Time-Frequency Transforms - Time-Frequency Transforms for Radar Imaging and Signal Analysis - page 42

For example, if we choose g ( t ¢ ) = (1/ | t ¢ | ) exp{ -a ( t ¢ ) 2 }, then the ker-
nel function is F ( u , t ¢ ) = exp{ -a ( t ¢ ) 2 } sinc( u t ¢ /2). In this case, the kernel
function is one along the u -axis and exp{ -a ( t ¢ ) 2 } along the t ¢ -axis where
a controls the decay. Figure 2.10 shows the CSD of the same test signal.
Again, the CSD reduces cross-term interference while nearly maintaining
the resolution of the WVD.

2.2.3 The TFDS

Another approach to overcoming the cross-term interference problem of the
WVD is the TFDS, proposed by Qian and Chen [22]. They suggested that
if the WVD can be decomposed into a sum of localized and symmetric
functions, it may be possible to suppress cross-term interference by selecting
only the low-order harmonics. This is accomplished by first decomposing
the original signal into the Gabor expansion

s ( t )= m n

C m , n h m , n ( t ) (2.33)

where

h m , n ( t )=( ps 2 ) - 1/4 exp H ( t - m D t ) 2 + jn Dv t J (2.34)

2 s 2

are time-shifted and frequency-modulated Gaussian basis functions. In the
above expression, m and D t are respectively the time sampling index and
time sampling interval, while n and Dv are the sampling index and sampling
interval in frequency. In other words, C mn represents the STFT of the
function s ( t ) using a Gaussian window and evaluated on a sampled grid.

Figure 2.10 The CSD of the test signal with four nonoverlapping, finite-duration sinusoids.