University of Saskatchewan Electrical & Computer Engineering CME 462 Digitall Signal Processing Laboratory

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1 University of Saskatchewan Electrical & Computer Engineering CME 462 Digitall Signal Processing Laboratory FIR/IIR Filters and Real Time FFT using TMS-320C6711 DSK (Digital Signal Processor Starter Kit) January 2008 Objective: This laboratory is to design digital filters, one FIR and one IIR type by using MATLAB digital filter design tool, then implement those on Texas Instrument Digital Signal Processor TMS320C6711 DSK (Starter Kit) to run in real time. This laboratory also studies the time-frequency relationship of simple waveforms; sine, square and triangular wave by the Fast Fourier Transform (FFT) implemented in real time on the TMS320C6711 processor. Reference: 1. TMS320C6711 DSK help, on line document included in DSK. 2. Getting Started With the Code Composer Studio Tutorial, on line tutorials included in Code Composer Studio. 1 Experiments 1. Design a 89 point band-pass FIR filter, which has a pass-band from 400 Hz to 1600 Hz. The digital filter operates at a sampling rate of 8 KHz. Calculate first the filter coefficients of this band-pass FIR digital filter by using MATLAB FIR filter design program, fir_window.m. Incorporate the Hamming window in the design. Use the provided sample program fir.c to realize this particular design. Measure the amplitude frequency response of the designed digital filter by applying sinusoidal wave of varying frequency to the analog input of the TMS-320C6711 DSK. Plot the frequency response with a spectrum analyzer. 2. Design a 5th order Butterworth low pass filter with a cut-off frequency of 1000Hz. Use the FIR program fir.c as reference, write a C-language program for this IIR type filter. Verify if the designed frequency response is achieved by your design and implementation. Use frequency prewarping if your design is done by the bilinear method. The MATLAB program iir_design.m calculates necessary filter coefficients to design this LP filter. 3. Make and run a FFT program FFT256.c written for TMS-320C6711 DSK as provided. This program take 256 samples from analog input (32 ms long at 8 KHz sampling rate) and immediately calculates its discrete Fourier Transform within a much shorter time than one sampling interval of 125µs. The calculated magnitude spectrum consisting of 256 points is output from the analog output. Thus, time trace and its magnitude spectrum can be observed simultaneously on an oscilloscope. Using a function generator as source, measure magnitude spectrum for three waveforms; sine, square and saw tooth wave at three different frequencies, 250 Hz, 500 Hz and 1000 Hz. Compare the results and theoretically known spectrum distributions. To help you understand the relationship between time scale and frequency scale, run MATLAB program DSPLab_FFT.m which simulates this experiment, concurrently while you are running FFT256.c. 1

