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Discrete Memory less Channel

Transmission rate over a noisy channel: Repetition code, Transmission rate.

DISCRETE MEMORYLESS CHANNEL:

 

·        Transmission rate over a noisy channel

 

Repetition code

 

Transmission rate

 

·        Capacity of DMC

 

Capacity of a noisy channel

 

Examples

Ø   All these transition probabilities from xi to yj are gathered in a transition matrix.

 

Ø   The (i ; j) entry of the matrix is P(Y = yj /jX = xi ), which is called forward transition probability.

 

Ø   In DMC the output of the channel depends only on the input of the channel at the same instant and not on the input before or after.

 

Ø   The input of a DMC is a RV (random variable) X who selects its value from a discrete limited set X.

 

Ø   The cardinality of X is the number of the point in the used constellation.

 

Ø   In an ideal channel, the output is equal to the input.

 

Ø   In a non-ideal channel, the output can be different from the input with a given probability.

 

·        Transmission rate:

Ø   H(X) is the amount of information per symbol at the input of the channel.

 

Ø   H(Y ) is the amount of information per symbol at the output of the channel.

 

Ø   H(XjY ) is the amount of uncertainty remaining on X knowing Y .

 

Ø   The information transmission is given by:I (X; Y ) = H(X) H(XjY ) bits/channel use

 

Ø   For an ideal channel X = Y , there is no uncertainty over X when we observe Y . So all the information is transmitted for each channel use: I (X;Y ) = H(X)

 

Ø   If the channel is too noisy, X and Y are independent. So the uncertainty over X remains the same knowing or not Y , i.e. no information passes through the channel: I (X; Y ) = 0.

 

·        Hard and soft decision:

 

Ø   Normally the size of constellation at the input and at the output are the same, i.e., jXj = jYj

 

Ø   In this case the receiver employs hard-decision decoding.

 

Ø   It means that the decoder makes a decision about the transmitted symbol.

 

Ø   It is possible also that jXj 6= jY j.

 

Ø   In this case the receiver employs a soft-decision.

 

ü   Channel models and channel capacity:


 

1.     The encoding process is a process that takes a k information bits at a time and maps each k-bit sequence into a unique n-bit sequence. Such an n-bit sequence is called a code word.

 

2. The code rate is defined as k/n.

 

3.     If the transmitted symbols are M-ary (for example, M levels), and at the receiver the output of the detector, which follows the demodulator, has an estimate of the transmitted data symbol with

 

(a). M levels, the same as that of the transmitted symbols, then we say the detector has made a hard decision;

 

(b). Q levels, Q being greater than M, then we say the detector has made a soft decision.

ü   Channels models:

 

1. Binary symmetric channel (BSC):

 

If (a) the channel is an additive noise channel, and (b) the modulator and demodulator/detector are included as parts of the channel. Furthermore, if the modulator employs binary waveforms, and the detector makes hard decision, then the channel has a discrete-time binary input sequence and a discrete-time binary output sequence.


Note that if the channel noise and other interferences cause statistically independent errors in the transmitted binary sequence with average probability p, the channel is called a BSC. Besides, since each output bit from the channel depends only upon the corresponding input bit, the channel is also memoryless.

 

2. Discrete memoryless channels (DMC):

 

A channel is the same as above, but with q-ary symbols at the output of the channel encoder, and Q-ary symbols at the output of the detector, where Q ³ q . If the channel and the modulator are memoryless, then it can be described by a set of qQ conditional probabilities

 

P (Y = y i  | X = x j ) º P ( y i  | x j ), i = 0,1,...,Q - 1; j = 0,1,..., q -1

 

Such a channel is called discrete memory channel (DSC).


If the input to a DMC is a sequence of n symbols u1 , u2 ,..., un selected from the alphabet X and the corresponding output is the sequence v1 , v 2 ,..., vn of symbols from the alphabet Y, the joint conditional probability is


the probability transition matrix for the channel.

 

3. Discrete-input, continuous-output channels:

 

Suppose the output of the channel encoder has q-ary symbols as above, but the output of the detector is unquantized (Q = ¥) . The conditional probability density functions

 

p ( y | X = x k ),      k = 0,1, 2,..., q -1

 

AWGN is the most important channel of this type.

 

Y = X + G


For any given sequence X i , i = 1, 2,..., n , the corresponding output is Yi , i = 1, 2,..., n

 

Yi  = X i + Gi , i = 1, 2,..., n

 

If, further, the channel is memoryless, then the joint conditional pdf of the detector‘s output is


 

4. Waveform channels:

 


If such a channel has bandwidth W with ideal frequency response C ( f ) = 1 , and if the bandwidth-limited input signal to the channel is x ( t) , and the output signal, y ( t) of the channel is corrupted by AWGN, then

y ( t ) = x ( t ) + n ( t)

 

The channel can be described by a complete set of orthonormal functions:



Since { ni } are uncorrelated and are Gaussian, therefore, statistically independent. So



ü   Channel Capacity:

Channel model: DMC

 

Input alphabet: X = {x0 , x1 , x 2 ,..., xq-1}

 

Output alphabet: Y = {y 0 , y1 , y 2 ,..., yq-1}

 

Suppose x j is transmitted,  yi is received, then

The mutual information (MI) provided about the event {X = x j } by the occurrence of the event


Hence, the average mutual information (AMI) provided by the output Y about the input X is


To maximize the AMI, we examine the above equation: 


(1). P ( y i) represents the jth output of the detector; 

(2). P ( y i  | x j ) represents the channel characteristic, on which we cannot do anything;

(3). P ( x j ) represents the probabilities of the input symbols, and we may do something or control them. Therefore, the channel capacity is defined by



 


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