Exercise A: FFT, Spectral Leakage, Zero Padding
Using a numerical computing software program like MATLAB, Octave, etc., implement the
following tasks. For each part, you need to prepare and send the source code, the image (.png,
.bmp, .pdf, …) of the required time-domain or frequency-domain plots, and your comments on
the result of each part, if needed. Name the files according to the numbers of corresponding parts,
and send all of them as a .zip file to the address below:
xamiri@fi.muni.cz
Deadline is April 15th
.
1) Generate two separate 64 length buffers of the two sinusoids:
𝑥1(𝑡) = 100 sin(2𝜋101.56𝑡) 𝑥2(𝑡) = 2 sin(2𝜋156.25𝑡)
Use a sampling frequency of 1 kHz.
2) Evaluate the FFT of each of these signals using a rectangular window, and determine which
frequency bins have the highest peaks.
3) Repeat the FFT evaluations done in part 2, but using a 1024 point FFT. (Applying a 1024 point
FFT to a 64 length data buffer has the effect of appending (1024 − 64) zeros to the end of the
data, which increases the number of points at which the spectrum is evaluated.)
4) The Hamming window is defined as:
𝑤(𝑛) = 0.54 − 0.46𝑐𝑜𝑠 (
2𝜋𝑛
𝑁
) 𝑓𝑜𝑟 𝑛 = 0,1,2, … , 𝑁 − 1
Generate and plot (stem) the Hamming window.
5) Now repeat parts 2 and 3, but applying first a Hamming window to the 64 length data buffers
before evaluating the FFTs. Observe the outputs of the FFTs.
6) Now add these two sine waves together, and apply both 64-length and 1024-length FFTs, with
both rectangular and Hamming windows. Observe how “spectral leakage” can mask a small
signal in a large one.
7) Now add Gaussian noise to the combined signals, with a variance of 10. Compare the results
of evaluating the 1024 point FFT on a 64-length data buffer (i.e. with zero padding), with a 1024
point FFT applied to a 1024-length data buffer (i.e. generate more data).
Hint: in MATLAB, to generate the noise, try: noise = sqrt (variance) * randn();