J-DSP Lab 5:  The Fast Fourier Transform (FFT)

 

Lab 5 concentrates on the Fast Fourier Transform (FFT).

 

Problem 5-1:  FFT Properties

 

Consider the symmetries in the following signals. We want to see how these symmetries affect FFT spectra.


1.       Generate the given signals in J-DSP and plot the FFT of size N=8.

 

Note:  In the Sig Gen block dialog box, set the “signal” to Self-Defined and an [Edit Signal] button will appear. Click the [Edit Signal] button and enter the index along with the desired value of the signal at that index. Click the [update] button for the change to take effect.  The new value is then shown in the table.

2.       For which of these FFT plots is the real (imaginary) part zero?

 

 

 

 

 

 

 

 

 

Problem 5-2:  The Rectangular Window

 

In this exercise we want to see the effect of truncation on the FFT spectra.  We will subsequently try tapered windows as well. Generate a sine wave of “gain” 1, “pulse width” 128 samples, and “time shift” 0, with ”frequency” p/10 = 0.1 p.

 

Window (truncate) the sine wave for both cases below and plot the FFT of size N=128 for both cases (use dB scaling).

 

i)                     A rectangular window of length 64 samples (what does this represent? zero padding?)

ii)                   A rectangular window of length 128 samples (is the sinusoid resolved exactly?)

 

Plot the FFT of size N=128 for both cases (use dB scaling).

 

iii)                  Repeat i) and ii) for “frequency” p/11.

 

Compare the outputs between each of the four cases. Explain the differences in the FFT magnitude plots. Think of the effects of the windows and zero padding; also try to figure out the frequencies that the 128-point FFT can resolve exactly.

 

 

 

Problem 5-4:  Window Tradeoffs

 

Generate the following signal

Window x(n) with

 

i)                     A rectangular window of length 128

ii)                   A Hamming window of length 128 

 

Using J-DSP, plot the FFT of size N=128 for both cases (use dB scaling).  Why is the shape of the FFT different?  Which window would you choose and why?