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Sampling and Phase Space 319

m and n terms. At this point the reader might fairly ask why one might
ever wish to use this incomplete description of the comb function in
any analysis. We can argue that this description is actually sufficient
for the analysis presented in this chapter. Those terms that we have
omitted in Fig. 10.3b do not manifest themselves in the marginals of the
WDF; i.e., if we integrate { T (x)}(x, k) over k, all those terms for even
m in Eq. (10.14) integrate to zero (those lines containing consecutive
positive and negative Dirac delta functions will integrate to zero).
Similarly, if we integrate { T (x)}(x, k) over x, all those terms for even
n in Eq. (10.14) integrate to zero. Importantly, it is possible to multiply
the comb function by the signal, thereby convolving their WDFs along
the k axis and to employ an incomplete version of the comb function’s
WDF in our analysis. This is true if (1) the signal’s bandwidth is less
than or equal to 1/T and (2) we intend only to view the signal from
the x or the k domain. For such a band-limited signal, the resultant
convolution will create copies of strips of the signal’s WDF that do
not overlap looking parallel to the x axis. Therefore there will be no
interference between adjacent copies in the Fourier domain, and the
cross-terms of the comb function can be ignored before the convolution
with the signal’s WDF. We note that viewing the signal from domains
other than the x or k domain will require us to include the cross-terms
from the outset. The idea becomes much clearer as we proceed to the
discussion on sampling in Sec. 10.4.
If the comb function defined in Eq. (10.12) is multiplied by a chirp
2
function exp(+ j x ), such as that illustrated in Fig. 10.2c, we get
∞
,
2
2
exp(+ j x ) T (x) = exp(+ j x ) (x − nT)
n=−∞
∞
, 2
= exp[+ j (nT) ] (x − nT) (10.15)
n=−∞
The result is a coordinate shift of the WDF given by Eq. (10.11). Thus
the actual WDF of a sheared comb function can be found by setting
k → k + x/ in Eq. (10.14). One may envisage the process as a con-
volution along k of the PSDs illustrated in Fig. 10.2c and Fig. 10.3.
The results shown in Fig. 10.4a and b correspond to Fig. 10.3a and b.
In Fig. 10.4a we show the actual WDF of the sheared comb function,
and in Fig. 10.4b we show the case when we have removed every
second delta function in both the x and k dimensions, just as we did
in Fig. 10.3b. We recall that in relation to Fig. 10.3b we stated that we
could use the incomplete comb function in our future analysis because
all those terms integrated to zero along the x and k projections. Thus
as long as our analysis is interested only in the spatial and FT distri-
butions, we may employ the incomplete WDF. We need to amend this