cost as that of a multiplication. By following these assumptions, comparisons are merged into the additions count and divisions plus floors are merged into the multiplications count, during the complexity evaluation process. Now Equations (15) and (16) can be written as follow:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ18_HTML.gif
(18)
By employing results of the example studied in the previous section, computational comparisons of the ARD and the ARR with the classical one are made in terms of additions and multiplications. The results are computed for different https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq330_HTML.gif time spans and are summarized in Tables 6 and 7.
Table 6

Computational gain of the ARD over the classical one for different x(t) time spans.

Signal part

Gain in additions

Gain in multiplications

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq331_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq332_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq333_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq334_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq335_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq336_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq337_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq338_HTML.gif

24.37

Whole signal

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq339_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq340_HTML.gif

Table 7

Computational gain of the ARR over the classical one for different https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq341_HTML.gif time spans.

Signal part

Gain in additions

Gain in multiplications

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq342_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq343_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq344_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq345_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq346_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq347_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq348_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq349_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq350_HTML.gif

Whole signal

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq351_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq352_HTML.gif

Gains in additions and multiplications of the proposed techniques over the classical one are clear from the above results. In the case of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq353_HTML.gif , where the resampling frequency and the filter order is the same as in the classical case (cf. Tables 4 and 5), a gain is achieved by using the proposed adaptive techniques. This is only due to the fact that the ASA correlates the window length to the activity (0.5 second), while the classic case computes during the total duration of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq354_HTML.gif  second. Gains are of course much larger in other windows, since the proposed techniques are taking benefit of processing the lesser samples along with the lower filter orders. When treating the whole https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq355_HTML.gif span of 20 seconds, the proposed techniques also take advantage of the idle https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq356_HTML.gif parts, which further induces additional gains compared to the classical case.

The above results confirm that the proposed filtering techniques lead toward a drastic reduction in the number of operations compared to the classical one. This reduction in operations is achieved due to the joint benefits of the AADC, the ASA and the resampling, as they enable to adapt the sampling frequency and the filter order by following the input signal local variations.

5.4. Complexity Comparison between the ARD and the ARR

The main difference between both proposed techniques occurs for the case when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq357_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq358_HTML.gif is fractional (cf. Section 3).

The ARD makes an increment in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq359_HTML.gif in order to keep it coherent with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq360_HTML.gif . Increase in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq361_HTML.gif causes to increase https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq362_HTML.gif and also to increase https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq363_HTML.gif . Thus, in comparison to the ARR, this technique increases the computational load of the post-filtering operation, while keeping the decimation process of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq364_HTML.gif simple.

The ARR performs https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq365_HTML.gif resampling at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq366_HTML.gif . Thus, in comparison to the ARD, this technique increases the complexity of the decimation process of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq367_HTML.gif while keeping the computational load of the post-filtering process lower.

In continuation to Section 5.3, a complexity comparison between the ARD and the ARR is made in terms of additions, and multiplications by employing Equations (17) and (18), respectively. It concludes that the ARR remains computationally efficient compared to the ARD, in terms of additions and multiplications, as far as the conditions given by expressions (19) and (20) remain true. Please note that Nr i and P i can be different for the ARD and the ARR (cf. Tables 4 and 5):

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ20_HTML.gif
(20)

For this studied example, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq368_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq369_HTML.gif are fractional ones, thus the ARD and the ARR proceed differently. Conditions (19) and (20) remain true for both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq370_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq371_HTML.gif (cf. Tables 4 and 5). Hence, the gains in additions and multiplications of the ARR are higher than those of the ARD for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq372_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq373_HTML.gif (cf. Tables 6 and 7). It shows that except for very specific situation the ARR technique will always remain less expensive than the ARD. The ARR achieves this computational performance by employing the fractional decimation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq374_HTML.gif , which may lead a quality compromise of the ARR compared to the ARD. This issue is addressed in the following section.

