Fundamentals of Digital
Communication
Digital communication system
Input
Signal
Analog/
Digital
Low
Pass
Filter
Sampler
Quantizer
Source
Encoder
Channel
Encoder
Multiplexer
Carrier
To Channel
Modulator
Pulse
Shaping
Filters
From Channel
DeModulator
Receiver
Filter
Line
Encoder
Detector
Carrier Ref.
Signal
at the
user end
Digital-to-Analog
Converter
Channel
Decoder
DeMultiplexer
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Noiseless Channels and Nyquist
Theorem
For a noiseless channel, Nyquist theorem gives the relationship
between the channel bandwidth and maximum data rate that can
be transmitted over this channel.
Nyquist Theorem
C  2B log 2 m
C: channel capacity (bps)
B: RF bandwidth
m: number of finite states in a symbol of transmitted signal
Example: A noiseless channel with 3kHz bandwidth can only transmit
a maximum of 6Kbps if the symbols are binary symbols.
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Nyquist minimum bandwidth
requirement

The theoretical minimum bandwidth needed for baseband
transmission of Rs symbols per second is Rs/2 hertz
h(t )  sinc( t / T )
H( f )
1
T
1
2T
?
0
1
2T
f
 2T  T
0
T 2T
t
4
Shannon’s Bound for noisy channels
There is a fundamental upper bound on achievable bandwidth efficiency.
Shannon’s theorem gives the relationship between the channel
bandwidth and the maximum data rate that can be transmitted over
a noisy channel .
Shannon’s Theorem
 B max
C
S
  log 2 (1  )
B
N
C: channel capacity (maximum data-rate) (bps)
B or W : RF bandwidth
S/N: signal-to-noise ratio (no unit)
5
Shannon limit …

Shannon theorem puts a limit on transmission
data rate, not on error probability:


Theoretically possible to transmit information at
any rate Rb , where Rb  C with an arbitrary small
error probability by using a sufficiently
complicated coding scheme.
For an information rate Rb > C , it is not possible
to find a code that can achieve an arbitrary small
error probability.
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Shannon limit …
C/W [bits/s/Hz]
Unattainable
region
Practical region
SNR [dB]
7
Shannon limit …
S

C  W log 2 1  
 N
 S  Eb C

 N  N 0W
 Eb C 
C

 log 2 1 
W
 N0 W 
C
As W   or
 0, we get :
W
Eb
1

 0.693  1.6 [dB]
N0
log 2 e


Shannon limit
There exists a limiting value of Eb / N 0 below which there can be no
error-free communication at any information rate.
By increasing the bandwidth alone, the capacity cannot be increased
to any desired value.
8
Shannon limit …
Practical region
W/C [Hz/bits/s]
Unattainable
region
-1.6 [dB]
Eb / N 0 [dB]
9
Bandwidth efficiency plane
R>C
Unattainable region
M=256
M=64
R=C
R/W [bits/s/Hz]
M=16
M=8
M=4
Bandwidth limited
M=2
M=4
M=2
R<C
Practical region
M=8
M=16
Shannon limit
Power limited
Eb / N 0 [dB]
MPSK
MQAM
MFSK
PB  105
10
Error probability plane
(example for coherent MPSK and MFSK)
M-PSK
bandwidth-efficient
M-FSK
power-efficient
k=5
Bit error probability
k=4
k=1
k=2
k=4
k=3
k=5
k=1,2
Eb / N 0 [dB]
Eb / N 0 [dB]
11
M-ary signaling

Bandwidth efficiency:
Rb log 2 M
1


W
WTs
WTb
[bits/s/Hz ]
W  1 / Ts  Rs [Hz]

Assuming Nyquist (ideal rectangular) filtering at baseband,
the required passband bandwidth is:
Rb / W  log 2 M [bits/s/Hz ]

M-PSK and M-QAM (bandwidth-limited systems)
 Bandwidth efficiency increases as M increases.
Rb / W  log 2 M / M [bits/s/Hz ]

MFSK (power-limited systems)
 Bandwidth efficiency decreases as M increases.
12
Power and bandwidth limited systems


Two major communication resources:
 Transmit power and channel bandwidth
In many communication systems, one of these resources is more
precious than the other. Hence, systems can be classified as:


Power-limited systems:
 save power at the expense of bandwidth
(for example by using coding schemes)
Bandwidth-limited systems:
 save bandwidth at the expense of power
(for example by using spectrally efficient modulation schemes)
13
Goals in designing a DCS

Goals:






Maximizing the transmission bit rate
Minimizing probability of bit error
Minimizing the required power
Minimizing required system bandwidth
Maximizing system utilization
Minimize system complexity
14
Limitations in designing a DCS





The Nyquist theoretical minimum bandwidth
requirement
The Shannon-Hartley capacity theorem (and the
Shannon limit)
Government regulations
Technological limitations
Other system requirements (e.g satellite orbits)
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