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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
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