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320 Chapter Ten

k k

1/T

T
x x
q

(a) (b)

FIGURE 10.4 WDF of a vertically sheared comb function, i.e., a comb
function that has been multiplied by a chirp function. (a) The actual WDF of
the vertically sheared comb function where we include all the interfering
terms. (b) Here, as in Fig. 10.3b, we show only those terms
x = mT/2,k = n/2T for all even integers m, n.

argument a little if we are to apply it to Fig. 10.4b. It is true that the
shearing of the comb function means that the projection of the WDF
along the k axis will be unchanged. However, the projection of the
sheared comb function along the x axis (i.e., into the FT domain) will
be altered significantly. This alteration is dependent on . Thus our
previous statement “as long as our analysis is interested only in the
spatial and FT distributions, we may employ the incomplete WDF”
is no longer valid. If we wish to argue a case for the use of this in-
complete sheared comb WDF, we alter this statement accordingly: As
long as our analysis is interested only in the spatial distribution and
that projection along the arrow line (at angle ), we may employ the
incomplete WDF in this future analysis. We note that the relevance
of this vertically sheared comb function PSD will become clear in the
context of sampling a signal that is bounded in some LCT domain.

10.2.5.2 Rect Functions
Another signal often encountered in sampling theory is the rectangu-
lar window function, denoted by rect. We often multiply the sampled
signal’s Fourier transform by a rect function to recover the original
continuoussignal.Thisprocessisofcourseequivalenttoaconvolution