6. Processing Error

6.1. Approximation Error

In the proposed techniques, the approximation error occurs due to two effects: the time quantization error which occurs due to the AADC finite timer precision and the interpolation error which occurs in the course of the uniform resampling process. After these two operations, the mean approximation error for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq375_HTML.gif can be computed by employing the following:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ21_HTML.gif
(21)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq376_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq377_HTML.gif resampled observation, interpolated with respect to the time instant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq378_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq379_HTML.gif is the original sample value which should be obtained by sampling https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq380_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq381_HTML.gif . In the studied example discussed in Section 4, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq382_HTML.gif is analytically known, thus it is possible to compute its original sample value at any given time instant. It allows us to compute the approximation error introduced by the proposed adaptive rate techniques by employing Equation (21).

The results obtained for each selected window for both the ARD and the ARR are summarized in Table 8.
Table 8

Mean approximation error of each selected window for the ARD and the ARR.

Selected window

W 1

W 2

W 3

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq383_HTML.gif for the ARD (dB)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq384_HTML.gif 23.71

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq385_HTML.gif 25.84

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq386_HTML.gif 26.35

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq387_HTML.gif for the ARR (dB)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq388_HTML.gif 23.71

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq389_HTML.gif 25.93

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq390_HTML.gif 26.63

Table 8 shows the approximation error introduced by the proposed techniques. This process is accurate enough for a 3-bit AADC. For the higher precision applications, the approximation accuracy can be improved by increasing the AADC resolution M and the interpolation order [6, 8, 37, 38]. Thus, an increased accuracy can be achieved at the cost of an increased computational load. Therefore, by making a suitable compromise between the accuracy level and the computational load, an appropriate solution can be devised for a specific application.

For a given M and interpolation order the approximation accuracy can be further improved by employing the symmetry during the interpolation process. It results into a reduced resampling error [38, 39]. The pros and cons of this approach are under investigation and a description on it is given in [40].

6.2. Filtering Error

In the proposed filtering techniques, a reference filter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq391_HTML.gif is employed and then it is online decimated for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq392_HTML.gif , depending on the chosen https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq393_HTML.gif . This online decimation can cause the filtering precision degradation. In order to evaluate this phenomenon on our test signal the following procedure is adapted.

A reference filtered signal is generated. In this case, instead of decimating https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq394_HTML.gif to obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq395_HTML.gif , a specific filter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq396_HTML.gif is directly designed for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq397_HTML.gif by using the Parks-McClellan algorithm. It is designed for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq398_HTML.gif by employing the same design parameters, summarized in Table 2. The signal activity corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq399_HTML.gif is sampled at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq400_HTML.gif with a high precision classical ADC. This sampled signal is filtered by employing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq401_HTML.gif . The filtered signal obtained in this way is used as a reference one for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq402_HTML.gif and its comparison is made with the results obtained by the proposed techniques.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq403_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq404_HTML.gif th reference-filtered sample and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq405_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq406_HTML.gif th filtered sample obtained by one of the proposed filtering techniques. Then, the mean filtering error for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq407_HTML.gif can be calculated by employing

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Equ22_HTML.gif
(22)
The mean filtering error of both proposed techniques is calculated, for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq408_HTML.gif activity by employing (22). The results are summarized in Table 9.
Table 9

Mean filtering error of each selected window for the ARD and the ARR.

Selected window

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq409_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq410_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq411_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq412_HTML.gif for the ARD (dB)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq413_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq414_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq415_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq416_HTML.gif for the ARR (dB)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq417_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq418_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq419_HTML.gif

Table 9 shows that the online decimation of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq420_HTML.gif in the proposed techniques causes a loss of the desired filtering quality. Indeed, the filtering error increases with the increase in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq421_HTML.gif . The measure of this error can be used to decide an upper bound to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq422_HTML.gif (by performing an offline calculation), for which the decimated and the scaled filters provide results with an acceptable level of accuracy. The level of accuracy is application-dependent. Moreover, for high precision applications, an appropriate filter can be online calculated for each selected window at the cost of an increased computational load. The process is clear from generating the reference filtered signal https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq423_HTML.gif , discussed above.

