Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu1/46 CHAPTER 4 BASEBAND PULSE TRANSMISSION

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu2/46 Outline 4.1 Introduction 4.2. Matched Filter 4.3 Error Due to Noise 4.4 Intersymbol Interference (ISI)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu3/ Introduction In Ch.3 – methods for digital transmission of analog information bearing signals In Ch.4 – 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

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu4/46

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu5/46 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

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu6/46 Outline 4.1 Introduction 4.2. Matched Filter 4.3 Error Due to Noise 4.4 Intersymbol Interference (ISI)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu7/ Matched Filter Basic task – detecting transmitted pulses at the front end of the receiver (corrupted by noise) Receiver model

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu8/46 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu9/46 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 This is equal to maximizing the peak signal-to-noise ratio: instantaneous power in the output signal average output noise

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu10/46 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu11/46 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

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu12/46 Thus the average power of the output noise n(t) is: Next step is to add the noise. What we know is that the power spectral density of white noise is:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu13/46 Substituting, So, given the function G(f), the problem is reduced to finding such an H(f) that would maximize η.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu14/46 We use Schwartz inequality which states that for two complex functions, satisfying the conditions: the following is true: and equality holds iff:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu15/46 In our case this inequality will have the form: and we can re-write the equation for the peak signal-to-noise ratio as:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu16/46 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:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu17/46 Let us denote the optimum value of H(f) by H opt (f). To find it we use equation 4.10: The result: Except for a scaling coefficient k exp(- 2πfT), the frequency response of the optimum filter is the same as the complex conjugate of the FT of the input signal.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu18/46 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…

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu19/46 Take inverse FT of H opt (f): and keeping in mind that for real signals G*(f) = G(-f):

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu20/46 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu21/46 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: h opt (t) = k. g(T-t) Frequency Domain: H opt (f) = kG*(f)exp(- j2πfT)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu22/46 Property 2: A matched filter is uniquely defined by the waveform of the pulse but for the: - time delay T - scaling factor k

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu23/46 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. using the inverse FT

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu24/46 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:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu25/46 we finally get the following expression: and

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu26/46 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu27/46 Outline 4.1 Introduction 4.2. Matched Filter 4.3 Error Due to Noise 4.4 Intersymbol Interference (ISI)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu28/ 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu29/46 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 ≤ T b can be represented as: amplitude white noise

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu30/46 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:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu31/46 Details: The filter is matched to a rectangular pulse of amplitude A and duration T b. The resulting matched filter output is sampled at the end of each signaling interval. The channel noise adds randomness to the matched filter output.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu32/46 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 σ 0 2 = N 0 T/2, where N 0 is the noise power in Watts/Hz.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu33/46 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: noise contribution symbol contribution

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu34/46 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)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu35/46 substituting we can write the expression for the probability of error as: complementary error function

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu36/46 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu37/46 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:

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu38/46 Figure 4.5 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu39/46 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 −5V to −12V and "zero" is +5 to +12V)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu40/46 Outline 4.1 Introduction 4.2. Matched Filter 4.3 Error Due to Noise 4.4 Intersymbol Interference (ISI)

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu41/ 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.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu42/46 Considered model: baseband binary PAM system incoming binary sequence b 0 consists of 1 and 0 symbols of duration T b Figure 4.7 Baseband binary data transmission system.

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu43/46 PAM changes this sequence into a new sequence of short pulses each with amplitude a k, represented in polar form as: applied to a transmit filter of impulse response g(t): transmitted signal

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu44/46 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

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu45/46 Receive filter output is: 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): should have a constant delay t 0 here set to 0

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu46/46 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 t i = iT b 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: contribution of the ith pulse residual effect due to other transmitted pulses

Chapter 4: Baseband Pulse Transmission Digital Communication Systems 2012 R.Sokullu47/47 Conclusion: In the absence of noise and ISI we get and ideal pulse (from (4.48) ) y(t i )=µa i The ISI can be controlled (reduced) by the proper design of the transmit and receive filter Subject discussed further in the following sections….