<< Chapter < Page Chapter >> Page >
You will investigate the effects of windowing and zero-padding on the Discrete Fourier Transform of a signal, as well as the effects of data-set quantities and weighting windows used in Power Spectral Density estimation.

Matlab exercise, part 1

Since the DFT is a sampled version of the spectrum of a digital signal, it has certain sampling effects. To explorethese sampling effects more thoroughly, we consider the effect of multiplying the time signal by different window functionsand the effect of using zero-padding to increase the length (and thus the number of sample points) of the DFT. Using thefollowing MATLAB script as an example, plot the squared-magnitude response of the following test cases overthe digital frequencies c 8 3 8 .

  • rectangular window with no zero-padding
  • hamming window with no zero-padding
  • rectangular window with zero-padding by factor of four ( i.e. , 1024-point FFT)
  • hamming window window with zero-padding by factor of four

Window sequences can be generated in MATLAB by using the boxcar and hamming functions.

1 N = 256; % length of test signals 2 num_freqs = 100; % number of frequencies to test 3 4 % Generate vector of frequencies to test 5 6 omega = pi/8 + [0:num_freqs-1]'/num_freqs*pi/4; 7 8 S = zeros(N,num_freqs); % matrix to hold FFT results 9 10 11 for i=1:length(omega) % loop through freq. vector 12 s = sin(omega(i)*[0:N-1]'); % generate test sine wave 13 win = boxcar(N); % use rectangular window 14 s = s.*win; % multiply input by window 15 S(:,i) = (abs(fft(s))).^2; % generate magnitude of FFT 16 % and store as a column of S 17 end 18 19 clf; 20 plot(S); % plot all spectra on same graph 21

Make sure you understand what every line in the script does. What signals are plotted?

You should be able to describe the tradeoff between mainlobe width and sidelobe behavior for the various window functions.Does zero-padding increase frequency resolution? Are we getting something for free? What is the relationship betweenthe DFT, X k , and the DTFT, X , of a sequence x n ?

Matlab exercise, part 2

Download and run the MATLAB file lab4b.m to observe direct and autocorrelation-based PSD estimates. A pseudo noise generator isfiltered with a fourth-order IIR filter and various PSD estimates are computed and plotted.

First plot

The first plot contains PSD estimates, using a 1024-point FFT,from the first 512 samples of the 1024-sample sequence. The direct method is to take the squared-magnitude of the FFT of the sequence. Theautocorrelation (AC) method is to take the magnitude of the FFT of the autocorrelation of the sequence. In this case rectangular windows wereused in both FFTs. Why do the estimates look exactly the same? Will theestimates be alike if all 1024 samples are used with a 1024-sample FFT, with all other conditions being equal? Why or why not?

Second plot

The second plot contains PSD estimates, using a 1024-point FFT, from the first 32 samples of the 1024-sample sequence. The directand AC estimates are computed in the same manner described above, except a hamming window has been applied to the sequence in thedirect-PSD estimate. Why are these estimates different? What will make these estimates identical?

Third plot

The third plot contains PSD estimates, using a 1024-point FFT, from all 32-sample blocks of the 1024-sample sequence. The direct-PSDestimate is computed by summing the hamming-windowed PSD estimates of each 32-sample block. The AC-PSD estimate is computed by taking themagnitude of the FFT of 63 samples of the autocorrelation of the entire 1024-point sequence. Why are 63 samples used in comparing theAC method to the direct method?

Fourth plot

The fourth plot contains the unfiltered spectrum of the PN sequence and the impulse response of the coloring filter.

Try various the block lengths ( Np ) and direct-PSD window types. Observe the changes. Are there anytradeoffs? Does either the direct method or the autocorrelation method have an advantage over the other? Is there a direct-method window thatresults in identical block-based estimates? For simplicity, the autocorrelation window in this lab isrectangular. Can we, however, consider this an unbiased autocorrelation estimate that is windowed by a rectangular window? (Hint: see theMATLAB documentation for the xcorr function.) If not, how would you create an unbiased autocorrelation estimate beforewindowing, and what is the effective window that we have applied to the unbiased autocorrelation?

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Digital signal processing laboratory (ece 420 55x). OpenStax CNX. Jan 18, 2010 Download for free at http://cnx.org/content/col10397/1.10
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing laboratory (ece 420 55x)' conversation and receive update notifications?

Ask