# Lab 3.0. Pulse Shaping and Rayleigh Channel. Faculty of Information Engineering & Technology. The Communications Department

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1 Faculty of Information Engineering & Technology The Communications Department Course: Advanced Communication Lab [COMM 1005] Lab 3.0 Pulse Shaping and Rayleigh Channel

2 1 TABLE OF CONTENTS 2 Summary Background:... 3 Pulse Shaping and Matched Filtering... 3 Rayleigh Channel System model In Lab Experiment... 6 Task 1: Square Waveform... 6 Task 2: Rayleigh Channel... 6 Task 3: Receiver... 6 Task 4: Equalization Post Lab Assignment:... 7 BER Curve:

4 3 BACKGROUND: PULSE SHAPING AND MATCHED FILTERING The objective of the course is to construct the digital signal processing blocks necessary to operate a wireless digital communication link. A digital signal cannot be transmitted over a communication channel. The digital signal needs to be mapped to an analog waveform that can easily be transmitted over the channel. The model for transmission and reception considered in Lab 2 is illustrated in Figure 1. The transmitter creates a complex pulse amplitude modulated signal of the following form: x(t) = E x s[n]g tx (t nt ). n= Conceptually, the mapping of bits to waveforms can be viewed in two steps. 1. Source bits to symbols: This involves mapping the bits emitted by the (digital) source to (possibly complex) symbols, s on a constellation, C determined by the modulation scheme. This was accomplished in Lab Symbols to pulses: This involves creating an analog pulse train to be transmitted over the wireless channel, using the complex symbols generated in the first step. This is done using a pulse-shaping filter g tx (t) at the transmit side. The receiver processing depends on the type of distortion in the communication channel. Assuming an additive white Gaussian noise (AWGN) channel, the receiver performs analogous operations as at the transmitter. 1. Pulses to symbols: In this step, the symbols are recovered from the received pulses using a matched filter at the receive side. 2. Symbols to bits: This involves mapping the symbols received to bits (depending on the modulation scheme chosen), using some form of a detection technique. This was accomplished in Lab 2 using the maximum likelihood detection algorithm. To implement the maximum likelihood detector as described in Lab 2 using single-shot detection, the pulse shaping filters g tx (t) and g rx (t) have to satisfy some properties. The transmit pulse shape is normalized such that g tx (t) 2 dt = 1. This is primarily for convenience in our notation to avoid additional scaling factors. The receive filter g rx (t) = g tx ( t), which simplifies to g rx (t) = g tx ( t) since the pulse-shaping function is usually real. This means that the receive filter is a matched filter. Effectively the filtering by g rx (t) becomes a correlation with g tx (t) thanks to the relationship between convolution and correlation. The choice of a 3

5 matched filter maximizes the received signal-to-noise ratio and thus gives the maximum likelihood detector its best performance. The composite filter g(t) = g rx (τ )g tx (t τ)dt is a Nyquist pulse shape. This means that it satisfies g(nt ) = δ[n]. The Nyquist pulse shape property eliminates intersymbol interference, allowing the detector to take a very simple form as described in Lab 2. Defining the bandwidth of the transmitted signal x(t) requires some mathematical subtleties. From the perspective of measuring the power spectrum of a signal, the following definition of power spectrum for the transmitted signal is sufficient: P x (f) E x G tx (f) 2 where it is assumed that the transmit constellation is normalized to unit energy. It is clear that G tx (f ) 2 determines the bandwidth of x(t), where G tx (f ) is the frequency response of the pulse-shape g tx (t). By normalizing the transmit pulse shape P x (f )df f as the power. = E x where E x is known as the symbol energy and E x /T A simple choice for g tx (t) is the rectangular pulse-shape. Unfortunately, such a pulse-shape would have poor spectral properties, since the Fourier transform of a rectangular function is a sinc function. This means that a lot of bandwidth would be required to transmit the signal. Another choice for g tx (t) is the sinc function. A sinc pulse has ideal spectral properties. For example, if g tx (t) = sin(πt/t ) then G πt/t tx (f ) = T for f [ 1/2T, 1/2T ] giving an absolute bandwidth of 1/2T. This is the smallest bandwidth a complex pulse amplitude modulated signal could have with symbol rate 1/T. Unfortunately the sinc pulse shaping filter has a number of problems in practice. Ideal implementations of g(t) do not exist in analog. Digital implementations require truncating the pulse shape, which is a problem since it decays in time with 1/t. Further the sinc function is sensitive to sampling errors (not sampling at exactly the right point). RAYLEIGH CHANNEL In Lab 2, BPSK in a simple AWGN channel was considered. Realistic wireless environments suffer not only from noise bit also from other effects like reflection, diffraction and scattering. Such events causes fading to the signal on different levels such as distance pathloss, large scale shadowing and small scale fading. In this lab we discuss the simple BPSK in a Rayleigh multipath channel. In a brief discussion on Rayleigh channel, wherein we stated that a circularly symmetric complex Gaussian random variable is of the form,, 4

8 5 POST LAB ASSIGNMENT: BER CURVE: Use the files you created in this lab to generate the BER curves for BPSK Modulation in AWGN channel and Rayleigh channel for Eb/No range between -5 and 35 db as shown in the below figure. Your output graph should include two curves one for AWGN channel only and one for Rayleigh and AWGN Channel. 7

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