2 2 Procedures Before hardware setup, down load the entire directory of DSPprojects from The directory DSPprojects contains three projects, FIR, IIR and fft to your own directory in H: drive. Make sure to keep the directory structure unchanged. Then, get necessary hardware and equipment. This laboratory requires a PC with network connection, TMS- 320C6711 DSK (mounted in aluminum case), function generator, oscilloscope and spectrum analyzer. Once all the equipment is ready, connect TMS-320C6711 DSK to the parallel port of PC and power the TMS-320C6711 DSK. Then, power your PC and log on. Start CCStudio from Texas Instruments > Code Composer Studio DSK Tools 2 > CCStudio, in an appropriate directory where this software is installed. CCStudio checks connection to TMS-320C6711 DSK and establish handshaking between TMS-320C6711 DSK and the PC. If connection fails, close CCStudio and check parallel port connection, then turn power for TMS-320C6711 DSK, OFF then ON. After TMS-320C6711 DSK is reset, restart CCStudio. In an event of broken connection or hang-up during the excution of a program, just reset TMS- 320C6711 DSK and restart CCStudio. Normally, no need for rebooting your PC. Do FIR filter first to get yourself familialized with basic functions of CCStudio. Run the provided programs in FIR project without modifications to make sure that the FIR filter (a band-pass filter to pass from 1000 Hz to 2000 Hz) works. Then, you need to modify only a filter coefficient file called bs2700.cof according to the filter design procedure described later. For IIR filter, you need to modify iir.c and bs2700.cof in IIR project. Procedures of this experiment is essentially the same as FIR filter. FFT project does not require programming. Measurements and interpretation of results are its objective. A MATLAB program is included for each of the three projects FIR, IIR and FFT to assist either your design or analysis of the results. 2.1 FIR filter: Making an excutable program on Code Composer Studio 1. Make sure that you have following files in the directory H:\DSPprojects\FIR. source include library etc. fir.c bs2700.cof C6xdsk.cmd c6xdskinit.c c6x.h rts6701.lib vectors_11.asm C6xdsk.h c6xdskinit.h c6xinterrupts.h 2. Connect the parallel port cable to TMS320C6711 DSK and to PC. Insert the power supply jack to TMS320C6711 DSK and power DSK before PC power is turned on. 3. Click on CCS-DSK ( C6000) to start Code Composer Studio. To quickly check if PC/DSK handshaking is properly established, go to the menu bar and click on GEL > Check DSK > QuickTest. If a message *****Target is Okay ******* shows up in the bottom window, PC/DSK connection is working all right. 4. Now we start a new project to build a FIR filter program from a set of provided program modules. We let the name of the new project be FIR which will create a project file FIR.pjt. Go to the menu bar and click on Project > New. Enter FIR to project name and H:\DSPprojects\ to location. Since there exists FIR project, you overwrite the existing FIR.pjt to start from scratch. 5. The next step is to add program files, header files, a command file and a library file to the project. There are three program files, fir.c, c6xdskinit.c, vectors_11.asm. Go to the menu bar and click on Project > Add Files to Project. Then, add those three files one by one from the directory H:\DSPprojects\FIR. 2

3 Figure 1: Code Composer Studio 6. Once all of the three files are added, you can see those under Source in File View window on your left. In order to bring header files with.h extension, go to the menu bar and click on Project > Scan All Dependencies. you will now see all of the necessary header files under Include in the File View window. 7. You have two more files to add to the project. Add C6xdsk.cmd. Then, add the run-time support library file rts6701.lib located in H:\DSPprojects\FIR. 8. If all of the necessary files are added to the project, the File View window should look like Fig. 1. Then, we are ready to compile and link all the program modules. Press the square icon with three downward arrows, Rebuild All. The Build windows will show up and the message Build Complete, 0 Errors, 0 Warnings, 0 Remarks should appear if all the previous steps are correct. This step has produced an excutable object file FIR.out in the directory H:\DSPprojects\FIR\Debug. 9. To load FIR.out to TMS320C6711 DSK and run it, go to the menu bar and click on File > Load Program... Select H:\DSPprojects\FIR\Debug\FIR.out and press Open. You will see a loading bar moving. The program is now loaded in DSK. You can start by pressing the icon of a running person and the similar icon with x symbol to stop. 2.2 FIR Filter: Understanding how program FIR.c works In spite of many program modules are used in the FIR project, only two programs written in C language describe the operation of the 89 point FIR digital filtering. The main program FIR.c with a include file BS2700.cof that holds filter coefficients plays a key role in this FIR filtering. FIR.c is a interrupt driven 3

