The

*K* th order PPS arriving on the

*m* th ULA sensor can be rewritten as follows:

${x}_{m}\left(n\right)=A\text{exp}\left\{j\left({a}_{K}{\left(n\mathrm{\Delta}\right)}^{K}+\sum _{k=0}^{K-1}\left({a}_{k}+\left(k+1\right){a}_{k+1}\psi m\right){\left(n\mathrm{\Delta}\right)}^{k}\right)\right\},\phantom{\rule{2.77695pt}{0ex}}m=0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}M-1.$

(7)

In order to estimate the DOA and PPS parameters of

*x*_{
m
}(

*n*), we propose to calculate the

*K* th higher-order instantaneous moment (HIM) of

*x*_{
m
}(

*n*) as [

14]

${\mathsf{\text{HIM}}}_{K}\phantom{\rule{0.3em}{0ex}}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right]={\mathsf{\text{HIM}}}_{2}[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{K-1}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right],\phantom{\rule{2.77695pt}{0ex}}\tau ,$

(8)

with the first two HIM orders given by

$\begin{array}{c}\mathsf{\text{HI}}{\mathsf{\text{M}}}_{1}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right]={x}_{m}\left(n\right),\\ \mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right]={x}_{m}\left(n\phantom{\rule{2.77695pt}{0ex}}+\tau \right){x}_{m}^{*}\left(n\phantom{\rule{2.77695pt}{0ex}}-\tau \right),\end{array}$

(9)

where

*τ* is the time lag parameter. Note that the HIM is implemented through recursive auto-correlations. The

*K* th order HIM of

*x*_{
m
}(

*n*) equals

$\begin{array}{ll}\hfill \mathsf{\text{HI}}{\mathsf{\text{M}}}_{K}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right]& ={A}^{{2}^{K-1}}\text{exp}\left\{j({2}^{K-1}K!{\tau}^{K-1}{\mathrm{\Delta}}^{K}{a}_{K}\left(n+m\psi /\mathrm{\Delta}\right)+{2}^{K-1}\left(K-1\right)!{\left(\tau \mathrm{\Delta}\right)}^{K-1}{a}_{K-1}\right\}\phantom{\rule{2em}{0ex}}\\ =B\text{exp}\left\{j{2}^{K-1}K!{\tau}^{K-1}{\mathrm{\Delta}}^{K}{a}_{K}n\right\}\text{exp}\left\{j{2}^{K-1}K!{\tau}^{K-1}{\mathrm{\Delta}}^{K-1}{a}_{K}\psi m\right\},\phantom{\rule{2em}{0ex}}\end{array}$

(10)

where

$m=0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}M-1,\phantom{\rule{2.77695pt}{0ex}}\left|n\right|\le \frac{N-1}{2}-\left(K-1\right)\tau ,$and

$B={A}^{{2}^{K-1}}\text{exp}\left\{j{2}^{K-1}\left(K-1\right)!{\left(\tau \mathrm{\Delta}\right)}^{K-1}{a}_{K-1}\right\}.$

HIM

_{K} [

*x*_{
m
}(

*n*),

*τ*] represents the product of two complex sinusoids, one with index

*n* and the other with

*m*. The frequency of the former is

*ω*_{
n
} = 2

^{K-1}*K*!

*τ*^{K-1}Δ

^{
K
} *a*_{
K
} , whereas the frequency of the latter is

*ω*_{
m
} = 2

^{K-1}*K*!

*τ*^{K-1}Δ

^{K- 1}*a*_{K} ψ. The estimation of the highest order parameter

*a*_{
K
} and DOA

*θ* therefore boils down to sinusoid frequency estimation. If we denote the frequency estimations of

*ω*_{
n
} and

*ω*_{
m
} as

${\widehat{\omega}}_{n}$ and

${\widehat{\omega}}_{m}$, respectively,

*a*_{
K
} and

*θ* can be estimated as

${\widehat{a}}_{K}=\frac{{\widehat{\omega}}_{n}}{{2}^{K-1}K!{\tau}^{K-1}{\mathrm{\Delta}}^{K}},$

(11)

