Unit IV BASEBAND Receiver 5202021 Digital Communication 146

Unit IV BASEBAND Receiver 5/20/2021 Digital Communication 1/46

Chapter 4: Baseband Pulse Transmission Outline • • 4. 1 Introduction 4. 2. Matched Filter 4. 3 Error Due to Noise 4. 4 Intersymbol Interference (ISI) 5/20/2021 Digital Communication 2/46

Chapter 4: Baseband Pulse Transmission 4. 1 Introduction • In Unit I – methods for digital transmission of analog information bearing signals • In Ch. II – methods for digital transmission of digital information using baseband channel • Digital data – broad spectrum; low-frequency components; • Transmission channel bandwidth – should accommodate the essential frequency content of the data stream • Channel is dispersive – channel is noisy – control over additive white noise (old problem. . ) – received signal pulses are affected by adjacent symbols (new problem) – intersymbol interference (ISI); major source of interference; – Distorted pulse shape (new problem) - channel requires control over pulse shape 5/20/2021 Digital Communication 3/46

Chapter 4: Baseband Pulse Transmission 5/20/2021 Digital Communication 4/46

Chapter 4: Baseband Pulse Transmission • Main issue to be discussed: – Detection of digital pulses corrupted by the effect of the channel • We know the shape of the transmitted pulse – Device to be used – matched filter 5/20/2021 Digital Communication 5/46

Chapter 4: Baseband Pulse Transmission Outline • • 4. 1 Introduction 4. 2. Matched Filter 4. 3 Error Due to Noise 4. 4 Intersymbol Interference (ISI) 5/20/2021 Digital Communication 6/46

Chapter 4: Baseband Pulse Transmission 4. 2 Matched Filter • Basic task – detecting transmitted pulses at the front end of the receiver (corrupted by noise) • Receiver model 5/20/2021 Digital Communication 7/46

Chapter 4: Baseband Pulse Transmission Details • The filter input x(t) is: • where – T is an arbitrary observation interval – g(t) is a binary symbol 1 or 0 – w(t) is a sample function of white noise, zero mean, psd N 0/2 The function of the receiver is to detect the pulse g(t) in an optimum manner, providing that the shape of the pulse is known and the distortion is due to effects of noise = To optimize the design of a filter so as to minimize the effects of noise at the filter output in some statistical sense. 5/20/2021 Digital Communication 8/46

Chapter 4: Baseband Pulse Transmission Designing the filter • Since we assume the filter is linear its output can be described as: • where – g 0(t) is the recovered signal – n(t) produced noise instantaneous power in the output signal • This is equal to maximizing the peak signal-to-noise ratio: 5/20/2021 Digital Communication average output noise 9/46

Chapter 4: Baseband Pulse Transmission So, • we have to define the impulse response of the filter h(t) in such a way that the signal-to-noise ratio (4. 3) is maximized. 5/20/2021 Digital Communication 10/46

Chapter 4: Baseband Pulse Transmission Let us assume that: - G(f) - FT of the signal g(t); - H(f) – frequency response of the filter then: FT of the output signal g 0(t)= H(f). G(f), or sampled at time t=T and no noise 5/20/2021 Digital Communication 11/46

Chapter 4: Baseband Pulse Transmission Next step is to add the noise. What we know is that the power spectral density of white noise is: Thus the average power of the output noise n(t) is: 5/20/2021 Digital Communication 12/46

Chapter 4: Baseband Pulse Transmission Substituting, So, given the function G(f), the problem is reduced to finding such an H(f) that would maximize η. 5/20/2021 Digital Communication 13/46

Chapter 4: Baseband Pulse Transmission We use Schwartz inequality which states that for two complex functions, satisfying the conditions: the following is true: and equality holds iff: 5/20/2021 Digital Communication 14/46

Chapter 4: Baseband Pulse Transmission In our case this inequality will have the form: and we can re-write the equation for the peak signal-to-noise ratio as: 5/20/2021 Digital Communication 15/46

Chapter 4: Baseband Pulse Transmission Remarks: • The right hand side of this equation does not depend on H(f). • It depends only on: – signal energy – noise power spectral density • Max value is for: 5/20/2021 Digital Communication 16/46

Chapter 4: Baseband Pulse Transmission • Let us denote the optimum value of H(f) by Hopt(f). To find it we use equation 4. 10: The result: Except for a scaling coefficient k exp(2πf. T), the frequency response of the optimum filter is the same as the complex conjugate of the FT of the input signal. 5/20/2021 Digital Communication 17/46

