SMS045 - DSP Systems in Practice. Lab 1 - Filter Design and Evaluation in MATLAB Due date: Thursday Nov 13, 2003

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1 SMS045 - DSP Systems in Practice Lab 1 - Filter Design and Evaluation in MATLAB Due date: Thursday Nov 13, 2003 Lab Purpose This lab will introduce MATLAB as a tool for designing and evaluating digital filters, prior to implementing them in a digital signal processor (DSP). A set of useful MATLAB commands will be introduced. Two versions of a low-pass filter will be designed: a FIR (Finite Impulse Response) representation and an IIR (Infinite Impulse Response) representation. Also, the effects of quantization of filter coefficients will be studied. Preparations Read through the lab instructions carefully. Required Hand-ins Hand in one report per group, consisting of the following: An introductory paragraph that summarizes what you have done in the lab (about 1/2 A4 page). Assignment 1: Your choice of filter order. Bode plot (magnitude and phase versus frequency) of the filter. A plot of the impulse response of the filter. Your MATLAB code. Assignment 2: Your choice of filter order. Bode plot of the filter. Pole-zero plot of the filter. Your MATLAB code. Assignment 3: Bode and pole-zero plots of the filter before and after coefficient quantization. Comment on the performance of the quantized filter, regarding the Bode and pole-zero plots. Your oppinions about the lab and the lab instructions. Too easy? Too difficult? Suggestions on changes.

2 Required Presentations Assignment 4: Demonstrate your filtered.wav file to the teacher. Include your names and addresses on the titlepage of the report. All plots must have titles and labeled axes. To include MATLAB plots in your report, use the print command to export MATLAB figures to a suitable file format. For example, print -deps firbodeplot.eps exports the current MATLAB figure to a.eps file called firbodeplot.eps. Remember: You can get information about a MATLAB command by typing: help <command name>. MATLAB codes that are included must begin with comments that show your names and an explanation of what the code does. This could for example look like: % Robert Plant E3, Ron Wood D4 % % Lab 1, Assignment 1 % Script for generating a LP FIR filter

3 Digital Filter Tutorial A digital filter is a discrete-time linear system, h[n], acting on an input signal, x[n], to produce an output signal, y[n]. This is illustrated in Figure 1, where the input x[n] is convolved with a filter with impulse response PSfrag h[n], replacements resulting in the output y[n]. Another way to treat discrete-time linear x[n] h[n] y[n] = x[n] h[n] Figure 1: A discrete-time linear system. systems is to transform them to z-domain respresentation. Going to z-domain representation will, in most cases, simplify analysis of discrete-time systems. The z-transform, X(z), of a discrete-time function, x[n], is defined as X(z) = x[n]z n, (1) n= where z is any complex number such that the sum on the right side is absolutely convergent, that is x[n]z n <. (2) n= The z-transform has many useful properties, similar to the Fourier or Laplace transforms, that simplifies analysis of discrete-time linear systems. For example: Linearity - For any complex numbers a, b: Time Shift - For any integer m: Time-Domain Convolution w[n] = ax[n] + by[n] W (z) = ax(z) + by (z). (3) w[n] = x[n m] W (z) = z m X(z). (4) w[n] = x[n] y[n] W (z) = X(z)Y (z) (5) Figure 2 shows PSfrag the z-domain replacements representation of the linear system from Figure 1. X(z) H(z) Y (z) = X(z)H(z) Figure 2: A linear system in z-domain representation. In the following assignments, digital low-pass filters will be designed using MATLAB. The ideal discrete-time low-pass filter attenuates frequency contents in the input signal above a specified 3 3rd November 2003

4 cut-off frequency, Ω c, and passes other frequencies through. The frequency response of an ideal low-pass filter is shown in Figure 3. The frequency range 0 < Ω < Ω c is called the pass band of the filter and the range Ω c < Ω < π is called the stop band. Recall that the discrete-time angular frequency, Ω is the angular frequency scaled by the sampling frequency, Ω = 2πf f s, (6) and that the frequency response of a discrete-time (sampled) linear system is periodic with period 2π. PSfrag replacements H(Ω) (db) 0 0 Ω c π 2π Ω c 2π Ω (rad/sample) Figure 3: The frequency response of an ideal digital low-pass filter. A typical realization of a low-pass filter would not exactly resemble the response in Figure 3. Figure 4 shows the frequency response of a real digital FIR low-pass filter designed in MATLAB, with Ω c = 0.3π. The magnitude in the pass band is not completely flat and does not vanish at Ω = Ω c. Instead, different design criterions are used to specify the magnitude response in the pass band. Furthermore, the stop band is typically defined as the range Ω stop < Ω < π for some specified Ω stop > Ω c. A certain stop band attenutation (e.g. -40dB) is often specified in the filter design. The frequency range Ω c < Ω < Ω stop is called the transition band. A monotonical decrease of the magnitude response in the transition band is often required Magnitude (db) Normalized Frequency ( π rad/sample) Figure 4: The frequency response of a digital FIR low-pass filter designed in MATLAB. 3rd November

