Digital Communication Systems Lecture-3, Prof. Dr. Habibullah Jamal Under Graduate, Spring 2008

Baseband Demodulation/Detection In case of baseband signaling, the received signal sis already in pulse-like form. Why is then is demodulator required? Arriving baseband pulses are not in the form of ideal pulse shapes, each one occupying its own symbol interval. The channel (as well as any filtering at the transmitter) causes intersymbol interference (ISI). Channel noise is another reason that may cause bit error is channel noise.

Effect of Channel Figure 1.16 (a) Ideal pulse. (b) Magnitude spectrum of the ideal pulse.

Figure 1. 17 Three examples of filtering an ideal pulse Figure 1.17 Three examples of filtering an ideal pulse. (a) Example 1: Good-fidelity output. (b) Example 2: Good-recognition output. (c) Example3: Poor-recognition output.

Effect of Noise

3.1.2 Demodulation and Detection Figure 3.1: Two basic steps in the demodulation/detection of digital signals The digital receiver performs two basic functions: Demodulation, to recover a waveform to be sampled at t = nT. Detection, decision-making process of selecting possible digital symbol

3.2 Detection of Binary Signal in Gaussian Noise For any binary channel, the transmitted signal over a symbol interval (0,T) is: The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse response of the channel hc(t), is (3.1) Where n(t) is assumed to be zero mean AWGN process For ideal distortionless channel where hc(t) is an impulse function and convolution with hc(t) produces no degradation, r(t) can be represented as: (3.2)

3.2 Detection of Binary Signal in Gaussian Noise The recovery of signal at the receiver consist of two parts Filter Reduces the effect of noise (as well as Tx induced ISI) The output of the filter is sampled at t=T.This reduces the received signal to a single variable z(T) called the test statistics Detector (or decision circuit) Compares the z(T) to some threshold level 0 , i.e., where H1 and H2 are the two possible binary hypothesis

Receiver Functionality The recovery of signal at the receiver consist of two parts: Waveform-to-sample transformation (Blue Block) Demodulator followed by a sampler At the end of each symbol duration T, predetection point yields a sample z(T), called test statistic (3.3) Where ai(T) is the desired signal component, and no(T) is the noise component Detection of symbol Assume that input noise is a Gaussian random process and receiving filter is linear (3.4)

Need to achieve maximum (S/N)T Then output is another Gaussian random process Where 0 2 is the noise variance The ratio of instantaneous signal power to average noise power , (S/N)T, at a time t=T, out of the sampler is: (3.45) Need to achieve maximum (S/N)T

3.2.2 The Matched Filter Objective: To maximizes (S/N)T Expressing signal ai(t) at filter output in terms of filter transfer function H(f) (Inverse Fourier transform of the product H(f)S(f)). (3.46) where S(f) is the Fourier transform of input signal s(t) Output noise power can be expressed as: (3.47) Expressing (S/N)T as: (3.48)

Now according to Schwarz’s Inequality: (3.49) Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate Associate H(f) with f1(x) and S(f) ej2 fT with f2(x) to get: (3.50) Substitute in eq-3.48 to yield: (3.51)

Or and energy E of the input signal s(t): Thus (S/N)T depends on input signal energy and power spectral density of noise and NOT on the particular shape of the waveform Equality for holds for optimum filter transfer function H0(f) such that: (3.54) (3.55) For real valued s(t): (3.56)

The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T. The filter designed is called a MATCHED FILTER Defined as: a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform

3.2.3 Correlation realization of Matched filter A filter that is matched to the waveform s(t), has an impulse response h(t) is a delayed version of the mirror image of the original signal waveform Signal Waveform Mirror image of signal waveform Impulse response of matched filter Figure 3.7

This is a causal system Recall that a system is causal if before an excitation is applied at time t = T, the response is zero for -  < t < T The signal waveform at the output of the matched filter is (3.57) Substituting h(t) to yield: (3.58) When t=T, (3.59)

The function of the correlator and matched filter are the same Compare (a) and (b) From (a)

From (b) But At the sampling instant t = T, we have This is the same result obtained in (a) Hence

Detection Matched filter reduces the received signal to a single variable z(T), after which the detection of symbol is carried out The concept of maximum likelihood detector is based on Statistical Decision Theory It allows us to formulate the decision rule that operates on the data optimize the detection criterion