$\widehat{\theta}=arcsin\left(\frac{c}{d}\widehat{\psi}\right)=arcsin\left(\frac{c}{d}\frac{{\widehat{\omega}}_{m}}{{2}^{K-1}K!{\tau}^{K-1}{\mathrm{\Delta}}^{K-1}{\widehat{a}}_{K}}\right).$

(12)

The frequency estimations ${\widehat{\omega}}_{n}$ and ${\widehat{\omega}}_{m}$ can be obtained using the periodogram maximization procedure [17]. The discrete Fourier transform (DFT) of the HIM is referred to as the HAF. The estimation of *a*_{
K
} thus requires the calculation of the DFT of HIM_{
K
} [*x*_{
m
}(*n*)*, τ*] with respect to index *n* and the DFT maximization. Similarly, the *θ* estimation requires the calculation of the DFT of HIM_{
K
} [*x*_{
m
}(*n*)*, τ*] with respect to index *m* and the DFT maximization. The DFT maximization requires 1D search [17].

The

*a*_{
K
} and

*θ* estimates can be improved by averaging results over

*m* and

*n*, respectively. Since for lower signal-to-noise ratios (SNRs) the estimation can be plagued by outliers, we propose to perform an

*α*-trimmed averaging instead of standard averaging. The

*α*-trimmed averaging does not take into consideration a percentage of extreme estimates, which most probably correspond to outliers. The

*α*-trimmed average of an

*N*-element array

**X** is defined as [

18]

$\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[X\right]=\frac{1}{N\left(1-2\alpha \right)}\sum _{k=\u230a\alpha N\u230b}^{\u2308\left(1-\alpha \right)N\u2309}{X}^{s}\left(n\right),$

(13)

where Trim_{
α
}[*·*] is the *α*-trimmed average operator, *α* the percentage of discarded elements, **X**^{
s
} represents the array **X** sorted in ascending/descending order, and ⌊·⌋ and ⌈·⌉ represent the round down and round up operators, respectively.

Lower order PPS parameters can be obtained from the dechirped signals

${\widehat{x}}_{m}^{d}\left(n\right)={x}_{m}\left(n\right)\text{exp}\left\{-j\left({\left(n\mathrm{\Delta}\right)}^{K}+Km{\left(n\mathrm{\Delta}\right)}^{K-1}\widehat{\psi}\right){\widehat{a}}_{K}\right\},\phantom{\rule{1em}{0ex}}m=0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}M-1,$

(14)

by repeating the procedure defined by (10) and (11).

In the sequel, we will explain in detail the estimation of DOA and parameters of second- and third-order PPSs.

### 3.1. Estimation algorithm for *K*= 2

Let us consider the second-order PPS, i.e., case

*K* = 2. Now we have

$\begin{array}{ll}\hfill {x}_{m}\left(n\right)& =A\text{exp}\left\{j\left({a}_{2}{\left(n\mathrm{\Delta}\right)}^{2}+\sum _{k=0}^{1}\left({a}_{k}+\left(k+1\right){a}_{k+1}\psi m\right){\left(n\mathrm{\Delta}\right)}^{k}\right)\right\}\phantom{\rule{2em}{0ex}}\\ =A\text{exp}\left\{j\left(\left({a}_{0}+{a}_{1}\psi m\right)+\left({a}_{1}+2{a}_{2}\psi m\right)n\mathrm{\Delta}+{a}_{2}{\left(n\mathrm{\Delta}\right)}^{2}\right)\right\}.\phantom{\rule{2em}{0ex}}\end{array}$

The second-order HIM equals

$\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\phantom{\rule{2.77695pt}{0ex}}\tau \right]=B\text{exp}\left\{j4\tau {\mathrm{\Delta}}^{2}{a}_{2}n\right\}\text{exp}\left\{j4\tau \mathrm{\Delta}{a}_{2}\psi m\right\},$

where

*m* = 0

*, . . . , M -* 1,

$\left|n\right|\le \frac{N-1}{2}-\tau ,$ and parameters

*a*_{2} and

*θ* can be estimated as

${\widehat{a}}_{2}=\frac{\text{arg}{\text{max}}_{{\omega}_{n}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{n}\left[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\tau \right]\right]\right\}}{4\tau {\mathrm{\Delta}}^{2}},$

(15)

$\widehat{\theta}=\mathsf{\text{arcsin}}\left(\frac{c}{d}\frac{\text{arg}{\text{max}}_{{\omega}_{m}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{m}\left[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\tau \right]\right]\right\}}{4\tau \mathrm{\Delta}{\widehat{a}}_{2}}\right).$

(16)

In (15) and (16), DFT_{
n
} [*·*] and DFT_{
m
} [*·*] represent the DFT operators with respect to *n* and *m*, respectively.

