George Mason University ECE 201: Introduction to Signal Analysis Spring 2017

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1 Assigned: March 20, 2017 Due Date: Week of April 03, 2017 George Mason University ECE 201: Introduction to Signal Analysis Spring 2017 Laboratory Project #6 Due Date Your lab report must be submitted on blackboard by 5:00 PM on the day of your lab on the week that it is due. Late lab reports will be penalized as noted in the Lab syllabus. Preparation Before going to lab you should, at a minimum, read the pre-lab to get an understanding of the operators and programs that you will be using. Ideally, you will spend some time using MATLAB to perform some of the exercises in the pre-lab either at home or using the GMU Virtual Commuting Lab (VCL) before your lab session. Get started early! Lab Report Your lab report for this lab will consist of answers and complete documentation of the questions and exercises in Section 3. The pre-lab is designed to get you ready for the problems in Section 3. Honor Code Forgeries and plagiarism are a violation of the honor code and any reasonable suspicion of an honor code violation will be reported. You are allowed to discuss lab exercises with other students, but the submitted work should be original and it should be your own work. 1 Introduction This lab is concerned with FIR filtering. In this lab you will also continue to gain more experience in writing MATLAB functions and code, and will be introduced to two new MATLAB functions, conv and filter. 2 Pre-Lab The primary goal of this lab is to learn how to implement FIR filters in MATLAB and study the response of FIR filters to various signals. You will also learn how filters can create effects such as echos and reverberation. 2.1 Overview of Filtering A simple yet very important class of discrete-time systems are those that are defined by an inputoutput relation given by y[n] = b k x[n k] (1)

2 where x[n] is the input and y[n] is the output. The constants {b k } are the filter coefficients that define the system. A simple example is y[n] = 1 3 x[n] x[n 1] + 1 3x[n 2] (2) where the n th value of the output is the average of the n th value of the input sequence x[n] and the two preceding values, x[n 1] and x[n 2]. In this case, the filter coefficients are b 0 = b 1 = b 2 = 1 3, and the impulse response is For the general case given in Eq. (1), the impulse response is h[n] = 1 3 δ[n] δ[n 1] + 1 3δ[n 2] (3) y[n] = b k δ[n k] Since the impulse response is finite in length, these systems are referred to as Finite-length Impulse Response (FIR) filters. When the input to an FIR filter is x[n], then the output is the convolution of x[n] with h[n], y[n] = x[n] h[n] If x[n] is stored in the vector x and h[n] in the vector h, then this convolution may be performed in MATLAB using conv as follows: >> y = conv(x,h); A more general filter than the one given in Eq. (1) is one where the output y[n] also depends on the previous output values, y[n] = b k x[n k] N a k y[n k] (4) where the constants a k and b k are the filter coefficients of the system. For these systems, the impulse response is infinite in length and are called Infinite-length Impulse Response (IIR) filters. A simple example is the system defined by k=1 y[n] = x[n] + y[n 1] In this case, it is not too hard to see that the impulse response is a step function, { 1; n 0 h[n] = 0; n < 0 Since the impulse response is infinite in length, it is not possible to use conv to find the response of the system to an input x[n]. Instead, one uses the filter command. The syntax of this command is as follows: >> yy = filter(bb,aa,xx); where bb and aa are the filter coefficients b k and a k, respectively, in Eq. (4). It is important to understand how the vector aa is defined and the sign convention that is used. Writing the difference equation as N y[n] + a k y[n k] = b k x[n k] k=1 the filter coefficients are then defined as follows:

