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Time Domain and Frequency Domain Signal Representation

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Time Domain and Frequency Domain Signal Representation ES442-Lab 1 ES440. Lab 1 Time Domain and Frequency Domain Signal Representation I. Objective 1. Get familiar with the basic lab equipment: signal generator, oscilloscope, spectrum analyzer, power supply. 2. Learn to observe the time domain signal representation with oscilloscope. 3. Learn to observe the frequency domain signal representation with oscilloscope. 4. Learn to observe signal spectrum with spectrum analyzer. 5. Understand the time domain representation of electrical signals. 6. Understand the frequency domain representation of electrical signals. II. Pre-lab 1) Let S1(t) = sin(2pf1t) and S1(t) = sin(2pf1t). Find the Fourier transform of S1(t) + S2(t) and S1(t) * S2(t). 2) Let S3(t) = rect (t/T), being a train of square waveform. What is the Fourier transform of S3(t)? 3) Assume S4(t) is a train of square waveform from +4V to -4V with period of 1 msec and 50 percent duty cycle, zero offset. Answer the following questions: a) What is the time representation of this signal? b) Express the Fourier series representation for this signal for all its harmonics (assume the signal is odd symmetric). c) Write the Fourier series expression for the first five harmonics. Does this signal have any DC signal? d) Determine the peak magnitudes and frequencies of the first five odd harmonics. e) Draw the frequency spectrum for the first five harmonics and clearly show the frequency and amplitude (in Volts) for each line spectra. 4) Given the above signal, assume it passes through a bandlimited twisted cable with maximum bandwidth of 10KHz. a) Using Fourier series, show the time-domain representation of the signal as it passes through the cable. How many components does it have? b) Ignoring any other distortion or attenuation, show the signal for one period t=[0 1000] microsecond. You must find v(t) for t=0, 62.6, 125, 250, 375, 437.5, 500, …. t (microsecond) V(t) Volt-Peak 0 62.5 III. Equipment: - MSO-X Scope with two probes, DEMO port working, and a signal generator option. - Memory stick to record your results - Coax cable with clips Spring 2014 Page 1 ES442-Lab 1 III. Procedures A. Time Domain Signal Representation In this section, we are going to examine the time domain properties of various signals with the oscilloscope. 1. Use function generator on the scope to generate a sinusoidal signal with Vpp =4V and frequency 10 KHz. Let’s denote this signal S1 (t) . Connect the signal to Ch1 of oscilloscope. Use the scope probe and an alligator/coax cable. Make sure the probe setting is 10:1. 2. Use DEMO pin on the scope to generate a square signal. We call this S2(t). What is the Vpp and frequency of this signal? 3. Adjust the oscilloscope until you get a stable display. Capture the display. 4. Take a snapshot of both S1 an S2 signals from the scope. Use your USB memory to record the signal. 5. Display the waveform of S1(t) – S2(t) by using the Math function of the oscilloscope. Capture the waveform (Ch1 and Ch2 should be turned off, and only capture S1(t)-S2(t)). 6. Display the waveform of S1(t)+S2(t) by using the Math function of the oscilloscope. Capture the waveform. 7. Display the waveform of S1(t) * S2(t) by using the Math function of the oscilloscope. Capture the waveform. Notice the sudden phase change introduced by the square waveform. 8. Measure the RMS of S1(t) and S2(t). Calculate the RMS manually. Are these numbers the same? B. Frequency Domain Signal Representation With FFT In this section, we are going to examine the frequency domain properties of the signals with the help of the FFT (fast Fourier transform) function of the oscilloscope. 1. Disconnect the signal from DEMO port. Change the signal frequency from the signal generator to 100KHz and set it to a Sinusoidal signal. (a) Push the “Math” button of the oscilloscope. (b) Then push the “FFT” softkey. (c) Press “Settings” softkey. Choose “Source” to be channel 1. (d) Set the Offset to 0dBV. (e) Set the Scale to50dBV/division and time resolution is set to about 100usec. (f) Change the sample rate to 25.0MSample/sec by using the Horizontal resolution knob. It should be note that the sample rate should be at least twice of the highest frequency component in the signal. What is the highest frequency component in the signal? Is the condition satisfied? (g) Make sure the center frequency is 100KHz and Span is 200KHz. (h) When the frequency span is 200KHz, what is the span of 1 horizontal division? Spring 2014 Page 2 ES442-Lab 1 (i) Capture the display. Reading out the frequency component and record the results. (j) Adjust the frequency of the signal to 50 KHz, 100 KHz, 150 KHz, what do you observe? What is the mathematical representation of this spike? What is the amplitude of the spike? (k) Take a snapshot of the FFT for the signal when it is set to 100KHz. 2. Disconnect the signal from DEMO port. Change the signal frequency from the signal generator to 100KHz and set it to a Square signal with no offset. a. Make sure your generated signal is actually a 4V square waveform with no offset. b. Capture the FFT of the signal. How does it look like? Take a snapshot. Capture at least the first 5 off harmonics. C. Matlab fundamentals In this section, we are going to learn the fundamentals of the software package Matlab, and use it to plot several time domain signals. 1. Open Matlab. Crate a new file and copy the following script to the file. Read and understand each line of the script. % Matlab cannot represent continuous-time functions % (e.g. sin(t)). To solve this problem, we use discrete-time % variables with very small time interval to approximate % continuous-time function. The small time interval % is called time domain resolution. % the time domain resolution we are going to use is 0.01 second. t_res = 0.001; % create vector starting from 0 and ending at 2, % the distance between consecutive samples is t_res % the discrete-time vector is used to approximate % continuous time from -2 sec to 2 sec t = [0:t_res:2]; % the freqeuncy is 2 Hz f = 2; % the initial phase is 0 theta_0 = 0; % define the function y1 = 2*sin(2*pi*f*t+theta_0); Spring 2014 Page 3 ES442-Lab 1 % draw the function with t on the x-axis and y on the y- axis. plot(t, y); % save the graph to your hard drive as a jpeg figure print –djepg sin1.jpg 2. Plot a sinusoidal signal with frequency 10 Hz and amplitude 1 (assign it to y2). 3. Plot the summation of the above two sinusoidal signals (y1+y2). 4. Plot the product of the above two sinusoidal signals (y1.*y2). You should obtain something like this: ES440 Lab#1 4 .
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