The parameter

*a*_{1} can be obtained by maximizing the DFT of

${\widehat{x}}_{m}^{d}\left(n\right)\phantom{\rule{2.77695pt}{0ex}}={x}_{m}\left(n\right)\phantom{\rule{2.77695pt}{0ex}}\text{exp}\left\{-j\left({\left(n\mathrm{\Delta}\right)}^{2}+2m\left(n\mathrm{\Delta}\right)\widehat{\psi}\right){\widehat{a}}_{2}\right\},$

with respect to index *n* and averaging the estimates obtained for all sensor indices *m* = 0,..., *M -* 1.

In the considered *K* = 2 case, instead of performing a 3D search as proposed in [9], we are able to estimate all the parameters by performing three 1D searches per sensor. The estimates are improved by averaging results obtained for all sensors, or by using the *α*-trimmed average (13).

The estimation algorithm is given below, followed by the calculation complexity analysis.

**for** *m* = 0 to *M -* 1

Calculate $\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\tau \right]={x}_{m}\left(n+\tau \right){x}_{m}^{*}\left(n-\tau \right)$, where *τ* = *N/* 4 [19].

Estimate

${a}_{2}^{m}$ from the DFT of HIM

_{2} [

*x*_{
m
}(

*n*)

*, τ*] calculated with respect to

*n*, i.e.,

${\widehat{a}}_{2}^{m}=\frac{\text{arg}{\text{max}}_{{\omega}_{n}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{n}\left[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\tau \right]\right]\right\}}{4\tau {\mathrm{\Delta}}^{2}}$

**end for**

Estimate

*a*_{2} using the

*α*-trimmed average operator as follows:

${\widehat{a}}_{2}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{a}}_{2}^{0},{\widehat{a}}_{2}^{1},\phantom{\rule{2.77695pt}{0ex}}\dots ,{\widehat{a}}_{2}^{M-1}\right]$

**for** *n* = *-* (*N -* 1)*/* 4 to (*N -* 1)*/* 4

Estimate

*θ*^{
n
} from the DFT of HIM

_{2} [

*x*_{
m
}(

*n*)

*, τ*] calculated with respect to

*m*, i.e.,

${\widehat{\theta}}^{n}=\mathsf{\text{arcsin}}\left(\frac{c}{d}\frac{\text{arg}{\text{max}}_{{\omega}_{m}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{m}\left[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{2}\left[{x}_{m}\left(n\right),\tau \right]\right]\right\}}{4\tau \phantom{\rule{0.3em}{0ex}}\mathrm{\Delta}{\widehat{a}}_{2}}\right)$

**end for**

Estimate

*θ* and

*ψ* as

$\begin{array}{c}\widehat{\theta}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{\theta}}^{-\frac{N-1}{4}},{\widehat{\theta}}^{-\frac{N-1}{4}+1},...,{\widehat{\theta}}^{\frac{N-1}{4}}\right]\\ \widehat{\psi}=\frac{d}{c}\text{sin}\left(\widehat{\theta}\right)\end{array}$

**for** *m* = 0 to *M -* 1

Dechirp the

*m* th signal

${\widehat{x}}_{m}^{d}\left(n\right)\phantom{\rule{2.77695pt}{0ex}}={x}_{m}\left(n\right)\phantom{\rule{2.77695pt}{0ex}}\text{exp}\left\{-j\left({\left(n\mathrm{\Delta}\right)}^{2}+2m\left(n\mathrm{\Delta}\right)\widehat{\psi}\right){\widehat{a}}_{2}\right\}$

Estimate

${a}_{1}^{m}$ from the DFT of

${\widehat{x}}_{m}^{d}\left(n\right)$ calculated with respect to

*n*, i.e.,

${\widehat{a}}_{1}^{m}=\text{arg}\underset{\omega}{\text{max}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{n}\left[{\widehat{x}}_{m}^{d}\left(n\right)\right]\right\}$

