Faculty of Information Engineering & Technology The Communications Department Course: Advanced Communication Lab [COMM 1005] Lab 3.0 Pulse Shaping and Rayleigh Channel
1 TABLE OF CONTENTS 2 Summary... 2 3 Background:... 3 Pulse Shaping and Matched Filtering... 3 Rayleigh Channel... 4 3.2.1 System model... 5 4 In Lab Experiment... 6 Task 1: Square Waveform... 6 Task 2: Rayleigh Channel... 6 Task 3: Receiver... 6 Task 4: Equalization... 6 5 Post Lab Assignment:... 7 BER Curve:... 7 1
2 SUMMARY In Lab 2, you implemented the digital modulation and detection parts of a baseband QAM modem. This lab will deal with pulse shaping in digital communication systems and the impact of Rayleigh fading channel. You will implement the pulse shaping and matched filtering blocks of the baseband QAM modem shown in Figure 1. The system considered in this lab is illustrated in Figure 1. This modem transmits symbols and applies a detection algorithm assuming only an additive white Gaussian noise (AWGN) channel. The transmitter maps bits to elements of a symbol constellation. The sequence of symbols is upsampled then filtered in discrete-time by a transmit pulse shape. The filtered sequence is then passed to the discrete-tocontinuous converter. The receiver samples the received signal and filters it with the receiver pulse shaping filter. The filtered symbols are passed to a detection block which determines the most likely transmitted symbol for that observation. The detected symbols are passed to an inverse symbol mapping block to produce a good guess of the transmitted bits. Figure 1: The system under consideration in this lab. The transmitter uses digital pulse-shaping and upsampling to create the transmit waveform. The baseband signal encounters additive white Gaussian noise during transmission. The receiver uses digital pulse-shaping and downsampling, followed by detection, to find a good guess of the transmitted symbols. We begin by discussing the motivation and mathematical model for pulse shaping. You will then build the pulse shaping, up-sampling, and matched filtering blocks of the modem in LabVIEW. You will be required to read the manual and answer the pre-lab questions and submit them in your lab session. You will also follow the instructions of section 5 to complete the tasks of the in-lab experiments. 2
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 2. 2. 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
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
where real and imaginary parts are zero mean independent and identically distributed (i.i.d) Gaussian random variables with mean 0 and variance. The magnitude which has a probability density, is called a Rayleigh random variable. This model, called Rayleigh fading channel model, is reasonable for an environment where there are large number of reflectors known as multipath environment causing small scale fading. 3.2.1 System model The received signal in Rayleigh fading channel is of the form,, where is the received symbol, is complex scaling factor corresponding to Rayleigh multipath channel, is the transmitted symbol (taking values +1 s and -1 s) and is the Additive White Gaussian Noise (AWGN) Assumptions: 1. The channel is flat fading In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. 2. The channel is randomly varying in time meaning each transmitted symbol gets multiplied by a randomly varying complex number. Since is modeling a Rayleigh channel, the real and imaginary parts are Gaussian distributed having mean 0 and variance 1/2. 3. The noise has the Gaussian probability density function with with and. 4. The channel is known at the receiver. Equalization is performed at the receiver by dividing the received symbol by the apriori known i.e. where is the additive noise scaled by the channel coefficient. The BER rate for BPSK channel 5
4 IN LAB EXPERIMENT TASK 1: SQUARE WAVEFORM - Start with a new VI and generate a bit stream modulated with BPSK. - Hint : use the LabVIEW built in waveform generator to generate the bits stream. - Shape the symbols by a square waveform. - Hint: use arrays and loops. - Use a numeric control to control the length of the square pulse - Use x-y graphs to show your transmitted waveform. TASK 2: RAYLEIGH CHANNEL - In this task you are going to add the effect of small scale Rayleigh fading to the channel. - Update your VI to generate both AWGN noise and Rayleigh coefficients and add it to modulated symbols. - The Power of the noise is controlled by the input variable E b /N o - The scale parameter of Rayleigh random variable b=1/sqrt(2). - Use indicators to show the impact of the channel on the constellation points. TASK 3: RECEIVER - Update your VI from task 2 to add the detection part at the receiver side. - The detection for the square waveform is based on the integrate and dump technique. - Calculate the BER. - Use indicators to show the calculated BER and the received bits. TASK 4: EQUALIZATION - Assume that the coefficients of the Rayleigh channel are perfectly known at the receiver side. - Equalize the effect of the Rayleigh channel - Hint: divide by the coefficients - Use indicators to show the new value of the calculated BER and the received bits. 6
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