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9.5. Sampling Theory 225
9.5.4 Dirac Impulses in Sampling Theory
The reason impulses are useful in sampling theory is that we can use them to talk
about samples in the context of continuous functions and Fourier transforms. We
represent a sample, which has a position and a value, by an impulse translated
to that position and scaled by that value. A sample at position a with value b is
represented by bδ(x −a). This way we can express the operation of sampling the
function f(x) at a as multiplying f by δ(x − a). The result is f(a)δ(x − a).
Sampling a function at a series of equally spaced points is therefore expressed
as multiplying the function by the sum of a series of equally spaced impulses,
called an impulse train (Figure 9.47). An impulse train with period T , meaning
that the impulses are spaced a distance T apart is
s
T
(x)=
∞
i=−∞
δ(x − Ti).
The Fourier transform of s
1
is the same as s
1
: a sequence of impulses at all
integer frequencies. You can see why this should be true by thinking about what
happens when we multiply the impulse train by a sinusoid and integrate. We wind
up adding up the values of the sinusoid at all the integers. This sum will exactly
cancel to zero for non-integer frequencies, and it will diverge to +∞ for integer
frequencies.
Because of the dilation property of the Fourier transform, we can guess that
the Fourier transform of an impulse train with period T (which is like a dilation
of s
1
) is an impulse train with period 1/T . Making the sampling finer in the space
domain makes the impulses farther apart in the frequency domain.
01
x
s
1
(x)
0
x
s
1/2
(x)
0
u
s
1
(u )
0
u
s
2
(u )
1
11
Figure 9.47. Impulse trains. The Fourier transform of an impulse train is another impulse
train. Changing the period of the impulse train in space causes an inverse change in the
period in frequency.