3 >> aa=[ 1, a1, a2,..., an]; >> bb=[b0, b1, b2,..., bm]; Note that the leading one in aa represents the coefficient a 0 that would multiply y[n], which is equal to one. For example, if y[n] = x[n] + 0.2x[n 1] + 0.5y[n 1] then with the input signal stored in the vector xx, to find the responses of the system to this input we would write >> bb=[1, 0.2]; >> aa=[1-0.5]; >> yy = filter(bb,aa,xx); For FIR filters, the vector aa is set equal to one, aa=1. For example, to implement a three-point averaging system given in Eq. (2) we would write: >> bb = 1/3*[1, 1, 1]; % Filter coefficients >> yy = filter(bb,1,xx); % Compute the output for an input xx In MATLAB all sequences have finite length because they are stored in vectors. If the input signal has, for example, L samples, we would normally only store the L samples in a vector, and would assume that x[n] = 0 for n outside the interval of L samples; i.e., we do not have to store any zero samples unless it suits our purposes. If we process a finite-length signal through (1), then the output sequence y[n] will be longer than x[n] by M samples. If conv is used to process x[n] according to the difference equation given in Eq. (1), we will find that length(yy) = length(xx)+length(bb)-1 However, if we use filter, then MATLAB returns the filtered data as a vector of the same size as x. The reason for the difference is that filter will not find the output values for values of n that go beyond the final index of the input (the filter is not allowed to run off the end of the input signal). 2.2 Filtering Warm-Up In this part of the pre-lab you will implement the three-point running-average filter given in Eq. (2). (a) Let x[n] be a unit amplitude pulse of length ten, { 1; n = 0, 1, 2,..., 9 x[n] = u[n] u[n 10] = 0; otherwise For this exercise, generate the length-10 pulse and put it inside of a longer vector with the statement >> xx = [ones(1,10),zeros(1,5)]; This produces a vector of length 15, which has 5 extra zero samples appended to the last value in the pulse. Now use conv to filter x[n] with the three-point running-average filter. Note that indexing in MATLAB can be confusing. The signal x[n] starts at index n = 0, but MATLAB starts its indexing at 1. Nevertheless, we can ignore the difference and pretend that MATLAB is indexing from zero, as long as we don t try to write x[0] in MATLAB.

4 (b) To see the effect of the 3-point averager on the input signal, it is informative to make a plot of the input and output signals together using the subplot function. Since x[n] and y[n] are discrete-time signals, a stem plot should be used: >> nn = first:last; % use first=1 and last=length(xx) >> subplot(2,1,1); >> stem(nn-1,xx(nn)) >> subplot(2,1,2); >> stem(nn-1,yy(nn), filled ) % Make black dots >> xlabel( Time Index (n) ) Here, it is assumed that xx is the input to the filter and yy is the output. Make a plot with first equal to the starting index of the input signal, and last equal to the last index of the input. In other words, the plotting range for both signals will be equal to the length of the input signal. Note that using nn-1 in the call to stem causes the x-axis to start at n = 0. (c) Explain the filtering action of the 3-point averager by comparing the plots in the previous part. This filter might be called a smoothing filter. Note how the transitions in x[n] from zero to one and from one back to zero have been smoothed. (d) Repeat parts (a) and (b) for the signal x[n] = ( 1) n ; n = 0, 1, 2,..., 10 Does the 3-point averager smooth the signal? Explain. How does your result change if you were to use a four-point averager? 3 Lab: FIR Filters In the following sections we will study how a filter can produce the following special effects: 1. Echo: FIR filters can produce echoes and reverberations because the filtering formula Eq. (1) contains delay terms. In an image, such phenomena would be called ghosts. 2. Deconvolution: One FIR filter can (approximately) undo the effects of another we will investigate a cascade of two FIR filters that distort and then restore a signal. This process is called deconvolution. 3.1 Deconvolution Experiment Use the function filter to implement the following FIR filter w[n] = x[n] 0.85x[n 1] (FIR Filter-1) (5) on the input signal x[n] that is created with the following MATLAB statement: >> xx = rem(0:99,30)>10; (a) Explain how the MATLAB command that generates the signal x[n] works. How would you modify this statement to create a sequence of ten zeros followed by ten ones, and have this repeat for five cycles?

