Time-Frequency Transforms - Time-Frequency Transforms for Radar Imaging and Signal Analysis - page 43

By taking the WVD of both sides of (2.33), we obtain

WVD ( t , v )= mn m ¢ n ¢ C m , n C m ¢ , n ¢ WVD h , h ¢ ( t , v )

(2.35)

where WVD h , h ¢ denotes the WVD between any pair of basis functions and
is available in closed form. Next, the above expression can be regrouped
based on the ''interaction distance''

D= | m - m ¢ | + | n - n ¢ | (2.36)

between the pairs of bases ( m , n ) and ( m ¢ , n ¢ ). This results in what is
termed the TFDS, also called the Gabor spectrogram:

v )= mn | C m , n | 2 WVD h , h ¢ ( t , v )

TFDS D ( t , ( D = 0 terms)

+ mn m ¢ n ¢ C m , n C m ¢ , n ¢ WVD h , h ¢ ( t , v ) D = 1 terms)
+ mn m ¢ n ¢ C m , n C m ¢ , n ¢ WVD h , h ¢ ( t , v ) D = 2 terms)
+ . . .

(2.37)

Clearly, if we take all the terms in the series ( D = ¥ ), the right-hand
side of (2.37) converges to the WVD of the original signal. This yields the
best resolution but is plagued by cross-term interference. At the other extreme,
if we take only the self-interaction terms in the series ( D = 0), the resulting
right-hand side is equivalent to the spectrogram of the signal using a Gaussian
window function. It has no cross-term interference problem but has the
worst resolution. As the order D increases, we gain in resolution but pay a
price in cross-term interference. It is often possible to balance the resolution
against cross-term interference by adjusting the order D . The optimal value
for D was reported to be around 2 to 4.
Figure 2.11 shows the effect of the order D on the frequency-hopping
signal discussed in earlier examples. For D = 0 [Figure 2.11(a)] the signal
has the least time-frequency resolution, but is devoid of cross-term effects.
Figure 2.11(b, c) show respectively the TFDS for D = 3 and D =6.We
see that at D = 3 it is possible to capture the most useful information in
the time-frequency plane without the degrading effect of the cross terms.