mathematical model for communication channels
DESCRIPTION
mathematical model for communication channelTRANSCRIPT
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….