5 Note: MATLAB normalizes the discrete-time angular frequency range 0 < Ω < 2π to 0 < Ω < 2, as seen from Figure 4. Usually, only the frequency range 0 < Ω < π (or 0 < Ω < 1 in MATLAB) is plotted. Another filter type that will be studied in this lab is band-pass filters. Figure 5 shows the magnitude response of an ideal digital band-pass filter. Frequency contents in the pass band, PSfrag replacements H(Ω) (db) 0 0 Ω 1 Ω 2 π 2π Ω 2 2π Ω 1 2π Ω (rad/sample) Figure 5: The frequency response of an ideal digital band-pass filter. Ω 1 < Ω < Ω 2, is passed through the filter, while frequencies outside the pass band are attenuated. As for low-pass filters, a typical realization of a band-pass filter would not exactly resemble the ideal response shown in Figure 5. FIR Low-Pass Filter Design The general structure of a finite impulse response (FIR) filter of order N, shown in Figure 6, consists of several delay blocks, each of which delays its input signal by one sample. The delayed samples are individually scaled by the coefficients b i and summed to produce the output signal y[n]. The characteristics and performance of the FIR filter at a certain sampling frequency is completely determined by the coefficients b i and the filter order N. Note that a filter of order N is represented by N + 1 filter coefficents. The structure from Figure 6 corresponds to the difference equation y[n] = b 0 x[n] + b 1 x[n 1] + b 2 x[n 2] b N x[n N] N = b i x[n i], (7) i=0 written in z-transform notation as Y (z) = b 0 X(z) + b 1 X(z) + b 2 X(z)z b N X(z)z N. (8) The transfer function H(z) of the filter is H(z) = Y (z) X(z) = b 0 + b 1 + b 2 z b N z N. (9) 5 3rd November 2003

6 PSfrag replacements x[n 1] x[n 2] x[n N] x[n] b 0 b 1 b 2 b N Σ y[n] Figure 6: FIR filter structure. There exist many methods for calculating the filter coefficients b i and the filter order N from specified filter criterions. For low filter orders, hand calculations may be used. However, as the filter order increases, hand calculations become impractical. The most common (and practical) way to design filters is by using computer-based tools, such as MATLAB. In MATLAB, filter coefficents are represented by arrays (vectors) of numbers. The FIR filter of order N = 3 with coefficients b 0 = 0.5 b 1 = 0.2 b 2 = 0.1 b 3 = 0.2 is represented in MATLAB by the vector b = [ ] The MATLAB command b = fir1(n,omega c) generates an N+1 long vector b with coefficents from a low-pass FIR filter with cut-off frequency Ω c =Omega c. The shape of the achieved frequency response will be greatly affected by the filter order N. The higher the order, the closer the response will be to an ideal one. However, a higher-order filter is more difficult to implement than a low-order filter due to higher computational complexity. Assignment 1 - Implementing a FIR Filter Use the fir1-command in MATLAB to construct a FIR low-pass filter with a cut-off frequency f c = 2 khz at a sampling rate of f s = khz (for the fir1 command, f c = 2 3rd November

7 khz means that the magnitude response at 2 khz is -6 db). For frequencies above f stop = 4 khz, the magnitude response must be less than -40 db. Use as low filter order as possible. Generate the Bode plot (magnitude and phase versus frequency) for the filter. Plot the impulse response of the filter. Hints: Nice Bode plots can be generated with the commands freqz and freqzplot. For example, [hplot,fplot]=freqz(b,1,1024, whole,11025); figure; freqzplot(hplot,fplot, Hz ); will generate the Bode plot for the FIR filter with coefficient vector b at the sampling frequency f s = khz. To plot the impulse response of a FIR filter vector b, use figure; stem(b); Use title to add a suitable title to your plots; and xlabel and ylabel to add labels to the x and y axis respectively. IIR Low-Pass Filter Design Figure 7 shows the structure of a general N th order digital infinite impulse response (IIR) filter. It resembles that of a FIR filter, only here, delayed and scaled previous output samples are also added to produce y[n]. The structure in Figure 7 corresponds to the difference equation y[n] = b 0 x[n] + b 1 x[n 1] + b 2 x[n 2] b N x[n N] = The z-transform of (10) is a 1 y[n 1] a 2 y[n 2]... a N y[n N] N N b i x[n i] a k y[n k]. (10) i=0 k=1 Y (z) + a 1 Y (z) + a 2 Y (z)z a N Y (z)z N = b 1 X(z) + b 2 X(z)z b N X(z)z N, (11) which then gives the transfer function H(z) for the general IIR filter as H(z) = Y (z) X(z) = b 0 + b b N z N B(z) 1 + a 1 = a N z N A(z). (12) 7 3rd November 2003

