Parameter estimation of linear and quadratic chirps by employing the fractional fourier transform and a generalized time frequency transform

Abstract

This paper is targeted towards a general readership in signal processing. It intends to provide a brief tutorial exposure to the Fractional Fourier Transform, followed by a report on experiments performed by the authors on a Generalized Time Frequency Transform (GTFT) proposed by them in an earlier paper. The paper also discusses the extension of the uncertainty principle to the GTFT. This paper discusses some analytical results of the GTFT. We identify the eigenfunctions and eigenvalues of the GTFT. The time shift property of the GTFT is discussed. The paper describes methods for estimation of parameters of individual chirp signals on receipt of a noisy mixture of chirps. A priori knowledge of the nature of chirp signals in the mixture – linear or quadratic is required, as the two proposed methods fall in the category of model-dependent methods for chirp parameter estimation.

A. Appendix

1.1 A.1 Proof of time shift property in GTFT

Suppose Y _α (u) is the GTFT of y(t) at angle α with f(t)=k t ³, for a constant k,

$$ \text{i.e. } Y_{\alpha}(u)={\int}_{-\infty}^{\infty}{y(t)K_{\alpha,f}(u,t)\text{d}t}. $$

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Our aim is to find the GTFT of y ¹(t)=y(t−t ₀) at same angle and with same function f(t).

$$ \begin{array}{lllll} Y_{\alpha}^{1}(u)&={\int}_{-\infty}^{\infty}{y(t-t_{0})K_{\alpha,f}(u,t)\text{d}t}\\ &={\int}_{-\infty}^{\infty}{y(t-t_{0})K_{\alpha,f}(u,t)\text{d}t}\\ &=\sqrt{\frac{1-j\cot \alpha}{2{\pi}}}{\int}_{-\infty}^{\infty}{y(t-t_{0})e^{j[\frac{u^{2}}{2}\cot(\alpha)+\frac{t^{2}}{2}\cot(\alpha)-ut\csc(\alpha)+ku^{3}-kt^{3}]}\text{d}t}. \end{array} $$

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Substitute t=t ₀+τ

$$ Y_{\alpha}^{1}(u)=\sqrt{\frac{1-j\cot \alpha}{2{\pi}}}{\int}_{-\infty}^{\infty}{y(\tau)e^{j[\frac{u^{2}}{2}\cot(\alpha)+\frac{(\tau+t_{0})^{2}}{2}\cot(\alpha)-u(\tau+t_{0})\csc(\alpha)+ku^{3}-k(\tau+t_{0})^{3}]}\text{d}\tau}. $$

(64)

Consider two new variables-

$$ \cot\beta=\cot\alpha-6kt_{0}. $$

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$$ \text{and \hspace{0.4cm} } \mu=\sin\beta(u\csc\alpha +3k{t_{0}^{2}}-t_{0}\cot\alpha)\\ $$

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Substituting these variables in (5), we get,

$$ \begin{array}{lllll} Y_{\alpha}^{1}(u)=\sqrt{\frac{1-j\cot \alpha}{2{\pi}}}{\int}_{-\infty}^{\infty}{y(\tau)e^{j[\frac{\mu^{2}}{2}\cot(\beta)+\frac{\tau^{2}}{2}\cot(\beta)-\mu\tau\csc(\beta)+k\mu^{3}-k\tau^{3}]}}\\ {\cdot}e^{j[\frac{u^{2}}{2}\cot\alpha+ku^{3}-ut_{0}\csc\alpha+\frac{{t_{0}^{2}}}{2}\cot\alpha -k{t_{0}^{3}}-\frac{\mu^{2}}{2}\cot\beta-k\mu^{3}]}\text{d}\tau. \end{array} $$

(67)

But, by definition

$$ K_{\beta,f}(\mu,\tau)=\sqrt{\frac{1-j\cot \alpha}{2{\pi}}}e^{j[\frac{\mu^{2}}{2}\cot(\beta)+\frac{\tau^{2}}{2}\cot(\beta)-\mu\tau\csc(\beta)+k\mu^{3}-k\tau^{3}]}. $$

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Therefore,

$$ \begin{array}{lllll} Y_{\alpha}^{1}(u)&={\int}_{-\infty}^{\infty}{y(\tau)K_{\beta,f}(\mu,\tau) e^{j[\frac{u^{2}}{2}\cot\alpha+ku^{3}-ut_{0}\csc\alpha+\frac{{t_{0}^{2}}}{2}\cot\alpha -k{t_{0}^{3}}-\frac{\mu^{2}}{2}\cot\beta-k\mu^{3}]}\text{d}\tau}\\ &=e^{j[\frac{u^{2}}{2}\cot\alpha+ku^{3}-ut_{0}\csc\alpha+\frac{{t_{0}^{2}}}{2}\cot\alpha -k{t_{0}^{3}}-\frac{\mu^{2}}{2}\cot\beta-k\mu^{3}]}{\int}_{-\infty}^{\infty}{y(\tau)K_{\beta,f}(\mu,\tau)\text{d}\tau}\\ &=e^{j[\frac{u^{2}}{2}\cot\alpha+ku^{3}-ut_{0}\csc\alpha+\frac{{t_{0}^{2}}}{2}\cot\alpha -k{t_{0}^{3}}-\frac{\mu^{2}}{2}\cot\beta-k\mu^{3}]}Y_{\beta}(\mu). \end{array} $$

(69)

This is the time shift property of the GTFT.