lecture 5

AWGN Channel

Digital Transmission over AWGN Channel

AWGN Channel

modulator at the transmitter performs the function of mapping the information sequence into signal waveforms.

These waveforms are transmitted over the channel, and a corrupted version of them is received at the receiver

Channels can suffer from a variety of impairments that contribute to errors.

Channel Impairments

Impairments include noise, attenuation, distortion, fading, and interference

Noise is present in all communication channels and is the major impairment in many communication systems

In this lecture we will deal with the design and performance characteristics of optimum receivers for the various modulation methods when the channel corrupts the transmitted signal by the addition of white Gaussian noise

VECTOR CHANNEL MODELS

The additive white Gaussian noise (AWGN) channel model is a channel whose sole effect is addition of a white Gaussian noise process to the transmitted signal.

is one of M possible signals

N0/2


receiver makes the optimal decision about which message was transmitted.

the decision rule minimizes the probability of disagreement between the transmitted message m and the detected message given by


Any orthonormal basis can be used for expansion of a zero-mean white Gaussian process, and the resulting coefficients of expansion will be iid (independent and identical destribution) zero-mean Gaussian random variables with variance N0/2

Optimal Detection

AWGN vector channel is given by

where all vectors are N -dimensional real vectors

vectors are selected from a set

Optimal Detection

A posteriori probability and Likelihood function

The posterior probability is the probability of the parameters given the evidence :

In contrasts with thelikelihood function, which is the probability of the evidence given the parameters: .

The two are related as follows:

Let us have apriorbelief that theprobability distribution functionis and observations with the likelihood , then the posterior probability is defined as

.

MAP and ML Receivers

MAP receiver can be simplified to

Maximum a posteriori probability (MAP) can be written as

MAP and ML Receivers

And Receiver is called the maximum-likelihood receiver, or ML receiver

It is important to note that the ML detector is not an optimal detector unless the messages are equiprobable.

The ML detector, however, is a very popular detector since in many cases having exact information about message if probabilities are difficult.

The Decision Regions

Any detector partitions the output space into

For a MAP detector we have

The Error Probability

is symbol error probability or message error probability

E X A M P L E 4.11

Preprocessing at the Receiver

the optimal detector based on the observation of makes the same decision as the optimal detector based on the observation of r.

In other words, an invertible preprocessing of the received information does not change the optimality of the receiver.

WAVEFORM AND VECTOR AWGN CHANNELS

WAVEFORM AND VECTOR AWGN CHANNELS

the second component cannot provide any information about the transmitted signal and therefore has no effect in the detection process and can be ignored without sacrificing the optimality of the detector

Optimal Detection for the Vector AWGN Channel

The MAP detector for this channel is given by

Where

as the bias term


It can further simplified as

The receiver receives r and looks among all sm to find the one that is closest to r using standard Euclidean distance.

Such a detector is called a nearest-neighbor, or minimum-distance, detector.

Also note that in this case, since the signals are equiprobable, the MAP and the

ML detector coincide, and both are equivalent to the minimum-distance detector.


In this case the boundaries of decisions Dm and are the set of points that are equidistant from sm and , which is the perpendicular bisector of the line connecting these two signal points

Optimal Detection for Binary Antipodal Signaling

The probabilities of messages 1 and 2 are p and 1 p, respectively.

Error Probability for Equiprobable Binary Signaling

Since Q( ) is a decreasing function, in order to minimize the error probability, the distance between signal points has to be maximized.

Optimal Detection for Binary Orthogonal Signaling

which is signal-to-noise ratio per bit, or SNR per bit, or simply the SNR

Optimal Detection for Binary Orthogonal Signaling

Implementation of the Optimal Receiver for AWGN Channels

The Correlation Receiver:

MAP decision rule is given by

Correlation Receiver

Matched Filter Receiver

In correlation receiver implementations we compute quantities of the form




OPTIMAL DETECTION AND ERROR PROBABILITYFOR BAND-LIMITED SIGNALING

Optimal Detection and Error Probability for ASK or PAM Signaling

Optimal Detection and Error Probability for PSK Signaling

Optimal Detection and Error Probability for QAM Signaling


The minimum distance between any two points is dmin which is given by

There are M 2 inner points and 2 outer points in the constellation

For the outer points, the probability of error is one-half of the error probability of an inner point since noise can







There is no advantage of the two-amplitude QAM signal set over M = 4-phase modulation.

(a) four-phase modulated signal

(b) QAM signal with two amplitude levels







lecture 5

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map detector

probability of disagreement

posteriori probability

optimal decision

transmitted message

receiver channels

receiverthe optimal

vector awgn channels