8 x[n 1] x[n 2] x[n N] x[n] PSfrag replacements b 0 b 1 b 2 b N Σ y[n] a N a 2 a 1 y[n N] y[n 2] y[n 1] Figure 7: IIR filter structure. The characteristics of the IIR filter is determined by the filter order N and the b i and a k coefficients. A higher-order filter generally has a better performance, but is more difficult to implement. The complex roots of the polynomial B(z) are called the zeros of H(z), and the complex roots of A(z) are called the poles of H(z). When designing IIR filters, a pole-zero plot is useful for determining the performance of the filter. The pole-zero plot of an IIR low-pass filter of order N = 3 is shown in Figure 8. A zero is denoted with a and a pole with a. A Nth order filter has N zeros and N poles. Pole-zero plots can be generated in MATLAB using the command zplane. Since an IIR filter includes a feedback loop (y[n] depends on former values y[n k]), stability is not guaranteed. Stability is easily checked from the pole-zero plot by verifying that all poles lies within the unit circle. If any pole fall outside the unit circle, the filter will be unstable. In MATLAB, a Nth order IIR filter is represented by two N + 1 long vectors, containing the coefficients b i and a i for the polynomials B(z) and A(z) respectively. Note: A MATLAB vector a containing the coefficients to an A(z) polynomial has a leading 1, corresponding to the 1 in the denominator in (12) (e.g. a = [ ]). The MATLAB command [b,a] = butter(n,omega c) generates the vectors b and a with coefficents to a Nth order Butterworth IIR low-pass filter with a cut-off frequency at Ω c =Omega c. 3rd November

9 Imz Rez PSfrag replacements Figure 8: Pole-zero plot of a third order IIR low-pass filter. Assignment 2 - Implementing an IIR Filter Use the butter command in MATLAB to construct an IIR low-pass filter with a cutoff frequency f c = 2 khz, using a sampling frequency f s = khz. The magnitude response in the pass band (0 < f < f c ) must not be smaller than -3 db. For frequencies above f stop = 4 khz, the magnitude response must be less than -40 db. Use as low filter order as possible. Generate the Bode plot for the filter. Generate the pole-zero plot for the filter. Hints: Use the buttord command to determine the minimum filter order N that will meet the specifications. Bode plots for an IIR filter can be generated with the commands freqz and freqzplot. For example: [hplot,fplot]=freqz(b,a,1024, whole,11025); figure; freqzplot(hplot,fplot, Hz ); will generate the Bode plot for the IIR filter with coefficient vectors b and a at the sampling frequency f s = khz. 9 3rd November 2003

10 IIR Filter Coefficient Quantization Filter coefficients generated in MATLAB are by default represented by double precision 64 bit floating point numbers. However, most digital signal processors will only handle fixed point numbers (a fixed number of decimals), typically using 16 bits to represent numbers. Therefore, the coefficients generated by MATLAB must be quantized to the correct number format before they can be used to implement filters in a DSP. Quantization of filter coefficients will most often have a negative effect on the performance. In MATLAB, an approximated form of quantization to N bit, fixed point representation (disregarding possible scaling of the coefficients), can be done for a coefficient b by multiplying b with 2 N 1 and then rounding of to the nearest integer (this is not an optimal quantization, but useful for the purposes of this lab). If the new number is then divided by 2 N 1, the effects of the quantization can be studied by comparing the original number with the quantized one. Assignment 3 - Filter Coefficient Quantization Use the butter command in MATLAB to design a band-pass filter of order N = 8 with a pass band from f 1 = 550 Hz to f 2 = 800 Hz, using a sampling frequency of f s = khz (type help butter for syntax help). Generate the Bode and pole-zero plots for this filter. Quantize the filter coefficients in MATLAB to 16 bits using the method described above. Generate the Bode and pole-zero plots for the quantized filter. Compare the performance between this filter and the previous (non-quantized) one. Filtering of a.wav File Since digital filters are easily used on audio signals, MATLAB contains routines for handling.wav files. The wavread command is used to read a.wav file from the hard drive and store it as a vector of samples (sound vector) in MATLAB. This vector can then be filtered using the filter command. The wavplay command can be used to play sound vectors. Assignment 4 - Filtering of a.wav File 1. Download the quimby.wav file from the course webpage. 3rd November

11 2. Use the MATLAB wavread command to store the contents of the.wav file as a sound vector in MATLAB (use the syntax x=wavread( quimby.wav )). 3. Play the sound vector in MATLAB using wavplay. 4. Filter the sound vector using your IIR low-pass filter from Assignment 2 and store the result as a new sound vector. 5. Play the filtered sound vector. Compare with the original one. 6. Demonstrate the filtered sound to the lab instructor. 11 3rd November 2003