STFT-based estimator of polynomial phase signals

doi:10.1016/j.sigpro.2012.05.015

Signal Processing

Volume 92, Issue 11, November 2012, Pages 2769-2774

https://doi.org/10.1016/j.sigpro.2012.05.015 Get rights and content

Abstract

The short-time Fourier transform (STFT) based instantaneous frequency (IF) estimator has been used for polynomial phase signal (PPS) parameters estimation. This estimator is biased but it is less sensitive to the noise influence than the higher-order techniques such as the high-order ambiguity function (HAF). Here, we have developed a method for estimation of the optimal window width in the STFT for the PPS estimation. Obtained results are then refined with the strategy proposed recently by O'Shea. In this way the resulting estimates of parameters outperform the HAF based ones.

Highlights

► The short-time Fourier transform based tool is proposed for polynomial phase signals parameter estimation. ► The estimator is based on the ICI algorithm, polynomial regression and refinement procedure. ► It outperforms the higher order ambiguity function.

Introduction

The most common strategy for estimation of the polynomial phase signals (PPSs) parameters is based on reducing problem dimensions, i.e., performing phase differentiation, that decreases the polynomial order in the signal phase. This strategy leads to numerically efficient techniques, such as the higher-order ambiguity function (HAF), product higher-order ambiguity function (PHAF), integrated generalized ambiguity function (IGAF) and cubic phase function (CPF) [1], [2], [3], [4]. These techniques are commonly designed to be unbiased estimators. However, the unbiasedness in the parametric estimation often leads to extremely high variance of the estimators. In the case of the PPSs, for example the HAF, as the simplest and the most commonly used technique in the field, has high mean squared error (MSE) comparing to the Cramer–Rao lower bound (CRLB), especially for higher-order PPS and low the signal-to-noise ratio (SNR). The estimation threshold approximately increases for 6 dB with each degree of the polynomial in the signal phase. This is the main motivation that, in this paper, we consider an alternative strategy with the techniques of a lower order, like the short-time Fourier transform (STFT). It is a biased estimator, but more robust to the noise influence. The STFT, when considered as an instantaneous frequency (IF) estimator, is concentrated close to the IF, being the first phase derivative [5]. Therefore, the IF estimate contains biased and noise-influenced information about the coefficients of the polynomial in the signal phase (excluding the lowest order coefficient corresponding to the initial phase). After an IF estimate is obtained we perform its polynomial interpolation and the coefficients of the polynomial in the signal phase are estimated. These coefficients are noisy and biased. In order to minimize these effects we have proposed an algorithm inspired by the intersection-of-the-confidence intervals (ICI) rule, for obtaining suboptimal window width in the STFT for each estimated coefficient [6]. For high SNR the obtained (biased) estimates are significantly worse than the ones resulting from the unbiased techniques. However, for low SNR the STFT-based estimator is better than the HAF and related techniques. In addition, the estimation obtained by the proposed procedure is precise enough that the recently defined refinement strategy can be applied. It can decrease the MSE above the SNR threshold to the CRLB. Thus, the proposed STFT-based adaptive IF estimation procedure, with polynomial regression of the IF estimate, can be treated as the rough (coarse) estimate for the next stage when a fine estimation is performed by using the O'Shea algorithm [7]. In this manner, we have obtained better estimation results than in the HAF in terms of the SNR threshold, for which the proposed procedure produces accurate results. This is a bit surprising, since for a long, in this field, it has been assumed that the biased technique are generally not useful and almost all efforts have been devoted to designing unbiased estimators.

The paper is organized as follows. The STFT-based IF estimator is described in Section 2 with asymptotic values for the bias and the variance. In Section 3 we discussed the algorithm for the PPS parameter estimation from the obtained STFT-based IF estimate. The ICI based algorithm is summarized in Section 4 with the refinement strategy. Numerical examples are given in Section 5 with conclusions and discussion in Section 6.

