A refined PCPF algorithm for estimating the parameters of multicomponent polynomial-phase signals

Abstract

This paper considers the parameter estimation of multicomponent polynomial-phase signals (mc-PPSs) with orders greater than two. The proposed method combines the product cubic phase function (PCPF) and high-order ambiguity function (HAF) when the mc-PPS orders exceed three. In the proposed method, the HAF is first applied to the observed mc-PPS to produce a cubic phase signal. Second, the algorithm is modified to estimate the parameters of mc-PPS using the CPF. To obtain accurate estimates of the two highest-order parameters (i.e., $a_{P}$ and $a_{P - 1}$ of each component), all possible $a_{P}$ and $a_{P - 1}$ must be obtained in this step in all combinations of the instantaneous frequencies; then, the maximum absolute value of all sum values must be identified by dechirping with all possible $a_{P}$ and $a_{P - 1}$. In addition, non-uniformly-spaced signal sample methods are used to employ fast Fourier transformation in the CPF. The proposed method is different from the PCPF–HAF method proposed by other researchers; it is referred to as the improved PCPF–HAF method and can remedy the shortcomings of the traditional method when estimating mc-PPS parameters. Additionally, the PCPF–HAF method cannot be used to treat multicomponent third-order polynomial-phase signals, but the proposed method can treat them using non-uniformly-spaced signal sample methods. The cross-terms can also be restrained more effectively than with the CPF, resulting in higher accuracy of the estimated parameters and a lower signal-to noise ratio threshold. Theoretical analysis and simulations are presented to support these claims.

Appendix

For the $P{\text{th-order}}$ mc-PPSs with $K$ components, after the initial parameter estimates,$\left\{ {\widehat{a}_{m}^{q} ,m = 1, \ldots ,P,q = 1, \ldots ,K} \right\}$,are obtained, the $k{\text{th}}$ component can be expressed as follows:

$$ y^{k} (n) = y_{r} (n) - \sum\limits_{q = 1,q \ne k}^{K} {\widehat{b}_{q} \exp \left\{ {j\sum\limits_{m = 0}^{P} {\widehat{a}_{m}^{q} } n^{m} } \right\}} $$

(29)

From (29) it follows that

$$ \begin{aligned} y^{k} (n) & = y_{r} (n) - \sum\limits_{q = 1,q \ne k}^{K} {\widehat{b}_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {\widehat{a}_{m}^{q} } n^{m} } \right)} \\ & = \sum\limits_{q = 1}^{K} {b_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {a_{m}^{q} } n^{m} } \right)} - \sum\limits_{q = 1,q \ne k}^{K} {\widehat{b}_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {\widehat{a}_{m}^{q} } n^{m} } \right)} + \upsilon (n) \\ & = b_{k} \exp \left( {j\sum\limits_{m = 0}^{P} {a_{m}^{k} } n^{m} } \right) + \sum\limits_{q = 1,q \ne k}^{K} {\left( \begin{aligned} b_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {a_{m}^{q} } n^{m} } \right) \hfill \\ - \widehat{b}_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {\widehat{a}_{m}^{q} } n^{m} } \right) \hfill \\ \end{aligned} \right)} + \upsilon (n) \\ \end{aligned} $$

(30)

According to the literatures (O’ Shea 2010), a simple M-point moving average filter is represented as follows:

$$ \begin{aligned} y_{0} (m) & = \frac{1}{M}\sum\limits_{n = mM - (M - 1)/2}^{mM + (M - 1)/2} {y_{d} (n)} \\ & \quad m = - Q, - Q + 1, \ldots , + Q \\ \end{aligned} $$

(31)

where $Q = \left\lfloor {(\tfrac{1}{2}((N/M) - 1))} \right\rfloor$, $M$ is the number of samples in the moving average filter, $y_{d} \left( n \right)$ is defined as

