Digital Signal Processing Reference
In-Depth Information
1
1
0.5
0
0.5
−0.5
0
0
500
1000
1500
0
0.5
1
(a) Freq
(b) Time, sec
1
1
0
0.5
−1
0
0
500
1000
1500
2000
2500
3000
0
0.5
1
(c) Freq
(d) Time, sec
1
1
0.5
0
0.5
−0.5
0
0
1000
2000
3000
4000
5000
6000
0
0.5
1
(e) Freq
(f) Time, sec
Figure 3.9:
(a) Chirp from 0 Hz to 1500 Hz, sampled at 3000 Hz; (b) Spectrogram (frequency versus
time) of (a); (c) Chirp from 0 Hz to 3000 Hz, sampled at 3000 Hz; (d) Spectrogram of (c); (e) Chirp
from 0 Hz to 6000 Hz, sampled at 3000 Hz; (f ) Spectrogram of (e).
•
Two sinusoids, sampled at different rates, that bear the same frequency relative to their respec-
tive Nyquist rates, produce sample sequences having the same frequency content.
•
Two sinusoids, sampled at different rates, that bear the same frequency and phase relative to
their respective Nyquist rates, produce essentially the same sample sequences.
Example 3.5.
Verify the above statements using MathScript.
The m-code
N=32; FrN2=0.5; t=0:1/N:1-1/N;
figure; stem(cos(2*pi*t*FrN2*(N/2)))
allows you to verify this by holding
FrN2
(a fraction of the Nyquist rate
N/2
, such as 0.5, etc.)
constant, while changing the sample rate
N
. In the above call, the phase angle (0) remains the same as
N
changes, and the resulting sequences will be identical except for total number of samples. Note that
FrN2
= 1 generates the Nyquist rate.
