Analysis of Digital Pulses

1. CSN08704
Data, Audio, Video and Images
http://asecuritysite.com/comms
Telecommunications
Prof Bill Buchanan
4. Analysis of Digital Pulses

2. CSN08704
Data, Audio, Video and Images
http://asecuritysite.com/comms
Telecommunications
Prof Bill Buchanan
4. Analysis of Digital Pulses: Frequency
and Time Domain

3. Signals
V t V ft( ) sin( ) 2 
Time domain Frequency domain
Vmax
f
T (1/f)
(1/T)
-V max
0
Vmax
AmplitudeAmplitude

4. Repetitive Signals
ω1 = 2.π.f1 where f1 is the fundamental frequency.
tNBtBtB
tNAtAtAAtf
N
N
11211
112110
sin...2sinsin
cos...2coscos)(





5. Repetitive Signals
• 1 1st, 0.4 2nd, 0.2 3rd.
• 0.4 4th, 0.05 5th.
• Link.

6. Repetitive Signals

7. Example
1
0.3
0.2
1/T 3/T 5/T f
T
0.9
-0.9
0
Time domain Frequency domain
V
f t t t t( ) sin( ) . sin( ) . sin( )    1 1 103 3 02 5

8. Even Symmetry
t
f t A A t A t A N tN( ) cos cos ... cos    0 1 1 2 1 12  

9. Odd Symmetry
t
f t B t B t B N tN( ) sin sin ... sin   1 1 2 1 12  

10. Half Symmetry
• No even harmonics ... just odd harmonics.
Time

11. CSN08704
Data, Audio, Video and Images
http://asecuritysite.com/comms
Telecommunications
Prof Bill Buchanan
4. Analysis of Digital Pulses: Pulse
Analysis

12. Repetitive Pulse Stream
t
+V
T
Duty Cycle
t
T

v t
V
T
V n f tn
n
n
( ) cos( ) 




2 1
1
V
V
T
Nx
x
n 
2 
.
sin
x
T



13. Sin(x)/x
0 10 20 30 40 50 60 70 80 90
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
V
V
T
Nx
x
n 
2 
.
sin
x
T


0
4 8 12 16 20 24 28
Harmonic
Amplitude
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1

14. Worked Example
5 V
0 V
25 s
5 s
v t
Vt
T
Vt
T
N t
T
N t
T
N t
N
( ) .
sin( )
cos( ) 










2
1
1



V V
t
T
VDC pk  5
5
25
1.

15. Worked Example
f
T
kHz
f kHz
f kHz
f kHz
f kHz
1 6
2
3
4
5
1 1
25 10
40
80
120
160
200
 







5 V
0 V
25 s
5 s

16. Worked Example
5 V
0 V
25 s
5 s
V
Vt
T
N t
T
N t
T
N 
2
.
sin( )

V
N
N
N
N
N 
 

2 5 5
25
0 2
0 2
318
0 63
.
sin( . )
.
.
.sin( . )


V
N f (kHz) V amplitude (Volts)
1
2
3
4
5
40
80
120
160
200
1.87
1.51
1.01
0.47
0

17. 5 V
0 V
25 s
5 s
v t t t t
t
i ( ) . sin( ) . sin( ) . sin( )
. sin( ) ......
   

1 187 151 2 101 3
0 47 4
1 1 1
1
  
+
t () V0 V1 V2 V3 
1 1.77 cost 1.51 cos2t 1.01 cos3t
45 1 1.32 0 –0.71 1.61
90 1 0 –1.51 0 –0.51
135 1 –1.32 0 0.71 0.39
180 1 –1.87 1.51 –1.01 –0.37
225 1 –1.32 0 0.71 0.39
270 1 0 –1.51 0 –0.51
315 1 1.32 0 –0.71 1.61
0,360 1 1.87 1.51 1.01 5.39

18. Worked Example
-1
0
1
2
3
4
5
6
0
45
90
135
180
225
270
315
360
Angle (deg.)
Voltage(V)
t () V0 V1 V2 V3 
1 1.77 cost 1.51 cos2t 1.01 cos3t
45 1 1.32 0 –0.71 1.61
90 1 0 –1.51 0 –0.51
135 1 –1.32 0 0.71 0.39
180 1 –1.87 1.51 –1.01 –0.37
225 1 –1.32 0 0.71 0.39
270 1 0 –1.51 0 –0.51
315 1 1.32 0 –0.71 1.61
0,360 1 1.87 1.51 1.01 5.39
5 V
0 V
25 s
5 s

19. Example
• Duty cycle=0.2.
• Link.

20. Example
• Duty cycle=0.3.
• Link.

21. CSN08704
Data, Audio, Video and Images
http://asecuritysite.com/comms
Telecommunications
Prof Bill Buchanan
4. Analysis of Digital Pulses