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} *|

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

If the
input to a DMC is a sequence of n symbols *u*_{1}
, *u*_{2} ,..., *u _{n}* selected from the alphabet
X and the corresponding output is the sequence

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} *),

AWGN is
the most important channel of this type.

*Y *=* X *+* G*

For any
given sequence *X* * _{i}* ,

*Y _{i}
*=

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 { *n _{i}* } are uncorrelated and are
Gaussian, therefore, statistically independent. So

^{}

^{ü} **Channel Capacity:**

Channel
model: DMC

Input
alphabet: *X* = {*x*_{0} , *x*_{1} , *x* _{2} ,..., *x _{q}*

Output
alphabet: *Y* = {*y* _{0} , *y*_{1} , *y* _{2} ,..., *y _{q}*

Suppose *x* * _{j}*
is transmitted,

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}* |

(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 |

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