Digital Transmission Through AWGN Channel

Reference: Proakis and Salehi, “Fundamentals of Communication Systems”, Ch. 8, 2nd ed. (2014)

GEOMETRIC REPRESENTATION OF SIGNAL WAVEFORMS

Binary modulation: The modulator maps each information bit to be transmitted into one of two possible distinct signal waveforms, say s₁(t) or s₂(t).

Nonbinary (M-ary) Modulation: The modulator may transmit k bits (k > 1) at a time by employing M = 2ᵏ distinct signal waveforms, say sm(t), 1 ≤ m ≤M.

Develop a vector representation of such digital signal waveforms provides a compact characterization of signal sets for transmitting digital information over a channel, and it simplifies the analysis oftheir performance.

Gram-Schmidt Orthogonalization Procedure

Suppose a set of M signal waveforms Sm (t), 1 ≤ m≤ M, which are to be used for transmitting information over a communication channel. From the set of M waveforms, construct a set of N≤M orthonormal waveforms, where N is the dimension of the signal space.

The first waveform

s₁(t) is assumed to have energy ε₁.

The second waveform

possess unit energy

In general, the orthogonalization of the kth function leads to

Example

(a) Original signal set

(b) Orthonormal waveforms

The M signals {sm(t)} can be expressed as a linear combinations of the {ψn(t)}.

Each signal waveform may be represented by the vector

or equivalently, as a point in N-dimensional signal space with coordinates {smi , i =1, 2, . . . , N}. The energy of the mth signal waveform is simply the square of the length of the vector or, equivalently, the square of the Euclidean distance from the origin tothe point in the N-dimensional space.

Inner product

Example: From the previous example

The set of basis functions {ψn(t)} obtained by the Gram-Schmidt procedure is not unique

Alternate set of basis functions

BINARY MODULATION SCHEMES

Binary Antipodal SignalingThe information bit 1 is represented by a pulse p(t) of duration T, and the information bit 0 is represented by -p(t).

Example: Binary PAM signals.

Unit energy basis function for binary PAM

A rectangular pulse of unit amplitude and duration Tb.

Geometric representation of binary PAM

Unit energy basis function for the antipodal signals

Example of Binary antipodal signals

Binary Amplitude-Shift KeyingBinary ASK is a special case of binary antipodal signaling in which two baseband signals ±p(t) are used to amplitude modulate a sinusoidal carrier signal cos 2Πfct

Binary Orthogonal Signaling

s₁(t) and s₂(t) have equal energy εb and are orthogonal

Geometric representation of binary orthogonal signal waveforms.

Binary Pulse Position ModulationTwo o pulses are employed that are different only in their location

Signal pulses in binary PPM (orthogonal signals)

Two orthonormal basis functions for binary PPM signals.

Binary Frequency-Shift Keying

k₁ and k₂ =distinct positive integer

Binary FSK signal waveforms

Spectral characteristics of binary PPM (left) and FSK (right) signals

OPTIMUM RECEIVER FOR BINARY MODULATED SIGNALS IN ADDITIVE WHITE GAUSSIAN NOISE

Additive White Gaussian Noise Channel

Model for the received signal passed through an AWGN channel

Receiver for digitally modulated signals

Correlation-Type Demodulator

Binary Antipodal Signals

Cross correlator for binary antipodal signals

The conditional probability density functions of the correlator output for binary antipodal signaling.

Binary Orthogonal Signals

Correlation-type demodulator for binary orthogonal signals

The conditional probability density functions of the outputs (y1, y2) from the cross correlators of two orthogonal signals.

Matched-Filter-Type Demodulator

Binary Antipodal Signals

•Pass the received signal r(t) through a linear time-invariant filter with impulse response

The output of the filter at t = Tb is exactly the same as the output obtained with cross correlator

Signal s(t) and filter matched to s(t)

Binary Orthogonal Signals•Two linear time-invariant filters are employed•The correlation-type demodulator and the matched-filter-type demodulate yield identical outputs at t = Tb.

Properties of the Matched Filter

If a signal s(t) is corrupted by AWGN, the filter with the impulse response matched to s(t) maximizes the output signal-to-noise ratio (SNR).

Note that the output SNR from the matched filter depends on the energy of the waveform s(t) but not on the detailedcharacteristics of s(t). This is another interesting property of the matched filter.

The Performance of the Optimum Detector for Binary Signals

The average probability of error for equiprobable messages

Binary antipodal Binary orthogonal

Probability of error for binary signals

For the same error probability P2, the binary antipodal signals require a factor of two (3 dB) less signal energy than orthogonal signals.

