Time-Frequency Transforms - Time-Frequency Transforms for Radar Imaging and Signal Analysis - page 37

Figure 2.7 The ADS of the test signal with four nonoverlapping, finite-duration sinusoids.

= E R ( t ¢ ) e - j v t ¢ dt ¢

| S ( v ) | 2

(2.16)

where the autocorrelation function is given by
R ( t ¢ )= E s ( t ) s *( t - t ¢ ) dt (2.17)

The power spectrum indicates how the signal energy is distributed in
the frequency domain. While the Fourier transform S ( v ) is a linear function
of s ( t ), the power spectrum is a quadratic function of s ( t ). Therefore, time-
frequency distributions derived directly from the Fourier transform, such as
those discussed in Section 2.1, can be classified as linear transforms, while
it is customary to call those distributions derived from the power spectrum
quadratic (or bilinear) time-frequency distributions. The main impetus for
quadratic time-frequency distribution is to define an appropriate time-
dependent power spectrum. In this section, we shall discuss three such time-
frequency transforms, the WVD, Cohen's class, and the TFDS.

2.2.1 The WVD

The most basic of the quadratic time-frequency representations, the WVD,
was first developed in quantum mechanics by Wigner in 1932 [14] and
later introduced for signal analysis by Ville [15]. In the WVD, a time-
dependent autocorrelation function is chosen as

R ( t , t ¢ )= s S t +

2 D s * S t - 2 D

t ¢ t ¢

(2.18)