The Story of Wavelets Theory and Engineering Applications

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Presentation transcript:

The Story of Wavelets Theory and Engineering Applications Time frequency representation Instantaneous frequency and group delay Short time Fourier transform –Analysis Short time Fourier transform – Synthesis Discrete time STFT

Time – Frequency Representation Why do we need it? Time info difficult to interpret in frequency domain Frequency info difficult to interpret in time domain Perfect time info in time domain , perfect freq. info in freq. domain …Why? How to handle non-stationary signals Instantaneous frequency Group Delay

Instantaneous Frequency & Group Delay Instantaneous frequency: defined as the rate of change in phase A dual quantity group delay defined as the rate of change in phase spectrum Frequency as a function of time Time as a function of frequency What is wrong with these quantities???

Time Frequency Representation in Two-dimensional Space TFR Linear STFT, WT, etc. Non-Linear Quadratic Spectrogram, WD

STFT Amplitude ….. ….. time t0 t1 tk tk+1 tn ….. ….. Frequency

The Short Time Fourier Transform Take FT of segmented consecutive pieces of a signal. Each FT then provides the spectral content of that time segment only Spectral content for different time intervals Time-frequency representation Time parameter Signal to be analyzed FT Kernel (basis function) Frequency parameter STFT of signal x(t): Computed for each window centered at t= (localized spectrum) Windowing function (Analysis window) Windowing function centered at t=

Properties of STFT Linear Complex valued Time invariant Time shift Frequency shift Many other properties of the FT also apply.

Alternate Representation of STFT STFT : The inverse FT of the windowed spectrum, with a phase factor

Filter Interpretation of STFT X(t) is passed through a bandpass filter with a center frequency of Note that (f) itself is a lowpass filter.

Filter Interpretation of STFT x(t) X X x(t)

Resolution Issues All signal attributes located within the local window interval around “t” will appear at “t” in the STFT Amplitude time k n Frequency

Time-Frequency Resolution Closely related to the choice of analysis window Narrow window  good time resolution Wide window (narrow band)  good frequency resolution Two extreme cases: (T)=(t) excellent time resolution, no frequency resolution (T)=1 excellent freq. resolution (FT), no time info!!! How to choose the window length? Window length defines the time and frequency resolutions Heisenberg’s inequality Cannot have arbitrarily good time and frequency resolutions. One must trade one for the other. Their product is bounded from below.

Time-Frequency Resolution

Time Frequency Signal Expansion and STFT Synthesis Basis functions Coefficients (weights) Synthesis window Synthesized signal Each (2D) point on the STFT plane shows how strongly a time frequency point (t,f) contributes to the signal. Typically, analysis and synthesis windows are chosen to be identical.

STFT Example 300 Hz 200 Hz 100Hz 50Hz

STFT Example

STFT Example a=0.01

STFT Example a=0.001

STFT Example a=0.0001

STFT Example a=0.00001

Discrete Time Stft