Chapter 4: Baseband Pulse Transmission Definition of the filter functions: • In the frequency domain: – knowing the input signal we can define the frequency response of the filter (in the frequency domain) as the FT of its complex conjugate. • In the time domain… 5/20/2021 Digital Communication 18/46

Chapter 4: Baseband Pulse Transmission Take inverse FT of Hopt(f): and keeping in mind that for real signals G*(f) = G(-f): 5/20/2021 Digital Communication 19/46

Chapter 4: Baseband Pulse Transmission • So, – in the time domain it turns out that the impulse response of the filter, except for a scaling factor k, is a time-reversed and delayed function of the input signal • This means it is “matched” to the input signal, that is why this type of time-invariant linear filters is known as “matched filter” • NOTE: The only assumption for the channel noise was that it is stationary, white, with psd N 0/2. 5/20/2021 Digital Communication 20/46

Chapter 4: Baseband Pulse Transmission Properties of Matched Filters Property 1: • A filter matched to a pulse signal g(t) of duration T is characterized by an impulse response that is time-reversed and delayed version of the input g(t): Time domain: hopt(t) = k. g(T-t) Frequency Domain: Hopt(f) = k. G*(f)exp(-j 2πf. T) 5/20/2021 Digital Communication 21/46

Chapter 4: Baseband Pulse Transmission Property 2: • A matched filter is uniquely defined by the waveform of the pulse but for the: - time delay T - scaling factor k 5/20/2021 Digital Communication 22/46

Chapter 4: Baseband Pulse Transmission Property 3: • The peak signal-to-noise ratio of the matched filter depends only on the ratio of the signal energy to the power spectral density of the white noise at the filter input. 5/20/2021 using the inverse FT Digital Communication 23/46

Chapter 4: Baseband Pulse Transmission • The integral of the squared magnitude spectrum of a pulse signal with respect to frequency is equal to the signal energy E (Rayleigh Theorem) so substituting in the previous formula we get: • After substitution of (4. 14) into (4. 7) we get the expression for the average output noise power as: 5/20/2021 Digital Communication 24/46

Chapter 4: Baseband Pulse Transmission we finally get the following expression: and 5/20/2021 Digital Communication 25/46

Chapter 4: Baseband Pulse Transmission Conclusion: • From (4. 20) we see that the dependence of the peak SNR on the input waveform g(t) has been completely removed by the matched filter. • So, in evaluating the ability of a matched-filter receiver to overcome/remove additive white noise we see that all signals with equal energy are equally effective. • We call the ratio E/N 0 signal_energy-to-noise ratio (dimensionless) • The matched filter is the optimum detector of a pulse of known shape in additive white noise. 5/20/2021 Digital Communication 26/46

Chapter 4: Baseband Pulse Transmission Outline • • 4. 1 Introduction 4. 2. Matched Filter 4. 3 Error Due to Noise 4. 4 Intersymbol Interference (ISI) 5/20/2021 Digital Communication 27/46

Chapter 4: Baseband Pulse Transmission 4. 3 Error Rate Due to Noise • In this section we will derive some quantitative results for the performance of binary PCM systems, based on results for the matched filter. • We consider the BER performance for a rectangular baseband pulse using and integrate and dump filter for detection. 5/20/2021 Digital Communication 28/46

Chapter 4: Baseband Pulse Transmission The model: • We assume – binary PCM system based on polar NRZ signaling (encoding) – symbols 1 and 0 are equal amplitude and equal duration – channel noise is modeled as AWGN w(t) with zero mean and psd N 0/2 – the received signal, in the interval 0 ≤ t ≤ Tb can be represented as: white noise amplitude 5/20/2021 Digital Communication 29/46

Chapter 4: Baseband Pulse Transmission • also it is assumed that the receiver has accurate knowledge of the starting and ending times of each pulse (perfect synchronization); • the receiver has to make a decision whether the pulse is a 1 or a 0. • the structure of the receiver is as follows: 5/20/2021 Digital Communication 30/46

Chapter 4: Baseband Pulse Transmission Details: • The filter is matched to a rectangular pulse of amplitude A and duration Tb. • The resulting matched filter output is sampled at the end of each signaling interval. • The channel noise adds randomness to the matched filter output. 5/20/2021 Digital Communication 31/46

Chapter 4: Baseband Pulse Transmission • For the channel model we are discussing (AWGN), integrating the noise over a period T is equal to creating noise with Gaussian distribution with variance σ02 = N 0 T/2, where N 0 is the noise power in Watts/Hz. 5/20/2021 Digital Communication 32/46