Table 9 shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq424_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq425_HTML.gif for the ARR are higher than that of the ARD. It is due to the fact of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq426_HTML.gif resampling for the ARR to deliver https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq427_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq428_HTML.gif . It makes to employ the interpolated coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq429_HTML.gif for filtering the resampled data, lies in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq430_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq431_HTML.gif respectively, which results in an increased filtering error of the ARR compared to the ARD. Similar to Section 6.1, this resampling error can also be reduced to a certain extent, by employing a higher order interpolator [37, 38]. In conclusion, a certain increase in the accuracy can be achieved at a certain loss of the processing efficiency.

7. Speech Signal as a Case Study

In order to evaluate performances of the ARD and the ARR for real life signals, a speech signal x(t) shown on Figure 6(a) is employed. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq432_HTML.gif is a 1.6 second, [50 Hz; 5000 Hz] band-limited signal corresponding to a three-word sentence. The goal is to determine the pitch (fundamental frequency) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq433_HTML.gif in order to determine the speaker's gender. For a male speaker, the pitch lies with the frequency range [100 Hz, 150 Hz], whereas for a female speaker, the pitch lies with the frequency range [200 Hz, 300 Hz] [41].
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_Fig6_HTML.jpg
Figure 6

On the top, the input speech signal (a), the selected signal with the ASA (b) and a zoom of the second window W 2 (c). On the bottom, a spectrum zoom of the filtered signal laying in W2 obtained with the reference filtering (d), with the ARD (e) and with the ARR (f), respectively.

The reference frequency is chosen as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq434_HTML.gif kHz, which is a common sampling frequency for speech. A 4-bit resolution AADC is used for digitizing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq435_HTML.gif and therefore we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq436_HTML.gif  kHz, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq437_HTML.gif  kHz. The amplitude range is always set to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq438_HTML.gif  V, which leads to a quantum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq439_HTML.gif  v. The amplitude of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq440_HTML.gif is normalized to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq441_HTML.gif  v in order to avoid the AADC saturation.

The studied signal is part of a conversation and during a dialog, the speech activity is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq442_HTML.gif of the total dialog time [42]. A classical filtering system would remain active during the total dialog duration. The proposed LCSS-based filtering techniques will remain active only during https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq443_HTML.gif of the dialog time span, which will reduce the system power consumption.

A speech signal mainly consists of vowels and consonants. Consonants are of lower amplitude compared to vowels [41, 43]. In order to determine the speakers pitch, vowels are the relevant parts of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq444_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq445_HTML.gif  v, consonants are ignored during the signal acquisition process, and are considered as low amplitude noise. In contrast, vowels are locally over-sampled like any harmonic signal [6, 10, 11]. This intelligent signal acquisition further avoids the processing of useless samples, within the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq446_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq447_HTML.gif activity, and so further improves the proposed techniques computational efficiency.

In order to apply the ASA, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq448_HTML.gif  seconds is chosen. It results in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq449_HTML.gif in this case (cf. Equation (9). The ASA delivers three selected windows, which are shown on Figure 6(b). The parameters of each selected window are summarized in Table 10.
Table 10

Summary of the selected windows parameters.

Selected window

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq450_HTML.gif (Second)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq451_HTML.gif (Samples)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq452_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq453_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq454_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq455_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq456_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq457_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq458_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq459_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq460_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq461_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq462_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq463_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq464_HTML.gif

Although the consonants are partially filtered out during the data acquisition process, yet for proper pitch estimation, it is required to filter out the remaining effect of high frequencies still present in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq465_HTML.gif . To this aim, a reference low pass filter is designed, with the standard Parks-McClellan algorithm. Its characteristics are summarized in Table 11.
Table 11

Summary of the reference filter paramete 1

Cut-off frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq466_HTML.gif

Transition band https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq467_HTML.gif

Pass-band ripples (dB)

Stop-band ripples https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq468_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq469_HTML.gif

P( order)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq470_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq471_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq472_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq473_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq474_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq475_HTML.gif

To find the pitch, we now focus on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq476_HTML.gif , which corresponds to the vowel "a". A zoom on this signal part is plotted on Figure 6(c). The condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq477_HTML.gif is valid, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq478_HTML.gif is fractional (cf. Equation (11).Thus, the filtering process for each proposed technique will differ, which makes it possible to compare their performances. The values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq479_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq480_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq481_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq482_HTML.gif for both techniques are given in Table 12.
Table 12