5 (b) Plot both the input and output waveforms x[n] and w[n] on the same figure, using the subplot and stem functions. (c) Explain why the output appears the way it does by figuring out (mathematically) the effect of the filter coefficients in Eq. (5) on x[n] Restoration Filter The following FIR filter y[n] = r k w[n k] (FIR Filter-2) can be used to undo the effects of the FIR filter in the previous section, and approximately restore the signal x[n]. Use the following steps to show how well this restoration filter works when r = 0.85 and M = 15. (a) Process the signal w[n] in Eq. (5) with FIR Filter-2 to obtain the output signal y[n]. (b) Make stem plots of w[n] and y[n] using a time-axis n that is the same for both signals. Put the stem plots in the same window for comparison using a two-panel subplot. (c) Since the objective of the restoration filter is to produce a y[n] that is almost identical to x[n], make a plot of the error between x[n] and y[n] over the range 0 n 100. (d) How well does the filter perform the restoration if you were to use a value of r = 0.84? Explain what you observe, and comment on what this means, in practice, if one wants to design a deconvolution filter. 3.2 Cascading Two Systems More complicated systems are often made up from simple building blocks. In the system of Fig. 1 two FIR filters are connected in cascade. Assume that the filters in Fig. 1 are described by the two equations: w[n] = x[n] q x[n 1] y[n] = r k w[n k] (FIR Filter-1) (FIR Filter-2) x[n] w[n] y[n] FIR FIR Filter #1 Filter #2 Figure 1: Cascade of two FIR filters.

6 3.2.1 Overall Impulse Response (a) Implement the system in Fig. 1 using MATLAB to get the impulse response of the overall cascaded system for the case where q = 0.85, r = 0.85 and M = 15. Here you will use two calls to filter. Plot the impulse response of the overall cascaded system. (b) Work out the impulse response h[n] of the cascaded system by hand to verify that your MATLAB result in part (a) is correct. (c) In a deconvolution application, the second system (FIR Filter-2) tries to undo the convolutional effect of the first. Perfect deconvolution would require that the cascade combination of the two systems be equivalent to the identity system: y[n] = x[n]. If the impulse responses of the two systems are h 1 [n] and h 2 [n], state the condition on h 1 [n] h 2 [n] to achieve perfect deconvolution, i.e., what must h 1 [n] h 2 [n] look like if h 2 [n] achieves perfect deconvolution? Does the cascade of FIR Filter-1 and Filter-2 perform perfect deconvolution? An Echo Filter The following FIR filter can be interpreted as an echo filter: y[n] = w[n] + αw[n P ] (FIR Echo) (6) Explain why this is a valid interpretation by working out the following: (a) Suppose that you have an audio signal sampled at f s = 8000 Hz and you would like to add a delayed version of the signal to simulate an echo. In order to produce an echo that is audible, the delay time has to be fairly long with respect to the sampling rate. A delay of one sample when f s = 8000 Hz is only 125 µsec. In order for an echo to be perceived by the human auditory system, you will need a delay of about 0.15 seconds. Determine the delay P needed in Eq. (6) to produce an echo of 0.15 seconds. Make sure that P is an integer. The quantity α will control the strength of the echo. (b) Implement the echo filter in Eq. (6) and filter the speech waveform v1 that is contained in the MATLAB file lab6dat.mat. Use the value of P determined in part (a), and set α = Since P may be a large number, the impulse response may be created easily using the upsample function as follows (here we assume that P = 100), >> h = upsample([1, 0.85], 100); Listen to the result, and compare it to the original to verify that you have produced an audible echo. Submit your speech echo along with your lab report. 3.3 Reverberation (Extra Credit) For this part, again use the speech file v1 in the MATLAB file lab6dat.mat. In this experiment you will be creating multiple echos to produce the effect called reverberation, which occurs when we have multiple echos. A reverberation that is created with three equallly spaced echos is y[n] = x[n] + α 1 x[n P ] + α 2 x[n 2P ] + α 3 x[n 3P ] where the α k determine the amplitude of each echo. The impulse response of this filter is h[n] = δ[n] + α 1 δ[n P ] + α 2 δ[n 2P ] + α 3 δ[n 3P ]

7 (a) Design a reverberation filter that will produce four echos with intensities α n where α = 0.8 and where P represents a delay of 0.1 seconds. (b) Make a plot of h[n]. (c) Filter the speech waveform with h[n] and describe the sound that you hear. (d) Submit your speech with reverberation along with your lab report.