Chapter 4: Baseband Pulse Transmission • If we consider a single symbol s(t) of voltage V passing through the detector with additive noise n(t) then the output of the integrator y(t) will be: symbol contribution noise contribution 5/20/2021 Digital Communication 33/46

Chapter 4: Baseband Pulse Transmission • Then the probability density of the integrated noise at the sampling point is: • A detection error will occur if the noise sample exceeds - VT/2. • The probability for such an event is calculated from (1) 5/20/2021 Digital Communication 34/46

Chapter 4: Baseband Pulse Transmission • substituting we can write the expression for the probability of error as: complementary error function 5/20/2021 Digital Communication 35/46

Chapter 4: Baseband Pulse Transmission • The probability of error can also be expressed in terms of the signal energy as follows: • As 0 and 1 are equiprobable the result for receiving a logic 0 in error will be the same only in this case the noise sample will be exceeding +VT/2. 5/20/2021 Digital Communication 36/46

Chapter 4: Baseband Pulse Transmission • For the special case we are discussing unipolar NRZ the average symbol energy is half that of the logic 1. So the symbol error probability will be: 5/20/2021 Digital Communication 37/46

Figure 4. 5 Chapter 4: Baseband Pulse Transmission Noise analysis of PCM system. (a) Probability density function of random variable Y at matched filter output when 0 is transmitted. (b) Probability density function of Y when 1 is transmitted. 5/20/2021 Digital Communication 38/46

Chapter 4: Baseband Pulse Transmission Unipolar vs Bipolar Symbols • For unipolar symbols as we said the average energy is equal to E 0/2. • For bipolar symbols (logic 1 is conveyed by +V, logic 0 – by –V volts) it can be proved that (An example of this is RS-232, where "one" is − 5 V to − 12 V and "zero" is +5 to +12 V) 5/20/2021 Digital Communication 39/46

Chapter 4: Baseband Pulse Transmission Outline • • 4. 1 Introduction 4. 2. Matched Filter 4. 3 Error Due to Noise 4. 4 Intersymbol Interference (ISI) 5/20/2021 Digital Communication 40/46

Chapter 4: Baseband Pulse Transmission 4. 4 PAM and Intersymbol Interference (ISI) • ISI is due to the fact that the communication channel is dispersive – some frequencies of the received pulse are delayed which causes pulse distortion (change in shape and delay). • Most efficient method for baseband transmission – both in terms of power and bandwidth - is PAM. 5/20/2021 Digital Communication 41/46

Chapter 4: Baseband Pulse Transmission Considered model: • baseband binary PAM system • incoming binary sequence b 0 consists of 1 and 0 symbols of duration Tb Figure 4. 7 Baseband binary data transmission system. 5/20/2021 Digital Communication 42/46

Chapter 4: Baseband Pulse Transmission • PAM changes this sequence into a new sequence of short pulses each with amplitude ak, represented in polar form as: applied to a transmit filter of impulse response g(t): transmitted signal 5/20/2021 Digital Communication 43/46

Chapter 4: Baseband Pulse Transmission • s(t) is modified by a channel with channel impulse response h(t) • random white noise is added (AWGN channel model) • x(t) is the channel output, the noisy signal arriving at the receiver front end • receiver has a receive filter with impulse response c(t) and output y(t) • y(t) is sampled synchronously with the transmitter (clock signal is extracted from the receive filter output) • reconstructed samples are compared to a threshold • decision is taken as for 1 or 0 5/20/2021 Digital Communication 44/46

Chapter 4: Baseband Pulse Transmission Receive filter output is: should have a constant delay t 0 here set to 0 The scaled pulse µp(t) can be expressed as double convolution: the impulse response of the transmit filter g(t), the impulse response of the channel filter h(t) (channel) and the impulse response of the receive filter c(t): 5/20/2021 Digital Communication 45/46

Chapter 4: Baseband Pulse Transmission • In the frequency domain we have: where P(f), G(f), H(f) and C(f) are FT of the respective p(t), g(t), h(t) and c(t) The received signal is sampled at times ti= i. Tb which, taking (4. 44) in mind and the norm. condition p(0) = 1 (µ is a scaling factor to account for amplitude changes), can be expressed as: residual effect due to other transmitted pulses contribution of the 5/20/2021 ith pulse Digital Communication 46/46

Chapter 4: Baseband Pulse Transmission Conclusion: • In the absence of noise and ISI we get and ideal pulse (from (4. 48) ) y(ti)=µai • The ISI can be controlled (reduced) by the proper design of the transmit and receive filter • Subject discussed further in the following sections…. 5/20/2021 Digital Communication 47/47