Values of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq483_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq484_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq485_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq486_HTML.gif for the ARD and the ARR.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq487_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq488_HTML.gif (Hz)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq489_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq490_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq491_HTML.gif

ARD

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq492_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq493_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq494_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq495_HTML.gif

ARR

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq496_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq497_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq498_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq499_HTML.gif

Computational gains of the proposed filtering techniques compared to the classical one are computed by employing Equations (14), (17), and (18). The results show 8.62 and 13.17 times gains in additions and 8.71 and 13.26 times gains in multiplications, respectively, for the ARD and the ARR, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq500_HTML.gif . It confirms the computational efficiency of the proposed techniques compared to the classical one. It is gained firstly by achieving an intelligent signal acquisition and secondly by adapting the sampling frequency and the filter order by following the local variations of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq501_HTML.gif .

Once more the conditions (19) and (20) remain true for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq502_HTML.gif so the ARR technique remains computationally efficient than the ARD one.

Spectra of the filtered signal laying in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq503_HTML.gif , obtained with the reference filtering (cf. Section 6.2), with the ARD and with the ARR techniques are plotted, respectively, on Figures 6(d), 6(e), and 6(f).

The spectra on Figure 6 show that the fundamental frequency is about 215 HZ. Thus, one can easily conclude that the analyzed sentence is pronounced by a female speaker. Although it is required to decimate the reference filter 3 times and 3.7 times, respectively, for the ARD and the ARR, yet spectra of the filtered signal, obtained with the proposed techniques are quite comparable to spectrum of the reference-filtered signal. It shows that even after such a level of decimation, results delivered by the proposed techniques are of acceptable quality for the studied speech application.

The above discussion shows the suitability of the proposed techniques for the low activity time-varying signals like electrocardiogram, phonocardiogram, seismic, and speech. Speech is a common, and easily accessible signal. Therefore, the proposed techniques performance is studied for a speech application, though it can be applied to other appropriate real signals like electrocardiogram, phonocardiogram, and seismic. The devised approach versatility lays in the appropriate choice of system parameters like the AADC resolution M, the distribution of level crossing thresholds, and the interpolation order. These parameters should be tactfully chosen for a targeted application, so that they ensure an attractive tradeoff between the system computational complexity and the delivered output quality.

8. Conclusion

Two novel adaptive rate filtering techniques have been devised. These are well suited for low activity sporadic signals like electrocardiogram, phonocardiogram and seismic signals. For both filtering techniques, a reference filter is offline designed by taking into account the input signal statistical characteristics and the application requirements.

The complete procedure of obtaining the resampling frequency https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq504_HTML.gif and the decimated filter coefficients https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq505_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq506_HTML.gif is described for both proposed techniques. The computational complexities of the ARD and the ARR are deduced and compared with the classical one. It is shown that the proposed techniques result into a more than one-order magnitude gain in terms of additions and multiplications over the classical one. It is achieved due to the joint benefits of the AADC, the ASA and the resampling as they allow the online adaptation of parameters ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq507_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F971656/MediaObjects/13634_2008_Article_2651_IEq508_HTML.gif ) by exploiting the input signal local variations. It drastically reduces the total number of operations and therefore, the energy consumption compared to the classical case.

A complexity comparison between the ARD and the ARR is also made. It is shown that the ARR outperforms the ARD in most of the cases. Performances of the ARD and the ARR are also demonstrated for a speech application. The results obtained in this case are in coherence with those obtained for the illustrative example.

Methods to compute the approximation and the filtering errors for the proposed techniques are also devised. It is shown that the errors made by the proposed techniques are minor ones, in the studied case. A higher precision can be achieved by increasing the AADC resolution and the interpolation order. Thus, a suitable solution can be proposed for a given application by making an appropriate tradeoff between the accuracy level and the computational load.

A detailed study of the proposed filtering techniques computational complexities by taking into account the real processing cost at circuit level is in progress. Future works focus on the optimization of these filtering techniques and their further employment in real life applications.

Authors’ Affiliations

(1)
TIMA, CNRS UMR 5159
(2)
Tiempo SAS

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© Saeed Mian Qaisar et al. 2009

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