4 program. Clock signal running at 8 KHz causes an interrupt every ms. When the interrupt occurs, excuting of a program is interrupted and forced to jump to c_int11() in FIR.c. Then, (a) one data sample is immediately input from A/D. (b) An output value from the FIR filter is calculated according to Eq. (1). (c) The calculated output value is output from D/A. These three steps (a), (b) and (c) are excuted and completed by the TMS320C6711 running at 150MHz or 900 MFLOPS very rapidly within a few µs. Then, it returns to the main routine main{} and idles there. The funcion of main{} is to initialize TMS320C6711 DSK and enable interrupt, and to provide an infinite loop to idle. The include file BS2700.cof has 89 FIR filter coefficients. To change the characteristics of the filter, you need to calculate the coefficients of your filter and replace them with newly caluculated coefficients. Program Listings: 1. //Fir.c FIR filter. Include coefficient file with length N #include "bs2700.cof" int yn = 0; short dly[n]; interrupt void c_int11() { short i; //coefficient file 2700Hz //initialize filter s output //delay samples //ISR } dly[0] = input_sample(); yn = 0; for (i = 0; i< N; i++) yn += (h[i] * dly[i]); for (i = N-1; i > 0; i--) dly[i] = dly[i-1]; output_sample(yn >> 15); return; //new beginning of buffer //initialize filter s output //y(n) += h(i)* x(n-i) end of buffer //update delays with data move //scale output filter void main() { comm_intr(); while(1); } //init DSK, codec, McBSP //infinite loop 2. //BS2700.cof FIR bandpass coefficients designed with MATLAB #define N 89 //number of coefficients short h[n]={-0, -34, -21, 7, -0, -9, 33, 66, -0, -87, -59, 20, -0, -26, 101, 196, -0, -250, -165, 54, -0, -67, 256, 486, -0, -598, -388, 126, -0, -155, 588, 1117, -0, -1401, -926, 308, -0, -412, 1666, 3458, -0, -5873, -5191, 3051, 8192, 3051, -5191, -5873, -0, 3458, 1666, -412, -0, 308, -926, -1401, -0, 1117, 588, -155, -0, 126, -388, -598, -0, 486, 256, -67, -0, 54, -165, -250, -0, 196, 101, -26, -0, 20, -59, -87, -0, 66, 33, -9, -0, 7, -21, -34, -0}; 2.3 FIR Filter: Measurements 1. You need a function generator and an oscilloscope to verify the operation of the FIR program. Connect a function generator to IN (Analog Input). Input voltage should be set 0.5 volt. The frequency range should be between 0-4 KHz (since the sampling rate is 8 KHz). Connect one channel of the oscilloscope to IN and the other channel to OUTof the DSK. Use a stereo phone jack for INand OUTterminals. The provided FIR filter is a band-pass filter with the pass band from 1000 Hz to 2000 Hz. Measure the cut-off frequencies where the gain drops down to -3 db (or of the pass band gain). Confirm that measured cut-off frequecies are close to 1000 Hz and 2000 Hz. 2. You can display the frequency response directly on the HP-3580A Spectum Analyzer. Feed the (frequency) sweep signal available from a BNC connnector located at the back of HP-3580A into 4

5 INand connect OUTto the input of HP-3580A. Hardcopy of the frequency response may be printed through the GPIB port. 2.4 FIR Filter: Filter Design by MATLAB program fir_window.m A MATLAB program fir_window.m is provided in H:\DSPprojects\FIR. Start MATLAB then connect to this directory. Then, enter fir_window to start this program. This program calculates FIR filter coefficients according to the window method. Enter your selection or value for each prompted question. Frequencies must be in the normalized frequency by the sampling rate (8 KHz in our case). Sample input dialogue is shown below. Once you get the filter sequence consisting of 89 coefficients as shown, you should cut and paste those values into your coefficient file BS2700.cof immediately. Remember, you are to design a band pass filter with pass-band from 400 Hz to 1600 Hz. >> fir_window What filter do you design, lp, hp, bp or bs? = bp Number of points in FIR filter = 89 Lower cut-off frequency f1 (0<f1<0.5) = 0.05 Upper cut-off frequency f2 (0<f1<0.5) = 0.2 Which window do you use? none, bartlett, hamming, hanning, blackman, kaiser : hamming ans = -36, -28, 0, 15, -0, -19, 0, 54, 84, 51, -0, 15, 91, 112, -0, -145, -153, -32, -0, -185, -392, -321, 0, 183, -0, -225, 0, 601, 907, 529, -0, 145, 860, 1034, -0, -1326, -1421, -311, -0, -2026, -4868, -4805, 0, 6689, 9830, 6689, 0, -4805, -4868, -2026, -0, -311, -1421, -1326, -0, 1034, 860, 145, -0, 529, 907, 601, 0, -225, -0, 183, 0, -321, -392, -185, -0, -32, -153, -145, -0, 112, 91, 15, -0, 51, 84, 54, 0, -19, -0, 15, 0, -28, -36, 2.5 IIR Filter: Filter Design by MATLAB program iir_design.m The second experiment is IIR filter. The algorithm of IIR filtering is given by Eq. (2). In contrast with FIR filters, IIR filter requires two sets of coefficients, one representing the numerator of a transfer function a(k) and the other representing the denominator b(k). Note that a(k) corresponds to h(k) in FIR filters, but (k) is missing in FIR filters. All the files in H:\DSPprojects\IIR are the duplicates from H:\DSPprojects\FIR. The file FIR.c was, however, renamed as IIR.c. There are two things to do to complete the IIR filter experiment. First, you must modify the renamed file IIR.c to add the second term of Eq. (2) related to b(k). Second, the filter coefficients a(k) and b(k) must be calculated by using the MATLAB program iir_design.m. You can copy the calculated a(k) and b(k) into BS2700.cof as FIR.c, or include both of the coefficients a(k) and b(k) directly in IIR.c without using #include "bs2700.cof" statement. To run iir_design.m, change working directory to in H:\DSPprojects\IIR and enter iir_design. Dialogues running this program are shown below. It is advised to use the bilinear transformation to convert an analog prototype filter to a digital filter as you can get the same number of coefficients for a(k) and b(k). When the order is set as N, the number of the coefficients is N + 1 including a constant term. For low pass digital filters of an unity gain, the sum of all a(k) should be equal to the sum of all b(k), since z = 1 at DC (frequency=0). >> iir_design Enter order of prototype filter =>5 Enter one of butt/bess/cheb1/cheb2/elli =>butt Enter freq-band conversion to lp/hp/bp/bs/none =>lp Enter new cutoff freq in Hz =>1000 Enter bilinear/impulse invariant (bil/imp) =>bil Enter sampling frequency in Hz =>8000 numerator coefficients ans = 27, 134, 268, 268, 134, 27, 5

