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Sampling and Phase Space 317

effect may be observed using the WDF. When the signal with PSD
shown in Fig. 10.1b is multiplied by the harmonic with PSD shown
in Fig. 10.2a the resultant PSD is given by the convolution of the two
along the k axis as defined in Eq. (10.5). The result of this convolution
is an exact replica of the signal’s WDF centred at k = k 0 . Chirp func-
2
tions are mathematically expressed as exp(+ j x ). In the paraxial
approximation such functions represent spherical waves with cur-
vature . The effect of a thin lens with focal length f is modeled
by multiplying by a chirp function where = 1/ f . In the case of
convex and concave lenses is negative and positive, respectively.
The WDF of the chirp signal can be shown to be (k − x). Thus at
any point x only one local frequency exists at k = x. The PSD for
the chirp signal is shown in Fig. 10.2c. In this case = 1/ tan(− ).
Again the arrows indicate that this Dirac delta line extends outward
infinitely.
From Eq. (10.4) we know that if a signal is convolved in space with
2
a chirp exp(+ j x ), their WDFs are also convolved along x. Hence
the PSD shown in Fig. 10.1d, is given by the convolution along x of
the PSDs shown in Fig. 10.1b and Fig. 10.2c. This can be interpreted
as the paraxial approximation of spherical waves with curvature .
This is actually equivalent to a LCT with A = D = 1,B = 1/ ,
and C = 0. Therefore if a signal is convolved with a chirp function
2
exp( j x ), the signal’s WDF undergoes the following coordinate
transformation.

{u(x)}(x, k) → {u(x)}(x + k/a, k) (10.10)

This follows from Eq. (10.9) and is known as a horizontal shearing. We
note that as = 1/ z in such a convolution, the result is the Fresnel
transform for a distance z.
Similarly from Eq. (10.5) we can see that if a signal is multiplied
2
in space with a chirp exp( j x ), their WDFs are convolved along k.
Hence the PSD shown in Fig. 10.1e is given by the convolution along
k of the PSDs in Fig. 10.1b and Fig. 10.2c. Again it is equivalent to a
LCT, this time with A = D = 1,C = 1/ , and = 0. Therefore if
2
a signal is multiplied with a chirp function exp( j x ), the signal’s
WDF undergoes the following coordinate transformation.
x

{u(x)}(x, k) → {u(x)} x, k + (10.11)
a
This is known as vertical shearing. We note that if = 1/ f ,
such a convolution, and the resultant WDF coordinate transforma-
tion, describes the result of a signal passing through a lens of focal
length f .