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180 Chapter 5 THE DISCRETE FOURIER TRANSFORM
The quantity |X(k)|2
N
is called the energy spectrum of finite-duration sequences.
Similarly, for periodic sequences, the quantity | ˜X(k)
N
|2 is called
the power spectrum.
5.5 LINEAR CONVOLUTION USING THE DFT
One of the most important operations in linear systems is the linear convolution.
In fact, FIR filters are generally implemented in practice using this
linear convolution. On the other hand, the DFT is a practical approach
for implementing linear system operations in the frequency domain. As we
shall see later, it is also an efficient operation in terms of computations.
However, there is one problem. The DFT operations result in a circular
convolution (something that we do not desire), not in a linear convolution
that we want. Now we shall see how to use the DFT to perform a linear
convolution (or equivalently, how to make a circular convolution identical
to the linear convolution). We alluded to this problem in Example 5.15.
Let x 1 (n) beanN 1 -point sequence and let x 2 (n) beanN 2 -point
sequence. Define the linear convolution of x 1 (n) and x 2 (n) by x 3 (n),
that is,
x 3 (n) =x 1 (n) ∗ x 2 (n)
∞∑
= x 1 (k)x 2 (n − k) =
k=−∞
N∑
1−1
0
x 1 (k)x 2 (n − k) (5.43)
Then x 3 (n) is a (N 1 + N 2 − 1)-point sequence. If we choose N =
max(N 1 ,N 2 ) and compute an N-point circular convolution x 1 (n) N○
x 2 (n), then we get an N-point sequence, which obviously is different
from x 3 (n). This observation also gives us a clue. Why not choose
N = N 1 + N 2 − 1 and perform an (N 1 + N 2 − 1)-point circular convolution?
Then at least both of these convolutions will have an equal
number of samples.
Therefore let N = N 1 + N 2 − 1 and let us treat x 1 (n) and x 2 (n) as
N-point sequences. Define the N-point circular convolution by x 4 (n).
x 4 (n) =x 1 (n) N○ x 2 (n) (5.44)
[ N−1
] ∑
= x 1 (m)x 2 ((n − m)) N R N (n)
m=0
[ N−1
]
∑ ∞∑
= x 1 (m) x 2 (n − m − rN) R N (n)
m=0 r=−∞
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