**end for**

Estimate

*a*_{1} as

${\widehat{a}}_{1}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{a}}_{1}^{0},{\widehat{a}}_{1}^{1},\phantom{\rule{2.77695pt}{0ex}}\dots ,{\widehat{a}}_{1}^{M-1}\right]$

In the algorithm's calculation complexity analysis, we will assume that the *N*-samples DFT calculation requires *N* log_{2} *N* complex additions and multiplications, and that the sorting of an *N*-samples long real valued sequence requires *N* log_{2} *N* comparison/exchange operations [20, Section 5.2.2]. In addition, 1D searches and scaling operations will not be accounted for in the analysis due to their low complexity.

For *τ* = *N/* 4, HIM_{2} [*x*_{
m
}(*n*)*, τ*] has *N/* 2 samples; therefore, the calculation complexity of HIM_{2} [*x*_{
m
}(*n*)*, τ*] and ${\widehat{a}}_{2}^{m}$ is *N/* 2 complex multiplications and *N/* 2 log_{2}(*N/* 2) complex additions and multiplications, respectively. The estimation of ${\widehat{a}}_{2}$ hence requires *MN/* 2 log_{2}(*N/* 2) complex additions, *MN/* 2 log_{2}*N* complex multiplications, plus *M* log_{2}*M* comparison/exchange operations and *M*(1 *-* 2*α*) real additions required for the trimming operation. Similarly, the calculation of $\widehat{\theta}$ requires *MN/* 2 log_{2}*M* complex additions, *MN/* 2 log_{2}*M* complex multiplications, *N/* 2 arcsine operations, as well as *N/* 2 log_{2}*N* comparison/exchange operations and *N/* 2(1 *-* 2*α*) real additions required for the trimming operation. The *ψ* estimation requires one sine operation. Finally, in the estimation of *a*_{1}, the calculation of ${\widehat{x}}_{m}^{d}\left(n\right)$ requires 2*N* real multiplications, *N* real additions and 2*N* sine/cosine operations for exp{·}, plus *N* complex multiplications for the product *x*_{
m
}(*n*) exp{·}. Therefore, the estimation of *a*_{1} requires 2*MN* real multiplications, *MN* real additions, 2*MN* sine/cosine operations, *MN* complex multiplications and *M* log_{2}*M* comparison/exchange operations and *M*(1 *-* 2*α*) real additions for the trimming operation.

Taking into consideration that one complex addition requires two real additions, and one complex multiplication requires four real multiplications and two real additions, we conclude that the joint estimation of {*a*_{2}, *a*_{1}, *θ*} requires 2*MN* log_{2}(2*MN*) + (1 *-* 2*α*)(*N/* 2 + 2*M*) real additions, 2*MN* log_{2}(8*MN*) real multiplications, 2*M* log_{2}*M* + *N/* 2 log_{2}(*N/* 2) comparison/exchange operations, *N/* 2 arcsine and 2*MN* + 1 sine/cosine operations. Alternatively, since trigonometric functions can be evaluated using the Taylor series expansion, the complexity of the proposed algorithm is *O*(*MN* log_{2}(*MN*)) operations (additions and multiplications), where *O* represents the big O notation. On the other side, the chirp beamformer requires the maximization of a 3D function *F*(**V**) (see (6)), and the complexity of evaluation of one point of *F*(**V**) is *O*(*MN*) operations. The overall complexity of the chirp beamformer is *O*(*M N N*_{
θ
} ${N}_{{a}_{1}}$ ${N}_{{a}_{2}}$) operations, where *N*_{
θ
}, ${N}_{{a}_{1}}$, and ${N}_{{a}_{2}}$ represent the number of points in the *θ*, *a*_{1}, and *a*_{2} grids, respectively, used in the maximization procedure. Clearly, the proposed approach offers a significant computational cost reduction with respect to the chirp beamformer.