Probabilities Review P[s1], P[s2]  a priori probabilities These probabilities are known before transmission P[z] probability of the received sample p(z|s1), p(z|s2) conditional pdf of received signal z, conditioned on the class si P[s1|z], P[s2|z]  a posteriori probabilities After examining the sample, we make a refinement of our previous knowledge P[s1|s2], P[s2|s1] wrong decision (error) P[s1|s1], P[s2|s2] correct decision

How to Choose the threshold? Maximum Likelihood Ratio test and Maximum a posteriori (MAP) criterion: If else Problem is that a posteriori probabilities are not known. Solution: Use Bay’s theorem:

MAP criterion: When the two signals, s1(t) and s2(t), are equally likely, i.e., P(s2) = P(s1) = 0.5, then the decision rule becomes This is known as maximum likelihood ratio test because we are selecting the hypothesis that corresponds to the signal with the maximum likelihood. In terms of the Bayes criterion, it implies that the cost of both types of error is the same

Substituting the pdfs

Hence: Taking the log of both sides will give

Hence where z is the minimum error criterion and  0 is optimum threshold For antipodal signal, s1(t) = - s2 (t)  a1 = - a2

This means that if received signal was positive, s1 (t) was sent, else s2(t) was sent

Detection of Binary Signal in Gaussian Noise The output of the filtered sampled at T is a Gaussian random process

Matched Filter and Correlation The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T. The filter designed is called a MATCHED FILTER and is given by:

Bay’s Decision Criterion and Maximum Likelihood Detector Hence where z is the minimum error criterion and  0 is optimum threshold For antipodal signal, s1(t) = - s2 (t)  a1 = - a2

Probability of Error Error will occur if s1 is sent  s2 is received The total probability of error is the sum of the errors

If signals are equally probable Numerically, PB is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s2) and is given by:

The above equation cannot be evaluated in closed form (Q-function) Hence,

Table for computing of Q-Functions

A vector View of Signals and Noise N-dimensional orthonormal space characterized by N linearly independent basis function {ψj(t)}, where: From a geometric point of view, each ψj(t) is mutually perpendicular to each of the other {ψj(t)} for j not equal to k.

Representation of any set of M energy signals { si(t) } as a linear combinations of N orthogonal basis functions where N  M. where:

Therefore we can represent set of M energy signals {si(t) } as: Waveform energy: Representing (M=3) signals, with (N=2) orthonormal basis functions

Question 1: Why use orthormal functions? In many situations N is much smaller than M. Requiring few matched filters at the receiver. Easy to calculate Euclidean distances Compact representation for both baseband and passband systems. Gram-Schmidt orthogonalization procedure. Question 2: How to calculate orthormal functions?

Examples

Examples (continued)

Generalized One Dimensional Signals One Dimensional Signal Constellation

Binary Baseband Orthogonal Signals Binary Antipodal Signals Binary orthogonal Signals

Constellation Diagram Is a method of representing the symbol states of modulated bandpass signals in terms of their amplitude and phase In other words, it is a geometric representation of signals There are three types of binary signals: Antipodal Two signals are said to be antipodal if one signal is the negative of the other The signal have equal energy with signal point on the real line ON-OFF Are one dimensional signals either ON or OFF with signaling points falling on the real line

With OOK, there are just 2 symbol states to map onto the constellation space a(t) = 0 (no carrier amplitude, giving a point at the origin) a(t) = A cos wct (giving a point on the positive horizontal axis at a distance A from the origin) Orthogonal Requires a two dimensional geometric representation since there are two linearly independent functions s1(t) and s0(t)

Typically, the horizontal axis is taken as a reference for symbols that are Inphase with the carrier cos wct, and the vertical axis represents the Quadrature carrier component, sin wct Error Probability of Binary Signals Recall: Where we have replaced a2 by a0.

To minimize PB, we need to maximize: or We have Therefore,

The probability of bit error is given by:

The probability of bit error for antipodal signals: The probability of bit error for orthogonal signals: The probability of bit error for unipolar signals:

Bipolar signals require a factor of 2 increase in energy compared to orthogonal signals Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than orthogonal

Comparing BER Performance For the same received signal to noise ratio, antipodal provides lower bit error rate than orthogonal

Relation Between SNR (S/N) and Eb/N0 In analog communication the figure of merit used is the average signal power to average noise power ration or SNR. In the previous few slides we have used the term Eb/N0 in the bit error calculations. How are the two related? Eb can be written as STb and N0 is N/W. So we have: Thus Eb/N0 can be thought of as normalized SNR. Makes more sense when we have multi-level signaling. Reading: Page 117 and 118.