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

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