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Introduction to the Short Time Fourier Transform, which includes it's definition and methods for its use.

Short time fourier transform

The Fourier transforms (FT, DTFT, DFT, etc. ) do not clearly indicate how the frequency content of a signal changes over time.

That information is hidden in the phase - it is not revealed by the plot of the magnitude of the spectrum.

To see how the frequency content of a signal changes over time, we can cut the signal into blocks and compute thespectrum of each block.
To improve the result,
  • blocks are overlapping
  • each block is multiplied by a window that is tapered at its endpoints.
Several parameters must be chosen:
  • Block length, R .
  • The type of window.
  • Amount of overlap between blocks. ( )
  • Amount of zero padding, if any.

Stft: overlap parameter

The short-time Fourier transform is defined as

X m STFT x n DTFT x n m w n n x n m w n n n 0 R 1 x n m w n n
where w n is the window function of length R .
  • The STFT of a signal x n is a function of two variables: time and frequency.
  • The block length is determined by the support of the window function w n .
  • A graphical display of the magnitude of the STFT, X m , is called the spectrogram of the signal. It is often used in speech processing.
  • The STFT of a signal is invertible.
  • One can choose the block length. A long block length will provide higher frequency resolution (because the main-lobeof the window function will be narrow). A short block length will provide higher time resolution because lessaveraging across samples is performed for each STFT value.
  • A narrow-band spectrogram is one computed using a relatively long block length R , (long window function).
  • A wide-band spectrogram is one computed using a relatively short block length R , (short window function).

Sampled stft

To numerically evaluate the STFT, we sample the frequency axis in N equally spaced samples from 0 to 2 .

k 0 k N 1 k 2 N k
We then have the discrete STFT,
X d k m X 2 N k m n 0 R 1 x n m w n n n 0 R 1 x n m w n W N k n DFT N n 0 R 1 x n m w n 0,0
where 0,0 is N R .

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

Match each signal to its spectrogram in .

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Spectrogram example

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

% 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

Narrow-band spectrogram: better frequency resolution

Wide-band spectrogram: better time resolution

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

Effect of window length r

Effect of l and n

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

  • L = time lapse between blocks.
  • N = FFT length (Each block is zero-padded to length N .)
In each case, the block length is 30 samples.

For each of the four spectrograms in can you tell what L and N are?

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L and N 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 120 256 1024 L 35 250 where

  • R = block length
  • L = time lapse between blocks.

For each of the four spectrograms in , match the above values of L and R .

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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.

Questions & Answers

what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
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Damian Reply
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Introduction about quantum dots in nanotechnology
Praveena Reply
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s. Reply
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
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Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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