# 7.7 Spectral Analysis of Doppler-Stretched Signals –Constant Radial Speed

In this section, the spectral analysis of the Doppler stretched signal received in the case of constant relative radial speed between TX and RX is addressed. The second-order statistical characterization is made in both time and frequency domains and both continuous-and discrete-time cases are considered. Sampling theorems are proved to obtain input/output relationships for discrete-time sampled signals which are formally analogous to their continuous-time counterparts. Proofs are reported in Section 7.10.

Let us consider the continuous-time Doppler-stretched signal (Section 7.3):

(7.261)

It is a time-scaled, delayed, and frequency-shifted version of the transmitted signal x_{a}(t). Its Fourier transform is given by (Section 7.10)

(7.262)

## 7.7.1 Second-Order Statistics (Continuous-Time)

Let x_{a}(t) exhibit (conjugate) cyclostationarity. Then, also y_{a}(t) exhibits (conjugate) cyclostationarity and its cyclic statistics are linked to those of x_{a}(t) as follows.

### 7.7.1.1 Cyclic Autocorrelation Function and Cyclic Spectrum

The cyclic autocorrelation function and the cyclic spectrum of y_{a}(t) are given by (Section 7.10)

(7.263)

(7.264)

Equivalently,

(7.265)

(7.266)

Thus, if x_{a}(t) exhibits cyclostationarity at cycle frequency ...