mathematical model for communication channels

MATHEMATICAL MODEL FOR COMMUNICATION CHANNELS

SAFEER V

MUHAMMED ASAD P T

DEPARTMENT OF ELECTRONICS ENGINEERING

PONDICHERRY UNIVERSITY

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

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

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

• 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

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 ...

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)

• There is absolute maximum of information capacity that can be transmitted

in a channel. This is called as (Shannon’s) channel capacity

• 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

In practice, a smaller bit rate can be achieved. Assume

22 log ( )b Tr B L

2log (1 )C B SNR

62 2log (1 ) 10 log (64) 6MbpsC B SNR

T4Mbps=2B log( ) 4br L L

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

ADDITIVE NOICE CHANNEL

• Simplest mathematical model

• Transmitted signal scorrupted by an additive random noise process n

• narise 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

when undergo attenuation then the received signal ,

r) =a*s+n

CHANNEL

n

rs+nS

Additive noise channel

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

impulse response of the system

denote the convolution

Linear filter

𝑛(𝑡 )CHANNEL

𝑠(𝑡) 𝑟 (𝑡 )=𝑐 (𝑡 )𝑠 (𝑡 )+𝑛(𝑡)

Linear filter channel with additive noise

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

• 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 ,

=

c( response of the channel at time t due to the impulse

applied at a time

Linear timeVariant filter

CHANNEL

𝑠(𝑡)

𝑛(𝑡 )

𝑟 (𝑡 )=𝑠 (𝑡 )𝑐 (𝜏 ;𝑡 )+𝑛(𝑡)

Linear time variant filter channel with additive noise

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.

• Correlation Demodulator:

• Decomposes the received signal and noise into a series of

• linearly weighted orthonormal basis functions.

• Equations for correlation demodulator:

dttftntsdttftrrk

T T

mkk )()()()()(0 0 Nk ,...2,1

,)()(0

dttftss k

T

mmk

,)()(0

dttftnn k

T

km

• Matched Filter Demodulator:

• Equation of a matched filter:

• Output of the matched filter is given by:

• k=1,2……N

),()( tTfth kk Tt 0

dthtrty k

T

k )()()(0

dtTftr k

T)()(

0

• 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

• m=1,2….M

N

nmnmn

N

nn

N

nnm ssrrsrD

1

2

11

2 2),(

,222

mm ssrr

22 mm ssr ),( msrD

Thank you….