Communication System
Elements of Communication System
SAMPLING
PART 1
Sampling Process
 Let g(t) be a finite-energy band-limited signal with bandwidth W.
 The ideal sampled signal g(t) obtained from g(t) is
g (t ) 


 g (nT ) (t  kT )  G ( f )  f  G( f  mf )
n  
s
s
s

m  
s

(  j 2nfTs )
g
(
nT
)
e
 s
n  
where Ts is the sampling period and fs=1/Ts is the sampling frequency (rate).
 From the expression of G(f), it is clear that sampling in time produces
periodicity in frequency. Also G(f) can be represented as the discrete-time
Fourier transform (DTFT) of the samples g(kTs).
 The DTFT may be viewed as a complex Fourier series representation of the
period frequency function G(f), where g(kTs) play the role of Fourier
coefficients.
(Continued)
 If Ts=1/2W (or fs=2W), then G(f) takes the form


 n   
G ( f )   g 
e
 2W 
n  
 f sG( f )  f s
jnf 

W 

 G( f
m  
m0
 mf s )
 The second equation shows that with the choice of fs=2W, the spectrum G(f)
may be separated from other periods of G(f), which is essential for the
reconstruction process.
 Under the two conditions: G(f)=0 for fW and fs=2W, G(f) can be written
as
1
G( f ) 
G ( f ),  W  f  W
2W
1

2W

 jnf 

W 
 n   
g
e

n    2W 
, W  f  W
• This latter expression means that G(f) is uniquely determined by
using the discrete-time Fourier transform (DTFT) of the samples
g(n/2W),
• taken at a sampling rate, fs=2W.
• This is true as long as fs2W.
Spectrum of band-limited signal g(t)
Spectrum of signal g(t) for a sampling period Ts=1/(2W)
Sampling Theorem
The sampling theorem for finite-energy band-limited signals
can be stated in two equivalent parts related to the transmitter
and receiver of a pulse modulation system:
1. A finite-energy band-limited signal with bandwidth W Hertz, is
completely described in terms of its samples taken at a rate
fs=1/Ts2W samples per second.
2. A finite-energy band-limited signal with bandwidth W Hertz, may
be completely recovered (reconstructed) from its samples taken
at a rate fs=1/Ts2W samples per second.
Note: The minimum sampling rate fs=2W is called Nyquist rate and
its reciprocal (inverse) Ts=1/2W is called Nyquist interval.
Aliasing Effect
• Aliasing effect can be eliminated by using an anti-aliasing filter
prior to sampling and using a sampling rate slightly higher than
Nyquist rate (fs=2W).
Anti-aliasing
g (t )
Filter
Sampler
g (kTs )
Sampling Rate
• The Nyquist Sampling Theorem
f s  2 f max
Where fs = minimum Nyquist sampling rate (Hz)
fm = maximum analog input frequency (Hz)
Example
Determine the Nyquist rates and Nyquist intervals used to sample the
following signals:
x(t)=sinc(200t), y(t)=sinc2(200t).
Solution
To determine the sampling rate of a signal, we have to know its frequency
spectrum. For x(t) and y(t), this may be done using Fourier transform.
1
 f 
X(f ) 
rect

200
200


f 
 1 

1

,


Y ( f )   200  200 
0,

f  200
f  200
From X(f) and Y(f), it is clear that for ;x(t) the maximum frequency
(bandwidth) W=100Hz; fs=2W=200Hz and Ts=1/fs=5ms.
and for y(t) the maximum frequency (bandwidth) W=200Hz;
fs=2W=400Hz and Ts=1/fs=2.5ms
Why Digital Transmission?
• Advantages
– Noise immunity
– Multiplexing
– Regeneration
– Simpler
• Disadvantages
– More bandwidth (costly & limited)
– Need additional circuitry (Encoder/Decoder)
PULSE MODULATION
PULSE MODULATION
• Sampling analog information signal
• Converting samples into discrete pulses
• Transport the pulses over physical transmission
medium.
• Four (4) Methods
1.
PAM
2.
3.
4.
PWM
PPM
PCM


Analog Pulse Modulation
Digital Pulse Modulation
Pulse Amplitude Modulation (PAM)
Modulation in which the amplitude of pulses is
varied in accordance with the modulating signal
Pulse Width Modulation (PWM)
Modulation in which the duration of pulses is varied
in accordance with the modulating signal
Pulse Width Modulation (PWM)
Pulse Position Modulation (PPM)
Modulation in which the temporal positions of the
pulses are varied in accordance with some characteristic of
the modulating signal.
How to encode analog waveforms ?
(from analog sources into baseband
digital signals)
Natural Sampling
Flat-top Sampling