### 3.2. Estimation algorithm for *K*= 3

When the PPS order is

*K* = 3, we have

$\begin{array}{c}{x}_{m}(n)=A\mathrm{exp}\left\{j({a}_{3}{(n\mathrm{\Delta})}^{3}+{\displaystyle \sum _{k=0}^{2}\left({a}_{k}+(k+1){a}_{k+1}\psi m){(n\mathrm{\Delta})}^{k}\right)}\right\}\\ =A\mathrm{exp}\left\{j\left({A}_{0}+{A}_{1}n\mathrm{\Delta}+{A}_{2}{(n\mathrm{\Delta})}^{2}+{A}_{3}{(n\mathrm{\Delta})}^{3}\right)\right\},\end{array}$

where

$\begin{array}{c}{A}_{0}={a}_{0}+{a}_{1}m\psi ,\\ {A}_{1}={a}_{1}+2{a}_{2}m\psi ,\\ {A}_{2}={a}_{2}+3{a}_{3}m\psi ,\\ {A}_{3}={a}_{3}.\end{array}$

(17)

For

*K* = 3, the HAF is not the optimal solution for the PPS estimation since the

*A*_{3} estimation requires the calculation of HIM

_{3} [

*x*_{
m
}(

*n*)

*, τ*], which incorporates two auto-correlations. Each auto-correlation increases the SNR threshold

^{a} by about 6 dB [

21]. Therefore, we will use the CPF which offers lower SNR threshold and more precise estimation [

15]. The CPF is defined as

$\mathsf{\text{CP}}{\mathsf{\text{F}}}_{m}\left(n,\mathrm{\Omega}\right)=\sum _{l=-\left(N-1\right)/2}^{\left(N-1\right)/2}{x}_{m}\left(n\phantom{\rule{2.77695pt}{0ex}}+l\right){x}_{m}\left(n\phantom{\rule{2.77695pt}{0ex}}-l\right){e}^{-j\mathrm{\Omega}{l}^{2}}.$

(18)

In noise-free case, the CPF is maximized at

$\mathrm{\Omega}={\varphi}_{m}\left(n\right)=2{A}_{2}+6{A}_{3}\left(n\mathrm{\Delta}\right)=2{a}_{2}+6{a}_{3}\left(n\mathrm{\Delta}\right)+6{a}_{3}\psi m,$

(19)

where ${\varphi}_{m}\left(n\right)$ represents the second-order phase derivative of *x*_{
m
}(*n*). Parameters *A*_{2} and *A*_{3} are estimated by locating maxima of the CPF calculated at two time instants and solving a set of two linear equations [15]. Therefore, in order to estimate *A*_{3}, we need to perform one auto-correlation less compared to the HAF. After estimating *A*_{3}, the DOA *θ* is estimated using (12), which entails the calculation of HIM_{3} [*x*_{
m
}(*n*)*, τ*] according to (10). The parameter *a*_{2} can then be estimated from *A*_{2} (see (17)). The estimates of *a*_{3}, *a*_{2}, and *θ* can be improved by averaging over all sensors (*a*_{3} and *a*_{2}) and time instants (*θ*).

Parameters

*A*_{0} and

*A*_{1} are estimated from the dechirped signals

${\widehat{x}}_{m}^{d}\left(n\right)={x}_{m}\left(n\right)\text{exp}\left\{-j\left({\widehat{a}}_{2}{\left(n\mathrm{\Delta}\right)}^{2}+{\widehat{a}}_{3}{\left(n\mathrm{\Delta}\right)}^{3}\right)\right\},\phantom{\rule{1em}{0ex}}m=0,\dots ,M-1.$

The estimates of *A*_{0}, *A*_{1}, *A*_{2}, and *A*_{3} can be refined using the method proposed in [22] and in turn used to refine *a*_{3}, *θ*, *a*_{2}, *a*_{1}, and *a*_{0}, respectively. The refinement method is outlined in Appendix 1.

Again, all the parameters are estimated via 1D searches, as opposed to the ML approach and polynomial-phase beamformer that require 5D and 4D searches, respectively.

The estimation algorithm follows, along with the calculation complexity summary.

