Baseband Receiver Receiver Design: Demodulation Matched Filter Correlator Receiver Detection Max. Likelihood Detector Probability of Error

Transmit and Receive Formatting

Sources of Error in received Signal  Major sources of errors: Thermal noise (AWGN)  disturbs the signal in an additive fashion (Additive)  has flat spectral density for all frequencies of interest (White)  is modeled by Gaussian random process (Gaussian Noise) Inter-Symbol Interference (ISI)  Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”.

Demodulation/Detection of digital signals Receiver Structure

Receiver Structure contd  The digital receiver performs two basic functions: Demodulation Detection  Why demodulate a baseband signal??? Channel and the transmitter’s filter causes ISI which “smears” the transmitted pulses Required to recover a waveform to be sampled at t = nT.  Detection decision-making process of selecting possible digital symbol

Important Observation  Detection process for bandpass signals is similar to that of baseband signals. WHY???  Received signal for bandpass signals is converted to baseband before detecting Bandpass signals are heterodyned to baseband signals Heterodyning refers to the process of frequency conversion or mixing that yields a spectral shift in frequency.  For linear system mathematics for detection remains same even with the shift in frequency

Steps in designing the receiver  Find optimum solution for receiver design with the following goals: 1.Maximize SNR 2.Minimize ISI  Steps in design: Model the received signal Find separate solutions for each of the goals.

Detection of Binary Signal in Gaussian Noise  The recovery of signal at the receiver consist of two parts Filter  Reduces the received signal to a single variable z(T)  z(T) is called the test statistics Detector (or decision circuit)  Compares the z(T) to some threshold level  0, i.e., where H 1 and H 0 are the two possible binary hypothesis

Receiver Functionality The recovery of signal at the receiver consist of two parts: 1. Waveform-to-sample transformation  Demodulator followed by a sampler  At the end of each symbol duration T, pre-detection point yields a sample z(T), called test statistic Where a i (T) is the desired signal component, and n o (T) is the noise component 2. Detection of symbol  Assume that input noise is a Gaussian random process and receiving filter is linear

Finding optimized filter for AWGN channel Assuming Channel with response equal to impulse function

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 h c (t), is Where n(t) is assumed to be zero mean AWGN process  For ideal distortionless channel where h c (t) is an impulse function and convolution with h c (t) produces no degradation, r(t) can be represented as:

Design the receiver filter to maximize the SNR  Model the received signal  Simplify the model: Received signal in AWGN Ideal channels AWGN

Find Filter Transfer Function H 0 (f)  Objective: To maximizes (S/N) T and find h(t)  Expressing signal a i (t) at filter output in terms of filter transfer function H(f) where H(f) is the filter transfer funtion and S(f) is the Fourier transform of input signal s(t) If the two sided PSD of i/p noise is N 0 /2  Output noise power can be expressed as: Expressing (S/N) T :

 For H(f) = H opt (f) to maximize (S/N) T use Schwarz’s Inequality:  Equality holds if f 1 (x) = k f* 2 (x) where k is arbitrary constant and * indicates complex conjugate  Associate H(f) with f 1 (x) and S(f) e j2  fT with f 2 (x) to get:  Substitute yields to:

 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 H 0 (f) such that: (3.55)  For real valued s(t):

 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

Matched Filter Output of a rectangular Pulse

Replacing Matched filter with Integrator

 A filter that is matched to the waveform s(t), has an impulse response h(t) is a delayed version of the mirror image (rotated on the t = 0 axis) of the original signal waveform Signal Waveform Mirror image of signal waveform Impulse response of matched filter Correlation realization of Matched filter

Correlator Receiver  This is a causal system 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  Substituting h(t) to yield:  When t=T So the product integration of rxd signal with replica of transmitted waveform s(t) over one symbol interval is called Correlation

Correlator versus Matched Filter The functions of the correlator and matched filter The mathematical operation of Correlator is correlation, where a signal is correlated with its replica Whereas the operation of Matched filter is Convolution, where signal is convolved with filter impulse response But the o/p of both is same at t=T so the functions of correlator and matched filter is same. Matched Filter Correlator

Implementation of matched filter receiver Bank of M matched filters Matched filter output: Observation vector

Implementation of correlator receiver Bank of M correlators Correlators output: Observation vector

Example of implementation of matched filter receivers Bank of 2 matched filters Tt Tt T T

Detection Max. Likelihood Detector Probability of Error

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

 P[s 0 ], P[s 1 ]  a priori probabilities These probabilities are known before transmission  P[z] probability of the received sample  p(z|s 0 ), p(z|s 1 ) conditional pdf of received signal z, conditioned on the class s i  P[s 0 |z], P[s 1 |z]  a posteriori probabilities After examining the sample, we make a refinement of our previous knowledge  P[s 1 |s 0 ], P[s 0 |s 1 ] wrong decision (error)  P[s 1 |s 1 ], P[s 0 |s 0 ] correct decision Probabilities Review

 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: How to Choose the threshold? This means that if received signal is positive, s 1 (t) was sent, else s 0 (t) was sent

1 Likelihood of S o and S 1

MAP criterion: When the two signals, s 0 (t) and s 1 (t), are equally likely, i.e., P(s 0 ) = P(s 1 ) = 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, both sides will give

 Hence where z is the minimum error criterion and  0 is optimum threshold  For antipodal signal, s 1 (t) = - s 0 (t)  a 1 = - a 0

Probability of Error  Error will occur if s 1 is sent  s 0 is received s 0 is sent  s 1 is received The total probability of error is sum of the errors

 If signals are equally probable  Hence, the probability of bit error P B, is the probability that an incorrect hypothesis is made  Numerically, P B is the area under the tail of either of the conditional distributions p(z|s 1 ) or p(z|s 0 )

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

Co-error function  Q(x) is called the complementary error function or co-error function  Is commonly used symbol for probability  Another approximation for Q(x) for x>3 is as follows:  Q(x) is presented in a tabular form

Co-error Table

To minimize P B, we need to maximize: or Where (a 1 -a 2 ) is the difference of desired signal components at filter output at t=T, and square of this difference signal is the instantaneous power of the difference signal i.e. Signal to Noise Ratio Imp. Observation