DSP3108 DSP Applications
03: Revision of DSP signals & spectra
There are a number of important basic digital signals used in DSP analysis. These include:
Unit Step: Defined as u(n)=1, n0 and zero elsewhere.
Unit Impulse: defined as (n)=1, n=0 and zero elsewhere.
Decaying Exponential: defined as g(n)=an, n0 and zero elsewhere where 0<a<1
Spectrum of the Unit Step: Defined as u(n)=1, n0 and zero elsewhere.
The unit step (pulse train) can be expressed using a Fourier series as:
p (t ) 
Ad  2 A
1

sin  nd / T  cos  n  2 f 0t , f 0 
T
T
n 1  n
Assume that the high level (A) of 1 V and a low level is 0 V. The period of the pulse train (T) is 125 us and its
high duration (d) is 31.25 us.
1
0.5V
DSP3108 DSP Applications
0V
V(S)
500mV
250mV
SEL>>
0V
0Hz
5KHz
10KHz
15KHz
20KHz
25KHz
30KHz
35KHz
40KHz
45KHz
50KHz
V(P)
Frequency
In this case d=T/4
1  2
1
1
p(t )    sin  n / 4  cos  n  2 f 0t , f 0  
 8 kHz
4 n1  n
T 125 us
n
1
2
3
4
5
6
sin(n*π/4)
0.707
1.000
0.707
0.000
-0.707
-1.000
cos  2  2 f 0t  0.707cos  3  2 f 0t  

0.707cos  2 f 0t  



1 2

2
3
p(t )   

4 
0.707cos  5  2 f 0t 

 0
.......


5


Problem 1:
An analogue signal consisting of 2 tones, of frequency 1 kHz and 2 kHz and of amplitudes 4 V and 2 V are
sampled using a pulse train ( p(t) ) with a high level (A) of 1 V and a low level of 0 V. The period of the pulse
train (T) is 125 us and its high duration (d) is 31.25 us. Plot the spectrum of p(t) and the sampled signal.
2
DSP3108 DSP Applications
1.2V
0.8V
0.4V
0V
0Hz
5KHz
V(SAMPLED)
10KHz
15KHz
20KHz
25KHz
30KHz
35KHz
40KHz
45KHz
50KHz
Frequency
Problem 2:
If the analogue signal in problem3 is replaced with a single tone of amplitude 2 V and frequency 6 kHz and is
sampled by this pulse train, sketch the spectrum of the sampled signal. Can the original signal be recovered
from the sampled signal? If not, what is the maximum input frequency that can be sampled and recovered
using this sampling frequency?
1.0V
0.5V
0V
V(S)
500mV
250mV
SEL>>
0V
0Hz
5KHz
10KHz
15KHz
20KHz
25KHz
30KHz
35KHz
40KHz
45KHz
50KHz
V(P)
Frequency
For an input frequency of 6 kHz an aliasing component of 2 kHz occurs, so the original signal cannot be
recovered. The maximum allowable input frequency is half the sampling frequency which is 8 kHz.
Digital sequences
A digital sequence is defined as a sum of scaled delayed impulses.
x( n )  a0 ( n )  a1 ( n  1)  a2 ( n  2)
N
In general x( n )   ak  ( n  k )
k 0
3
DSP3108 DSP Applications
Problem3:
Write down the digital sequence for x(n) if N=4 and ak=2,3,4,5,6 for k=0 to 4. Draw a sketch of this sequence.
Problem4:
The digital sequence in Problem 3 is added to the following sequence:
6
y ( n )   ak  ( n  k ) , ak  b k
k 0
Sketch the combined signal if:
(i)
b=0.8
(ii)
b=2.4
4