**for** *m* = 0 to *M -* 1

Estimate

${A}_{3}^{m}$ and

${A}_{2}^{m}$ from the CPF calculated at two time instants

*n* = 0 and

$n={n}_{1}^{\mathsf{\text{b}}}$, i.e.,

$\begin{array}{c}{\mathrm{\Omega}}_{0}=\text{arg}\underset{\mathrm{\Omega}}{\text{max}}\left|\mathsf{\text{CP}}{\mathsf{\text{F}}}_{m}\left(0,\mathrm{\Omega}\right)\right|\\ {\mathrm{\Omega}}_{1}=\text{arg}\underset{\mathrm{\Omega}}{\text{max}}\left|\mathsf{\text{CP}}{\mathsf{\text{F}}}_{m}\left({n}_{1},\mathrm{\Omega}\right)\right|\\ {\widehat{a}}_{2}^{m}={\mathrm{\Omega}}_{0}/2\\ {\widehat{a}}_{3}^{m}=\left({\mathrm{\Omega}}_{1}-{\mathrm{\Omega}}_{0}\right)/\left(6{n}_{1}\mathrm{\Delta}\right)\end{array}$

Calculate HIM_{3} [*x*_{
m
}(*n*)*, τ*], where *τ* = *N/* 6 [19].

**end for**

Estimate

*A*_{3} as

${\widehat{a}}_{3}={\widehat{a}}_{3}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{a}}_{3}^{0},{\widehat{a}}_{3}^{1},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{\widehat{a}}_{3}^{M-1}\right]$

(20)

**for** *n* = *-* (*N -* 1)*/* 6 to (*N -* 1)*/* 6

Estimate

*θ*^{
n
} as

$\begin{array}{c}{\widehat{\theta}}^{n}=\mathsf{\text{arcsin}}\left(\frac{c}{d}\frac{\text{arg}{\text{max}}_{{\omega}_{m}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{m}\left[\mathsf{\text{HI}}{\mathsf{\text{M}}}_{3}\left[{x}_{m}\left(n\right),\tau \right]\right]\right\}}{24{\tau}^{2}{\mathrm{\Delta}}^{2}{\widehat{a}}_{3}}\right)\end{array}$

**end for**

Estimate

*θ* and

*ψ* as

$\begin{array}{c}\widehat{\theta}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{\theta}}^{-\frac{N-1}{6}},{\widehat{\theta}}^{-\frac{N-1}{6}+1},\dots ,{\widehat{\theta}}^{\frac{N-1}{6}}\right]\\ \widehat{\psi}=\frac{d}{c}\text{sin}\left(\widehat{\theta}\right)\end{array}$

**for** *m* = 0 to *M -* 1

Estimate ${a}_{2}^{m}$ from ${\widehat{a}}_{2}^{m}$ as ${\widehat{a}}_{2}^{m}={\widehat{a}}_{2}^{m}-3{\widehat{a}}_{3}m\widehat{\psi}$

**end for**

Estimate

*a*_{2} as

${\widehat{a}}_{2}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{a}}_{2}^{0},{\widehat{a}}_{2}^{1},\phantom{\rule{2.77695pt}{0ex}}\dots ,{\widehat{a}}_{2}^{M-1}\right]$

**for** *m* = 0 to *M -* 1

Dechirp the

*m* th signal as

${\widehat{x}}_{m}^{d}\left(n\right)={x}_{m}\left(n\right)\text{exp}\left\{-j\left({\widehat{a}}_{2}^{m}{\left(n\mathrm{\Delta}\right)}^{2}+{\widehat{a}}_{3}^{m}{\left(n\mathrm{\Delta}\right)}^{3}\right)\right\}$

Estimate

${A}_{1}^{m}$ from the DFT of

${\widehat{x}}_{m}^{d}\left(n\right)$ calculated with respect to

*n*${\widehat{a}}_{1}^{m}=\text{arg}\underset{\omega}{\text{max}}\left\{\mathsf{\text{DF}}{\mathsf{\text{T}}}_{n}\left[{\widehat{x}}_{m}^{d}\left(n\right)\right]\right\}$

Estimate ${a}_{1}^{m}$ from ${\widehat{a}}_{1}^{m}$ as ${\widehat{a}}_{1}^{m}={\widehat{a}}_{1}^{m}-2{\widehat{a}}_{2}m\widehat{\psi}$

**end for**

Estimate

*a*_{1} as

${\widehat{a}}_{1}=\mathsf{\text{Tri}}{\mathsf{\text{m}}}_{\alpha}\left[{\widehat{a}}_{1}^{0},{\widehat{a}}_{1}^{1},\dots ,{\widehat{a}}_{1}^{M-1}\right]$

*REFINEMENT STAGE*

**for** *m* = 0 to *M -* 1

Refine parameters ${A}_{1}^{m}$ ${A}_{2}^{m}$, and ${A}_{3}^{m}$ using the approach outlined in Appendix 1.