M-ARY DIGITAL MODULATION

Relationship between the symbol interval and the bit interval

The input sequence to the modulator is subdivided into k-bit blocks, called symbols, and each of the M = symbols is associated with a corresponding signal waveform from the set {sm(t), m = 1 , 2, .. , M}.

The channel is assumed to corrupt the signal by the addition of white Gaussian noise. The received signal

The Signal DemodulatorThe receiver is subdivided into two parts: the signal demodulator and the detector. The function of the signal demodulator is to convert the received waveform r(t) into an N –dimensional vector N is the dimension

of the transmitted signal waveforms

The function of the detector is to decide which of the M possible signal waveforms was transmitted based on observation of the vector y.

The M-ary signal waveforms (each is N-dimensional)

ψk(t) and k = 1, 2, . . . , N are N orthonormal basis waveforms that span the N-dimensional signal space.

Correlation-type demodulator

Matched-filter-type demodulator

PDFs for M = 4 received PAM signals in additive white Gaussian noise

Example: 4-PAM Signalling

The Optimum DetectorDesign a signal detector that makes a decision on the transmitted signal in each signal interval based on the observation of the vector y in each interval, such that the probability of a correct decision is maximized.

Decision rule is based on the computation of the posterior probabilities

The receiver chooses the Sm that maximizes

Bayes's rule

This decision criterion is called the maximum a posteriori probability (MAP) criterion.

conditional PDF of the observed vector given Sm

P(sm) = Priori probability of the mth signal being transmitted

When the priori probabilities P(sm) are all equal, P(sm) = 1 / M for all M.

The decision rule based on finding the signal that maximizes

is equivalent to finding the signal that maximizes

The conditional PDF

is usually called the likelihood function. The decision criterion based on the maximum of over the M signals is called the maximum-likelihood (ML) criterion.

We observe that a detector based on the MAP criterion and one that is based on the criterion make the same decisions, as long as the a priori probabilities P(sm) are all equal; in other words, the signals {sm} are equiprobable.

M-ARY PULSE AMPLITUDE MODULATION

normalizedversion of p(t)

p(t) is a lowpass pulse signal of duration T

p(t) = gT(t) for a rectangular pulse shape

Rectangular pulse gT(t) and basis function ψ(t) for M-ary PAM.

M = 4 PAM signal waveforms

In order to minimize the average transmitted energy and to avoid transmitting signals with a DC component, we want to select the M signal amplitudes to be symmetric about the origin and equally spaced

The average energy

Example of M = 4 PAM signal waveforms

Carrier-Modulated PAM (M-ary ASK)

The transmitted signal waveforms

(for Bandpass Channels)

m = 1, 2, ... ,M.

Amplitude modulation of the sinusoidal carrier

Spectra of (a) baseband and (b)DSB-SC amplitude-modulated signal

Demodulation and Detection of Amplitude-Modulated PAM Signals

The transmitted signal

The received signal

Cross correlating the received signal r(t) with the basis function

The average probability of error

Probability of a symbol error for PAM

PHASE-SHIFT KEYING

where p(t) is a baseband signal of duration T and ɸm is determined by the transmitted message.

Example of a four-phase PSK (quadrature PSK (QPSK)) signal)

Block diagram of a digital-phase modulator

Geometric Representation of PSK Signals

The digital phase-modulated signals can be represented geometrically as two-dimensional vectors with components

PSK signal constellations

Demodulation and Detection of PSK Signals

The received signal may be correlated with the two basic functions

For binary-phase Modulation, the error probability

The symbol error probability for M = 4

P4

Probability of a symbol error for PSK signals

Differential Phase Encoding (Differential Phase –Shift Keying)

In differential encoding, the information is conveyed by phase shifts between any two successive signal intervals. For example, in binary-phase modulation, the information bit 1 may be transmitted by shifting the phase of the carrier by 1 80° relative to the previous carrier phase, while the information bit 0 is transmitted by a zero-phase shift relative to the phase in the preceding signaling interval. In four-phase modulation, the relative phase shifts between successive intervals are 0°, 90°, 1 80°, and 270°, corresponding to the information bits 00, 0 1 , 1 1, and 10, respectively.

Block diagram of a DPSK demodulator

Probability of error for binary PSK and DPSK

QUADRATURE AMPLITUDE-MODULATED DIGITAL SIGNALS

Impress separate information bits on each of the quadrature carriers

M = 16 QAM signal constellation

Functional block diagram of a modulator for QAM

Probability of a symbol error for QAM

SNR ADVANTAGE (in dB) OF M -ARY QAM OVER M-ARY PSK

1- Probability of Error for DPSK

2- Probability of Error for QAM