Section snippets

STFT-based IF estimator

Assume a signal model: $x (t) = A \exp (j ϕ (t)) + ν (t), t \in [- T / 2, T / 2),$ where A is the signal amplitude, $ϕ (t)$ is the signal phase, T is the signal duration while $ν (t)$ is a white complex zero mean Gaussian noise with independent real and imaginary parts and with variance $σ^{2}$ . The signal is sampled with the sampling rate $Δ t = T / N$ under assumption that the sampling theorem condition is satisfied, i.e., $Δ t \leq 1 / f_{\max}$ where $f_{\max} = \max | ϕ' (t) | / 2 π$ . The discretized signal is denoted as $x (n) = x (n Δ t), n \in [- N / 2, N / 2) .$ The STFT is

Polynomial approximation of the STFT-based IF estimation

Assume that signal (1) has a polynomial phase: $ϕ (t) = \sum_{m = 0}^{M} a_{m} t^{m} / m,$ and that the order of the polynomial is known. The IF of this signal is given as $ω (t) = \sum_{m = 1}^{M} a_{m} t^{m - 1} .$ The STFT-based IF estimation bias, for this signal, is proportional to $ω^{(2)} (t)$ , i.e., $\sum_{m = 3}^{M} (m - 1) (m - 2) a_{m} t^{m - 3}$ .

If we have accurate IF estimate, then it is possible to estimate parameters of the PPS ${a_{m}, m = 1, \dots, M}$ using the classical polynomial regression. With different windows, we create series of the IF estimates ${\hat{ω}}_{h_{j}} (n), n \in [(- T / 2 + h / 2) / Δ t, (T / 2 -$

ICI algorithm for the parameters estimation

Now we will use the ICI algorithm for the estimation of the coefficients ${a_{m}$ , $m = 1, \dots, M}$ . For each coefficient we can assume that a very narrow window will produce low bias and large estimation variance. By increasing the window width the bias will increase and the variance will decrease, for each coefficient estimate. Thus, we can look for an adaptive window width with the bias–variance trade-off in each coefficient, as the widest window from the considered set H satisfying: $| {\hat{a}}_{m, h_{j}} - {\hat{a}}_{m, h_{j - 1}} | \leq κ [σ_{m}$

Examples

Here, we will consider two examples. The signal is assumed in the form $f (t) = \exp ({ja}_{5} t^{5} / 5 + {ja}_{4} t^{4} / 4 + {ja}_{3} t^{3} / 3 + {ja}_{2} t^{2} / 2 + {ja}_{1} t) .$ In the first example we have set $a_{5} = 0$ and considered signal of the fourth order. Parameters $a_{m}, m = 1,2, 3,4$ are selected randomly in each trial in the specific range $a_{4} \in [40 π, 56 π]$ , $a_{3} \in [31.5 π, 40.5 π]$ , $a_{2} \in [- 56 π, - 40 π]$ , $a_{1} \in [6 π, 10 π]$ . In the second example, the first four parameters are selected according to the previous rule, while a₅ is selected randomly within the range $a_{5} \in [0,2 π]$ .

Conclusion

In this paper we have shown that the lower order transform such as the STFT, in combination with the adaptive ICI algorithm, can be used for estimation of the polynomial phase signals. This transform is biased but it has significantly lower SNR-threshold than the HAF, that is commonly used as the state-of-the art in the field. In addition, we are using the recently proposed refinement strategy that is able to reach the CRLB for biased coarse estimate. In this way, the proposed estimator has

Acknowledgment

This research has been supported in part by the Ministry of Science of Montenegro and Montenegrin Academy of Sciences and Arts (CANU).

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A large number of estimators for PPSs have been studied and implemented. In recent years, Igor Djurović et al. proposed the quasi-maximum likelihood (QML) estimator [12,24,25] as an effective technique for the parameter estimation of PPSs. Simulation results show that the performance of the QML estimator is better than those of all existing approaches [26].
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STFT-based estimator of polynomial phase signals

Abstract

Highlights

Introduction

Section snippets

STFT-based IF estimator

Polynomial approximation of the STFT-based IF estimation

ICI algorithm for the parameters estimation

Examples

Conclusion

Acknowledgment

Signal Processing

Historia Mathematica

Signal Processing

The discrete polynomial phase transform

IEEE Transactions on Signal Processing

Product high-order ambiguity function for multicomponent polynomial-phase signal modeling

IEEE Transactions on Signal Processing

Analysis of polynomial-phase signals by the integrated generalized ambiguity function

IEEE Transactions on Signal Processing