Short Time Fourier Transform (STFT)CS474/674 Prof. Bebis

Fourier AnalysisFourier analysis expands signals or functions in terms of sinusoids (or complex exponentials).It reveals all frequency components present in a signal.where:(inverse DFT)(forward DFT)

Examples

Examples (contd)F1(u)F2(u)F3(u)

Fourier Analysis Examples (contd)F4(u)

Limitations of Fourier Analysis (contd)1. Cannot not provide simultaneous time and frequency localization.

2. Not useful for analyzing time-variant, non-stationary signals.

3. Not appropriate for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

Fourier Analysis Examples (contd)F4(u)Provides excellent localization in the frequency domain but poor localization in the time domain.

Limitations of Fourier Analysis (contd)1. Cannot not provide simultaneous time and frequency localization.

2. Not useful for analyzing time-variant, non-stationary signals.

3. Not appropriate for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

Stationary vs non-stationary signalsStationary signals: time-invariant spectra

Non-stationary signals: time-varying spectra.

Stationary vs non-stationary signals

F4(u)Stationary signal:Three frequency components,present at all times!

Stationary vs non-stationary signals (contd)Perfect knowledge of what frequencies exist, but no Information about where these frequencies arelocated in time!F5(u)Non-stationary signal:

Limitations of Fourier Analysis (contd)1. Cannot not provide simultaneous time and frequency localization.

2. Not useful for analyzing time-variant, non-stationary signals.

3. Not appropriate for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

Representing discontinuities or sharp corners

Representing discontinuities or sharp corners (contd)FT

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)Reconstructed

Representing discontinuities or sharp corners (contd)ReconstructedOriginal

Short Time Fourier Transform (STFT)Need a local analysis scheme for a time-frequency representation (TFR).Windowed F.T. or Short Time F.T. (STFT)Segmenting the signal into narrow time intervals (i.e., narrow enough to be considered stationary).Take the Fourier transform of each segment.

Short Time Fourier Transform (STFT) (contd)Steps :(1) Choose a window function of finite length(2) Place the window on top of the signal at t=0(3) Truncate the signal using this window(4) Compute the FT of the truncated signal, save results.(5) Incrementally slide the window to the right (6) Go to step 3, until window reaches the end of the signal

Short Time Fourier Transform (STFT)For each time location where the window is centered, we obtain a different FTEach FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information

Short Time Fourier Transform (STFT) (contd)STFT of f(t):computed for each window centered at t=t

Time parameterFrequencyparameterSignal to be analyzedWindowingfunctioncentered at t=t

Short Time Fourier Transform (STFT) (contd)STFT maps 1-D time domain signals to 2-D time-frequency signals (i.e., in u and t )

Example

f(t)[0 300] ms 100 Hz sinusoid[300 600] ms 50 Hz sinusoid [600 800] ms 25 Hz sinusoid [800 1000] ms 10 Hz sinusoid

Example

W(t)f(t)scaled: t/20

Choosing Window W(t)What shape? Rectangular, Gaussian, Elliptic?

How wide? Window should be narrow enough to make sure that the portion of the signal falling within the window is stationary.Very narrow windows do not offer good localization in the frequency domain.

STFT Window SizeW(t) infinitely long: STFT turns into FT, providing excellent frequency localization, but no time information.

W(t) infinitely short: gives the time signal back, with a phase factor, providing excellent time localization but no frequency information.

STFT Window Size (contd)

Wide window good frequency resolution, poor time resolution.

Narrow window good time resolution, poor frequency resolution.

Exampledifferent size windows(four frequencies, non-stationary)

Example (contd)scaled: t/20

Example (contd)scaled: t/20

Multiresolution AnalysisThe issue of choosing the right window size in STFT leads to the idea of analyzing a signal using windows of different size!

This is called multi-scale or multi-resolution analysis which is the core of wavelets.

Multiresolution Analysis (contd)Many signals or images contain features at various levels of detail (i.e., scales).Small size objects shouldbe examined at a high resolution.

Large size objects shouldbe examined at a low resolution.

Heisenberg (or Uncertainty) PrincipleTime resolution: How well two spikes in time can be separated from each other in the transform domain.

Frequency resolution: How well two spectral components can be separated from each other in the transform domain

Heisenberg (or Uncertainty) Principle (contd)One cannot know the exact time-frequency representation of a signal.We cannot precisely know at what time instance a frequency component is located. We can only know what interval of frequencies are present in which time intervals.

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