Digital Signal Processing Reference
In-Depth Information
Fig. 2.14 I/O relations in a
digital system
Digital System
x
(
n
)
y
(
n
)
H
(
k
) = DFT [
h
(
n
) ]
X
(
k
)
Y
(
k
)
y
(
n
) =
x
(
n
)
h
(
n
)
Y
(
k
) =
X
(
k
) .
H
(
k
)
FFT algorithm and implement the filtering in the frequency domain instead. This
alternative is discussed in the following paragraphs (Fig.
2.14
).
Since convolution in the time domain is equivalent to multiplication in the
frequency domain, it is tempting to
1. take the FFTs of x(n), and h(n) to obtain X(k) and H(k), respectively,
2. multiply X(k) and H(k) together to obtain Y(k), and
3. then take the inverse FFT to obtain y
c
(n).
This three-step procedure, however, would yield the wrong result in general. To
see why the procedure is flawed, imagine that x(n) and h(n) are both N samples
long. Then X(k), H(k), Y(k) and y
c
(n) would all be N samples long as well. Since
y
c
(n) is supposed to be the convolution of x(n) and h(n), however, it should be
N ? N - 1 samples long rather than N samples.
The reason for the flaw in the three step procedure above is that multiplication in
the DFT (or FFT) domain implicitly corresponds to circular convolution in the time
domain. For true filtering to occur, one must have linear convolution rather than
circular convolution, and so one must zero-pad both x(n) and h(n) to be of at least
length L = N
x
? N
h
- 1, where N
x
is the length of x(n) and N
h
is the length of h(n).
If one performs this zero-padding, the circular convolution inherently implemented
in the FFT domain becomes equivalent to linear convolution in the time domain.
2.5 Signal Correlation, Power, and Energy
2.5.1 Definitions
This subsection presents formulae for correlation, power, and energy of discrete-
time signals. These formulae are analogous to those for analog signals, but with
integrations changed to summations. The formulae are presented below.
2.5.1.1 Autocorrelation of Non-Periodic Discrete-Time Energy Signals
R
x
ð
k
Þ¼
X
1
x
ð
n
Þ
x
ð
n
þ
k
Þ:
n
¼1
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