6 denominator coefficient ans = 10000, , 30272, , 6112, -825, When a new IIR.c is completed, rebuild the excutable file IIR.out and run the program. Follow the same procedure as given in FIR filter to verify that your low pass filter is working properly. 2.6 Real Time Spectrum Analysis by FFT: Running FFT project FFT.out The FFT project to perform real time spectrum analysis on TMS-320C6711 DSK is found in H:\DSPprojects\FFT. Necessary program modules to build the excutable program are listed in the following table. The executable program FFT.out has been built and is ready to run. For this project, go to Project > Open then open FFT.pjt. Click on File > Load Program... to select Select H:\DSPprojects\FFT\Debug\FFT.out then press Open. source include library etc. FFT.c c6x.h C6xdsk.cmd FFT256c.c c6xdsk.h rts6701.lib c6xdskinit.c vectors_11.asm C6xdsk.h c6xdskinit.h c6xinterrupts.h To run this FFT project successfully, precise setting of input voltage and frequency is important. Keep input peak voltage exactly at 0.5 V. When input voltage comes close to 1 V and exceeds A/D converter s input range, input values do not properly represent the input voltage. Also, set frequency to be 500 Hz first. We change this to 250 Hz and 1000 Hz later. You may vary frequency from 200 Hz to 2000 Hz to observe the fundamental frequency to slide along the frequency axis. Try sine wave first because it is simple. Then, try square wave and saw tooth wave. You need your oscilloscope to observe two channels, one for input and the other for output. Set trigger on the output. FFT256c.c sets x1[0] = to provide a trigger to display magnitude spectrum easily on the scope. The output voltage from TMS320C6711 DSK is inverted, so display the output in inverted mode if available. After you stabilize the traces on the scope by properly adjusting the trigger level on the output signal, you can observe time trace on the input channel and frequency trace on the output channel. 2.7 Real Time Spectrum Analysis by FFT: MATLAB simulation by DSPLab_FFT.m The 256 point FFT program FFT.out to run on TMS-320C6711 DSK samples 256 input data, then immediately apply 256 point FFT. Beause of the speed of TMS-320C6711 DSK, 900 MFLOPS (Mega floating point operations per second), it takes only several micro-seconds to complete 256 point FFT. So, as soon as 256 input data are accumulated, immediately its magnitude spectrum is calculated and ready for output. Therefore, as one sample of the next 256 input data is acquired, one point of the calculated 256 point magnitude spectrum is output synchronously. That is why time data and frequency spectrum can be displayed simultaneously. Though the program FFT.out works well as long as the input voltage is kept within the range of ±0.5 V and frequency range is between 200 Hz and 2000 Hz, this program does not provide the frequency scale. Time scale can be easily calibrated on the scope. MATLAB program DSPLab_FFT.m simulate exactly what FFT.out does. DSPLab_FFT.m displays magnitude spectrum properly scaled to the frequency in Hz by taking the sampling rate of 8 KHz and the data size of 256 points. One cycle of the magnitude spectrum display is 8 KHz whereas its time span is 32 ms. Run DSPLab_FFT.m concurrently with FFT.out so that the expected magnitude spectrum is readily known for the frequency and the waveform of your input. An example of dialogue with this program is shown as follows. The expected spectra for sine, square and saw tooth wave are also shown in Fig. 2. 6

