Digital Signal Processing Reference
In-Depth Information
the DTFT plot. The three DFS frequencies are 2
πk/
3 where
k
= 0,1, and 2. To get both the DTFT and
the three DFS samples onto the same plot, the following code evaluates the DTFT from 0 to 2
π
radians,
allowing all three DFS frequencies to be plotted:
x = [1 0 1]; N = length(x); n = 0:1:N-1;k=n;
W = exp(-j*2*pi/N); DFS = x*(W.ˆ(n'*k));
w = 0:0.01:2*pi; DTF T = 1+exp(-j*2*w);
figure(8); clf; hold on; plot(w/(pi),abs(DTF T));
stem(2*[0:1:2]/3,abs(DFS),'ro');
xlabel('Norm Freq (Units of pi)'); ylabel('Magnitude')
Figure 3.1 illustrates the essential results from the code above.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Normalized Freq (Units of
π
)
Figure 3.1:
A dense grid of samples of the DTFT of the sequence [1 0 1], with a stem plot of the three
DFS samples.
From the above it is possible to see that the DFS coefficients of a sequence
x
[
n
]
(computed using
one period
x
[
n
]
of
x
[
n
]
)
are essentially equally spaced samples of the DTFT of
x
[
n
]
. Assuming that
the ROC of the
z
-transform of
includes the unit circle, then it is also possible to say that the DFS
coefficients are essentially equally spaced samples of the
z
-transform of
x
[
n
]
evaluated along the unit
circle. Prior to proceeding to the DFT, we briefly investigate the effect of sampling the
z
-transform.
x
[
n
]
3.4
SAMPLING IN THE Z-DOMAIN
The
z
-transform of an absolutely summable sequence
x
[
n
]
, which may be finite or infinite in extent, is
defined as
