Home | | Digital Communication | Discrete Memory less Channel

# 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

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
Communication Theory : Information Theory : Discrete Memory less Channel |