DSP3108 DSP Applications
02: Sampling & digital frequency
Nyquist rate
All signals must be sampled at a least twice their maximum frequency. This sampling rate is known as the
nyquist rate.
Discrete Sinusoidal Signal
In discrete form the tone is defined as:
x (n )  A sin(a nT ), T 
1
, f s  sampling freq
fs
For a sampled signal time ‘t’ is replaced with t=nT where T is the sampling period.
x(n)  A sin(2 nf a / f s )
 fa 
   , the digital frequency
f
 s
The factor 2 
It represents the angle that the analogue signal rotates through during one sampling period.
x (n )  A sin(n )
If for example the analogue signal is a single tone of frequency 1 kHz and the sampling frequency is 8kHz, then
=2×1/8=/4.
Tone of digital frequency pi/32
1
0.9
0.8
Amplitude (V)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
Time (sec s)
1
15
DSP3108 DSP Applications
Tone of digital frequency pi/16
1
0.8
0.6
Amplitude (V)
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10
15
Time (sec s)
Prob 1: A 4 kHz sinusoidal signal is sampled using a digital frequency of π/8. What is the equivalent analogue
sampling frequency? Plot approximately, the analogue signal and the resultant sampled signal.
Prob 2:
A 4 kHz signal and an 8 kHz signal are processed as follows:
v1=10sin(2π×4000t)
fs=32 kHz
Sampler
+
y(n)
v2=5sin(2π×8000t)
Sampler
Sketch approximately, the 1st 6 output sample levels. (Hint: Find Ѳ1 and Ѳ2 for the 2 inputs v1 and v2.
You can then define v1(n) and v2(n) and combine these 2 signals.)
Prob 3:
Set up a table in Excel that calculates the signals and then combines them for the 1 st 20 samples.
Then modify the Matlab program above to give the same results.
2
DSP3108 DSP Applications
Extra Discussion
Draw analogue sine, specify amplitude, period and frequency. Then draw digital sine, specifying the sampling
duration and sampling frequency relationship. Then derive the digital tone equation culminating with a
summary of digital sine equation and the equation for digital frequency. These are the essentials.
3