Experiment 1

Description: In this experiment, you will generate a 1 MHz square-wave, and extract its first harmonic using a 7-th order Butterworth filter with cut-off frequency of 2 MHz. You will also extract its third harmonic using a bandpass Butterworth filter centered around 3 MHz.

Instructions

• I. Initialization: Copy the initialization files for the experiment into your Matlab working directory. It would be a good idea to browse through the help file before starting

  The following constants should be set by the startup file. Use who in the matlab workspace to check if the constants are all there.

  • FREQ = 2*pi*1e6 ----------------signal frequency in rad/sec
  • T_PERIOD = 1e-6 ---------------Time period of square wave
  • T_SAMPLE = 1e-6/32 ------------Sampling period used in Simulink
  • FFT_NUM_PTS = 4096 ------------Number of points used for FFT
  • T_SIM = FFT_NUM_PTS*T_SAMPLE --Time of simulation
  • NUM_SAMPLES = T_SIM/T_SAMPLE --Total number of samples
  • F_CUTOFF = 4*pi*1e6 ------------Cutoff frequency of lowpass Butterworth filter
  • F_BAND_LOW = 2.7*2pi*1e6 -------Lower cutoff of bandpass Butterworth filter
  • F_BAND_HIGH = 3.3*2pi*1e6 ------Higher cutoff of bandpass Butterworth filter
• II. Testing (This part is a test to make sure things are set up properly.)
  1. From Simulink/Sources, select a Sine wave generator and copy it to your model window. Set the frequency to FREQ.
  2. From ece359lib copy a Fourier transform block and link it to the output of the Sine wave block.
  3. Click on the menu item Simulation in your model window and select Parameters. Within the Solver sub-menu, set Stop time to T_SIM and Max step size to T_SAMPLE.
  4. Attach a Scope from Simulink/Sinks to the signal and zoom in to observe the signal shape.
  5. Start the simulation. You should see a window with the Fourier transform of the sine wave. As expected a spike is seen at FREQ=2*pi*1e06 rad/sec
  6. If you are successful, move on to the next step. Fourier transform computation may take a long time (1 min on a 200MHz Pentium cpu). If your computer is slower, or has less memory, try reducing FFT_NUM_PTS by a factor of two.
• III. Constructing a Butterworth filter
  1. Use help butter at the Matlab prompt to get help on Butterworth filter design. We will be designing a continuous time low-pass filter.
  2. Construct a Butterworth filter with cutoff frequency at F_CUTOFF=4*pi*10^6 rad/sec and order 7. Store the coefficients of the numerator polynomial in B and denominator in A, i.e. use
     [B,A] = butter(7,F_CUTOFF,'s')
  3. Use bode(B,A) to look at the transfer function of the Butterworth filter.
  4. From the Simulink/Linear library copy a Transfer Fcn block into a new model window, and set Numerator to B and Denominator to A. Note: Simulink will get the values of the vectors A and B from the Matlab workspace, assuming you have done step 2.
• IV. Lowpass filtering the first harmonic
  1. From the Simulink/Sources library, get a Signal Generator block. Under waveform select square. Set the amplitude to 1 and frequency to FREQ. Make sure that the frequency is in rad/sec
  2. Pass the square wave output through the filter.
  3. Attach a Fourier transform block at both the input and the output of the filter. Set the first three parameters in the block to FFT_NUM_PTS and set the Sample time to T_SAMPLE. Do not use C-click to create multiple copies of the Fourier transform block. Instead, copy the block from ece359lib each time you wish to use it.
  4. Start the simulation, two windows with the input and the output Fourier transforms should appear.
  5. Use Scopes from simulink/sinks to look at the signals at the input and the output in time domain. Zoom in sufficiently to observe the waveforms.
• V. Bandpass filtering the third harmonic
  1. Design a bandpass Butterworth filter with lower and higher frequency limits F_BAND_LOW and F_BAND_HIGH respectively. Use order 7. (Again do help butter.m at the Matlab prompt for help).
  2. Repeat part IV with this filter. You only need to update A and B in the Matlab workspace to do this. Other blocks in the experiment can remain unchanged.

Questions

1. Submit a printout of the block diagram in part IV.
2. Submit the Fourier transforms plots at the input and output of the Butterworth filter for part IV and at the output only for part V.
3. Does the lowpass filter achieve the goal of extracting the first harmonic of the square wave? Explain.
4. Does the bandpass filter achieve the goal of extracting the third harmonic of the square wave? Explain.
5. Write an expression for the Fourier transform of a square wave with period T_PERIOD. (We did this in class; you don't have to rederive the expression).
6. Write an expression for the Fourier transform of a square wave with period T_PERIOD, truncated to the interval [0,T_SIM]. (Agian, we did this in class; you don't have to rederive the expression).
7. Use the above expressions, to predict the height of the spikes observed in the Fourier transform of the square wave input. Do they match what you observe?
8. Use the expression for the frequency response of a 7-th order lowpass Butterworth filter to predict the height of the spike at the output of the lowpass filter in part IV. Does this match what you observe? Calculate the attenuation produced by the lowpass filter at the frequency of the third harmonic.