$$ \begin{aligned} y_{d} \left( n \right) & = y^{k} (n)\exp \left( { - j\sum\limits_{m = 1}^{P} {\widehat{a}_{m}^{k} n^{m} } } \right) \\ & = \left( {\begin{array}{*{20}l} {b_{k} \exp \left( {j\sum\limits_{m = 0}^{P} {a_{m}^{k} } n^{m} } \right)} \hfill \\ { + \sum\limits_{q = 1,q \ne k}^{K} {\left( \begin{aligned} b_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {a_{m}^{q} } n^{m} } \right) \hfill \\ - \widehat{b}_{q} \exp \left( {j\sum\limits_{m = 0}^{P} {\widehat{a}_{m}^{q} } n^{m} } \right) \hfill \\ \end{aligned} \right)} + \upsilon (n)} \hfill \\ \end{array} } \right)\exp \left( { - j\sum\limits_{m = 1}^{P} {\widehat{a}_{m}^{k} n^{m} } } \right) \\ & = b_{k} \exp \left( {j\left( {a_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {a_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right)} \right) \\ & \quad + \sum\limits_{q = 1,q \ne k}^{K} {\left( {b_{q} \exp \left( {a_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {a_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right)} \right)} \\ & \quad - \sum\limits_{q = 1,q \ne k}^{K} {\left( {\widehat{b}_{q} \exp \left( {\widehat{a}_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {\widehat{a}_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right)} \right)} \\ & \quad + \upsilon (n)\exp \left( { - j\sum\limits_{m = 1}^{P} {\widehat{a}_{m}^{k} n^{m} } } \right) \\ \end{aligned} $$

(32)

Here define $\upsilon '(n)$ and $\delta$ as

$$ \left\{ \begin{aligned} & \upsilon '(n) = \upsilon (n)\exp \left( { - j\sum\limits_{m = 1}^{P} {\widehat{a}_{m}^{k} n^{m} } } \right) \\ & \delta = \sum\limits_{q = 1,q \ne k}^{K} {\left( \begin{aligned} b_{q} \exp \left( {a_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {a_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right) \hfill \\ - \widehat{b}_{q} \exp \left( {\widehat{a}_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {\widehat{a}_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right) \hfill \\ \end{aligned} \right)} + \upsilon '(n) \\ \end{aligned} \right. $$

(33)

So

$$ y_{d} (n){ = }b_{k} \exp \left( {j\left( {a_{0}^{k} + \sum\limits_{m = 1}^{P} {\left( {a_{m}^{k} - \widehat{a}_{m}^{k} } \right)} n^{m} } \right)} \right) + \delta $$

(34)

According to (29), $\delta$ is the sum of interference terms, and it is very small when SNR is greater than SNR threshold. In addition, from (27) the refined estimates for the phase parameters can be obtained, as follows.

$$ \left\{ \begin{aligned} \widehat{a}_{{k_{f} }} & = \widehat{a}_{k} + \widehat{\delta }a_{k} /M^{k} ,k = 1,2, \ldots ,P \\ \widehat{a}_{{0_{f} }} & = \widehat{a}_{0} \\ \end{aligned} \right. $$

(35)

where $\widehat{\delta }a_{k}$ can be computed according to

$$ \left[ {\widehat{a}_{0} ,\widehat{\delta }a_{1} ,\widehat{\delta }a_{2} , \ldots ,\widehat{\delta }a_{p} } \right]^{\text{T}} = ({\mathbf{G}}^{\text{T}} {\mathbf{G}})^{ - 1} {\mathbf{G}}^{\text{T}} {\mathbf{V}} $$

(36)

where

$$ \left\{ {\begin{array}{*{20}l} {{\mathbf{G}} = \left[ {\begin{array}{*{20}l} 1 \hfill & { - Q} \hfill & {( - Q)^{2} } \hfill & {( - Q)^{3} } \hfill \\ 1 \hfill & { - Q + 1} \hfill & {( - Q + 1)^{2} } \hfill & {( - Q + 1)^{3} } \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ 1 \hfill & Q \hfill & {Q^{2} } \hfill & {Q^{3} } \hfill \\ \end{array} } \right]} \hfill \\ {{\mathbf{V}} = \{ V_{i} (m),m = - Q, \ldots ,Q\} = {\text{unwrap}}(y_{0} (m))} \hfill \\ \end{array} } \right. $$

(37)

From (31)–(33) ${\mathbf{G}}$ is determined by $Q \,$,$Q \,$ is determined by $M$, $M$ is a fixed value, and it is entirely unrelated to the initial estimated parameters, ${\mathbf{V}}$ is related to the initial estimated parameters. Therefore the MSEs of the refined estimates, $\widehat{a}_{{k_{f} }} ,\;\widehat{a}_{{0_{f} }}$,for the phase parameters are determined by the initial estimates for the phase parameters.