This document discusses mathematical models for communication channels. It begins by showing a block diagram of a basic digital communication system and defines the key components. It then discusses different types of communication channels and mediums that can be used to transmit signals, including wires, wireless spectra, and optical fibers. The rest of the document discusses several common mathematical models used to represent communication channels, including additive noise channels, linear filter channels, and linear time-variant filter channels. It also discusses parameters that characterize channels and limits on data transmission rates. Finally, it covers optimum receivers for signals corrupted by additive white Gaussian noise.

1. MATHEMATICAL MODEL FOR
COMMUNICATION CHANNELS
SAFEER V
MUHAMMED ASAD P T
DEPARTMENT OF ELECTRONICS ENGINEERING
PONDICHERRY UNIVERSITY

2. BLOCK DIAGRAM OF A DIGITAL
COMMUNICATION SYSTEM
Information
source and
input
transducer
Channel
decoder
Output
transducer
Channel
Digital
modulator
Channel
encoder
Source
encoder
Source
decoder
Digital
demodulator

3. COMMUNICATION CHANNELS AND MEDIUM
• A physical medium is an inherent part of a communications system
• Wires (copper, optical fibers) , wireless radio spectra
• Communications systems include electronic or optical devices that are part of the
transmission path followed by a signal Equalizers, amplifiers, signal conditioners
(regenerators)
• Medium determines only part of channels behavior. The other part is
determined how transmitter and receiver are connected to the medium
• Therefore, by telecommunication channel we refer to the combined end-to-end
physical medium and attached devices
• Often term “filter” refers to a channel, especially in the context of a specific
mathematical model for the channel. This is due to the fact that all
telecommunication channels can be modeled as filters. Their parameters can be
deterministic ,random, time variable, linear/nonlinear

4. COMMUNICATION CHANNEL
• A medium for sent the signal
• Provide a connection between the transmitter and receiver
• Wireless transmission --- atmosphere
• Wire line transmission --- twisted pair wire , coaxial cable , optical fibre

5. • Wire line channel carry electrical signal
• Optical fibre carries information on modulated light beam
• Under water – information transmitted acoustically
• Free space -- information bearing signal transmitted by antenna

6. CHANNELS PARAMETERS
• Characterized by
• attenuation , transfer function
• impedance matching
• bandwidth , data rate
• Transmission impairments change channel’s effective properties
• system internal/external interference
• cross-talk - leakage power from other users
• channel may introduce inter-symbolic interference (ISI)
• channel may absorb interference from other sources
• wideband noise
• distortion, linear (uncompensated transfer function)/nonlinear (non-linearity in
circuit elements)
• Channel parameters are a function of frequency, transmission length,
temperature ...

7. DATARATE LIMITS
• Data rate depends on: channel bandwidth, the number of levels in
transmitted signal and channel SNR (received signal power)
• For an L level signal with theoretical sinc-pulse signaling transmitted maximum bit
rate is (Nyquist bit rate)
 2 2 log ( ) b T r B L
• There is absolute maximum of information capacity that can be transmitted in a
channel. This is called as (Shannon’s) channel capacity
C  Blog2(1SNR)
• Example: A transmission channel has the bandwidth
and SNR = 63. Find the appropriate bit rate and number of signal levels. Solution:
Theoretical maximum bit rate is
    6
2 2 C Blog (1 SNR) 10 log (64) 6Mbps
In practice, a smaller bit rate can be achieved. Assume
   T 4Mbps=2B log( ) 4 b r L L

8. WHY DO WE GO FOR A MATHEMATICAL MODEL
FOR COMMUNICATION CHANNELS?
• Mathematical model reflect the most important characteristic of the system
• Channel mathematical model help to design channel encoder and modulator
at receiver and channel decoder and demodulator at receiver side

9. ADDITIVE NOICE CHANNEL
• Simplest mathematical model
• Transmitted signal s(푡) corrupted by an additive random noise process n(푡)
• n(푡) arise from electrical components
• If noise is introduced primarily at receiver side by components, it may be
characterized as thermal noise. this type of noise is characterized as Gaussian
noise process. hence mathematical mode of this channel is called additive
Gaussian noise channel

10. CHANNEL
S(푡) r(푡) =s(푡)+n(푡)
n(푡)
Additive noise channel
when undergo attenuation then the received signal ,
r(푡) =a*s(푡)+n(푡)

11. LINEAR FILTER CHANNEL
• In wire line channel the signal do not exceed specified bandwidth
• Channel characterized mathematically as linear filter (for limit the bandwidth) with additive
noise
푟 푡 = 푠 푡 푐 푡 + 푛(푡)
∞
−∞
푐 휏 푠 푡 − 휏 푑휏 + 푛(푡)
푐(푡) impulse response of the system
denote the convolution

12. Linear filter
푠(푡) 푟 푡 = 푐 푡 푠 푡 + 푛(푡)
푐(푡)
푛(푡)
CHANNEL
Linear filter channel with additive noise

13. LINEAR TIME VARIANT FILTER CHANNEL
• Under water acoustic channel
is characterized as a multipath
channel due to signal reflection
from the surface and bottom of
the sea

14. • Because of water motion, signal multipath component undergo time time varying
propagation delay
• So channel modelled mathematically as a linear filter characterized by time variant channel
impulse response
• The output signal ,
푟 푡 = 푠 푡 푐 휏; 푡 + 푛(푡)
∞
= −∞
푐 휏; 푡 푠 푡 − 휏 푑휏 + 푛(푡)
c(휏; 푡) response of the channel at time t due to the impulse
applied at a time 푡 − 휏

15. Linear time
Variant filter
푐(휏; 푡)
CHANNEL
푠(푡)
푛(푡)
푟 푡 = 푠 푡 푐 휏; 푡 + 푛(푡)
Linear time variant filter channel with additive noise

16. OPTIMUM RECEIVERS CORRUPTED BY ADDITIVE WHITE GAUSSIAN
NOISE
• General Receiver:
r(t)=Sm(t)+n(t)
Sm(t)
n(t)
Receiver is subdivided into:
• 1. Demodulator.
• (a) Correlation Demodulator.
• (b) Matched Filter Demodulator.
• 2. Detector.

17. • Correlation Demodulator:
• Decomposes the received signal and noise into a series of
• linearly weighted orthonormal basis functions.
• Equations for correlation demodulator:
r r t f t dt s t n t f t dt
     k 1,2,...N
0 0 k
T T
k k m ( ) ( ) ( ) ( ) ( )
T
mk m  
s s t f t dt k
( ) ( ) ,
0
T
km  
n n t f t dt k
( ) ( ) ,
0

18. • Matched Filter Demodulator:
• Equation of a matched filter:
h (t) f (T t), k k   0  t  T
• Output of the matched filter is given by:
T
y ( t ) r ( t ) h ( t  )
d k k
  
0
• k=1,2……N
T
( ) ( )
0
r t f T t  d k
   

19. • Optimum Detector:
• The optimum detector should make a decision on the
transmitted signal in each signal interval based on the observed
vector
• Optimum detector is defined by
N
N
N
2 ( , ) 2
  
  
D r s  r  r s 
s
m n 2
mn mn
n
n
n
n
1
1 1
2 2
m m  r  r  s  s
2 ,
• m=1,2….M
2
2 m m ( , )    r  s  s m D r s

20. Thank you….