<< Chapter < Page Chapter >> Page >
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 .

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The dft, fft, and practical spectral analysis' conversation and receive update notifications?

Ask