7 >> DSPLab_FFT Magnitude must be set to be 0.25 V or 0.5 Vp-p Sampling frequency is internally set at 8000 samples/sec. Specify Input waveform, sine, square, triangle or random >square Specify frequency in KHz 0<f<4KHz >0.5 Specify time delay in number of sampling intervals, 0=<n<256 >0 Figure 2: Comparision of Time-Frequency Display of Waveforms 3 Mathematical Background 3.1 Digital Filters An N point FIR digital filter is given by the following transfer function in the z-transform. N 1 H(z) = h(k)z k. For a given input sequence X (z), the output from the filter Y (z) is N 1 Y (z) = H(z)X (z) = h(k)z k X (z). 7

8 Taking the inverse z-transform, we have its time domain equivalence: y(n) = N 1 h(k)x(n k) (1) = h(0)x(n) + h(1)x(n 1) + h(2)x(n 2) + + h(n 2)x(n N + 2) + h(n 1)x(n N + 1) This equation of convolution between the filter sequence h(k) and the past N input data x(k) for 0 k < N is used in the C-language program FIR.c. In this program, both of the filter sequence h(k) and the input x(k) are defined as short integer (16 bits), and y as integer (32 bits). Therefore, the filter sequence calculated by MATLAB program must be scaled for the short integer by multiplying them by or = The filter coefficients generated by MATLAB program by the window method or frequency sampling method, are generally scaled to make pass pand gain to be unity and they are fractional numbers. For IIR filters, the FIR filter program must be modified to included poles as well as zeros (already included in FIR). The general form of the Nth order IIR filter transfer function H(z) is given by H(z) = 1 + N a k z k. N b k z k The input-output equation, Y (z) = H(z)X (z), is expressed in a equivalent difference equation, y(n) = k=1 N N a(k)x(n k) b(k)y(n k). (2) In the case of IIR filters, N means the order of the filter rather than the length of the filter. Since Nth order polynomial has the highest order for a constant, and the term z N for the most negative power of N, the total number of terms involved is N + 1. The Nth order IIR filter has N + 1 points (0 through N) for both numerator and denominator. Also, the coefficients a(k) and b(k) should be scaled by a single scale factor to implement the filter in short integers. k=1 3.2 Fourier Series of Simple waveforms In this laboratory, We are dealing with the Discrete Fourier Transform (DFT) applied for periodic waveforms looked through a short time window of 256 samples. What you observe as magnitude spectrum for the three kinds of waveforms, sine wave, square waveform and saw tooth waveform are better interpreted by the Fourier series. When you switch from one waveform to another, you will immediately notice the difference in higher harmonics. The Fourier series of these three wave forms are given here to recall that (1) sine wave has only one spectrum at the frequency of the sine wave. (2) square wave has only odd harmonics and their magnitude decreases inversely proportional to the harminic number (odd only), (3) saw tooth wave has odd and even harmonics and their magnitude decreases inversely proportional to the harmonic number. Keep in mind that the fundamental frequency varies when you change the frequency of the input wave and their hamonics appear at the mulpiple of the fundamental frequency. So, the Fourier series actually expands or contrants as you increase or decrease the frequencey. The Fourier series of those three waveforms that occupy 2π to fit one cycle are shown below. 1. Sine Wave f (t) = sin kt for π < t < +π b n = δ(n k) 8

9 2. Square Wave f (t) = { 1 for 0 < t < π 1 for pi < t < 0 b 2n+1 = 4 π(2n + 1) for n 0 3. Saw Tooth Wave f (t) = t π for π < t < +π b n = 2 πn ( 1)n 1 for n 1 The E N D 9