Sampled STFT

To numerically evaluate the STFT, we sample the frequency axis ωω in NN equally spaced samples from ω=0 ω 0 to ω=2⁢π ω 2 .

In this definition, the overlap between adjacent blocks is R−1 R 1 . The signal is shifted along the window one sample at a time. That generates more points than is usually needed, so we also sample the STFT along the time direction. That means we usually evaluate X d ⁢kL⁢m X d k L m where LL is the time-skip. The relation between the time-skip, the number of overlapping samples, and the block length is Overlap=R−L Overlap R L

Exercise 1

Match each signal to its spectrogram in Figure 2.

Figure 2

(a) (b)

Solution

[ Show Solution ] [ Hide Solution ]

Spectrogram Example

Figure 3

Figure 4

The matlab program for producing the figures above (Figure 3 and Figure 4).

	  

	  % LOAD DATA
	  load mtlb;
	  x = mtlb;

	  figure(1), clf
	  plot(0:4000,x)
	  xlabel('n')
	  ylabel('x(n)')

	  % SET PARAMETERS
	  R = 256;               % R: block length
	  window = hamming(R);   % window function of length R
	  N = 512;               % N: frequency discretization
	  L = 35;                % L: time lapse between blocks
	  fs = 7418;             % fs: sampling frequency
	  overlap = R - L;

	  % COMPUTE SPECTROGRAM
	  [B,f,t] = specgram(x,N,fs,window,overlap);

	  % MAKE PLOT
	  figure(2), clf
	  imagesc(t,f,log10(abs(B)));
	  colormap('jet')
	  axis xy 
	  xlabel('time')
	  ylabel('frequency')
	  title('SPECTROGRAM, R = 256')

	  
	

Effect of window length R

Figure 5

Narrow-band spectrogram: better frequency resolution

Figure 6

Wide-band spectrogram: better time resolution

Here is another example to illustrate the frequency/time resolution trade-off (See figures - Figure 5, Figure 6, and Figure 7).

Figure 7

Effect of Window Length R
(a) (b)

Effect of L and N

A spectrogram is computed with different parameters: L∈110 L 1 10 N∈32256 N 32 256

Exercise 2

For each of the four spectrograms in Figure 8 can you tell what LL and NN are?

Figure 8

(a) (b)

Solution

[ Show Solution ] [ Hide Solution ]

LL and NN do not effect the time resolution or the frequency resolution. They only affect the 'pixelation'.

Effect of R and L

Shown below are four spectrograms of the same signal. Each spectrogram is computed using a different set of parameters. R∈1202561024 R 120 256 1024 L∈35250 L 35 250 where

Exercise 3

For each of the four spectrograms in Figure 9, match the above values of LL and RR.

Figure 9

Solution

[ Show Solution ] [ Hide Solution ]

If you like, you may listen to this signal with the soundsc command; the data is in the file: stft_data.m. Here is a figure of the signal.

Figure 10