**end for**

Repeat the steps starting from (20), now using the refined estimates of ${A}_{1}^{m}$ ${A}_{2}^{m}$, and ${A}_{3}^{m}$.

According to (18), it can be shown that the evaluation of one CPF sample requires 6*N* real additions and 8*N* real multiplications. The calculation of vector ${e}^{-j\mathrm{\Omega}{l}^{2}}$, where $l=-\frac{N-1}{2},\dots ,\frac{N-1}{2}$, is not included in the complexity analysis since it can be calculated only once and such used for all values of *m*. The estimation of *A*_{3} therefore requires 12*QMN* real additions and 16*QMN* real multiplications for the calculation of ${\widehat{a}}_{3}^{m}$ and ${\widehat{a}}_{2}^{m}$, *m* = 0, 1,..., *M -* 1, plus *MN* complex multiplications for the HIM_{3} calculation, and *M* log_{2}*M* comparison/exchange operations and *M*(1 *-* 2*α*) real additions for the trimming operation. Herein, *Q* represents the number of elements in the Ω grid where the CPF is calculated. The HIM_{3} calculation complexity is *N* complex multiplications (with *τ* = *N/* 6, HIM_{3} [*x*_{
m
}(*n*)*, τ*] has *N/* 3 samples). The calculation complexities of $\widehat{\theta}$, $\widehat{\psi}$, ${\widehat{a}}_{2}$ and ${\widehat{a}}_{1}$ are determined analogously to the *K* = 2 case and we will give only the final expression for calculation complexity of joint estimation of {*a*_{3}*, a*_{2}*, a*_{1}*, θ*} which amounts to 12*QMN* + (1 *-* 2*α*)(3*M* + *N/* 3) + *M*(2 + 5*N*) + 4*MN/* 3 log_{2}(*MN*^{3}) real additions, 16*QMN* + (10*N* + 6)*M* + 4*MN/* 3 log_{2}(*MN*^{3}) real multiplications, 3*M* log_{2} *M* + *N/* 3 log_{2}(*N/* 3) comparison/exchange operations, *N/* 3 arcsine and 2*MN* + 1 sine/cosine operations. Alternatively, using the big O notation, the complexity of the proposed algorithm is *O*(*QMN*) operations. The proposed method clearly outperforms the polynomial-phase beamformer since the beamformer requires *O*(*M N N*_{
θ
} ${N}_{{a}_{1}}$ ${N}_{{a}_{2}}$ ${N}_{{a}_{3}}$) operations, where *N*_{
θ
} , ${N}_{{a}_{1}}$, ${N}_{{a}_{2}}$, and ${N}_{{a}_{3}}$ represent the number of points in the *θ*, *a*_{1}, *a*_{2}, and *a*_{3} grids, respectively, used in the maximization procedure.

### 3.3 Estimation for higher and unknown orders

Underwater acoustic signals can be modeled by PPSs of order higher than three [23]. The proposed algorithm for joint estimation of the PPS parameters and DOA, presented in Section 3, works with an arbitrary PPS order. The highest order parameter *a*_{
K
} and DOA are estimated using (11) and (12), respectively, while lower order PPS parameters are estimated from the dechirped signal (14). Keep in mind, however, that the SNR threshold in the HAF-based approach increases with the PPS order [21]. Then, if the underlying application requires estimation at lower SNR values, we could use the product HAF (PHAF) [19] or the hybrid CPF-HAF approach [24], instead of the HAF. These approaches are characterized by lower SNR threshold for higher PPS orders.

When the PPS order is unknown, we could use the strategy of increasing the HIM order until the DC component is obtained [19]. When the HIM order and that of the PPS coincide, a complex sinusoid with frequency proportional to the highest order parameter is obtained [14, 19]. If, however, the HIM order exceeds the PPS order, a DC component is obtained. Another approach for determining unknown PPS order is presented in [25].