Digital Signal Processing Reference
In-Depth Information
We see that the DFT can give a good approximation of the Fourier transform.
The errors in the Fourier transform approximation can be reduced by choosing a
windowing function that causes less spectrum-leakage distortion. Increasing the
sampling rate and increasing the number of samples used in the calculation also
tend to decrease errors. Problems at the end of this chapter provide the opportunity
for the student to investigate various ways of decreasing errors when using the DFT
to approximate the Fourier transform.
The convolution of two discrete-time signals is described by the equation
q
m=-
q
[eq(10.16)]
x[n]
*
h[n] =
x[m]h[n - m].
We call this
linear convolution
. In this section, we discuss another convolution oper-
ation for discrete-time sequences, called
circular convolution
.
First, we determine the time-domain process corresponding to the product of
two discrete Fourier-transform functions. If
Y[k] = X[k]H[k],
then what is the relationship among and We approach the answer
to this question by beginning with the definition of the DFT:
y[n],
x[n],
h[n]?
N- 1
x[n]e
-j2pkn/N
;
X[k] =
[x[n]] =
a
n= 0
[eq(12.30)]
N- 1
1
N
a
-1
[X[k]] =
X[k]e
j2pkn/N
.
x[n] =
k= 0
Using (12.30), we write the transform equation for
H[k].
We have
N- 1
h[m]e
-j2pkm/N
, m = 0, 1, 2, Á , N - 1;
H[k] =
a
m= 0
N- 1
N- 1
B
x[n]e
-j2pkn/N
RB
h[m]e
-j2pkm/N
R
Y[k] =
;
a
a
n= 0
m= 0
N- 1
N- 1
N- 1
1
N
a
B
x[l]e
-j2pkl/N
RB
h[m]e
-j2pkm/N
R
e
j2pkn/N
;
y[n] =
a
a
k= 0
l = 0
m= 0
N- 1
N- 1
N- 1
1
N
a
B
e
j2pk(n- l -m)/N
R
y[n] =
x[l]
a
h[m]
.
a
l = 0
m= 0
k= 0
