The Story of WaveletsTheory 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

domainFrequency info difficult to interpret in time

domainPerfect 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

)(2

1)( tx

dt

dtfx

)(2

1)( fX

df

dftx

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

LinearSTFT, WT, etc.

QuadraticSpectrogram, WD

Non-Linear

STFT

….. …..

time

Am

plit

ude

Fre

quen

cy …..…..

t0 t1 tk tk+1 tn

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 onlySpectral content for different time intervalsTime-frequency representation

t

tjx dtetWtxSTFT )()(),(

STFT of signal x(t):Computed for each window centered at t=(localized spectrum)

Time parameter Frequency

parameter

Signal to be analyzed

Windowingfunction

(Analysis window)Windowing function

centered at t=

FT Kernel(basis function)

Properties of STFT

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

Alternate Representation of STFT

deXetSTFT

dfefffXeftSTFT

tjtjx

ftj

f

tfjx

)~()()~,(

)~

()()~

,(

*~)(

2*~

2)(

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

)~

()( * fffX

Filter Interpretation of STFT

ftj

ftj

f

ettxfffX

dfefffXfffXF

2*

2**1

)()()~

()(

)~

()()~

()(

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

f~

Filter Interpretation of STFT

x(t) ftjet 2)( X

ftje 2

),()( ftSTFTx

x(t) )( t ),()( ftSTFTxX

ftje 2

Resolution Issues

time

Am

plit

ude

Fre

quen

cy

k

All signal attributes located within the local window intervalaround “t” will appear at “t” in the STFT

)( kt

n

)( kt

Time-Frequency Resolution

Closely related to the choice of analysis windowNarrow window good time resolutionWide 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

Fre

quen

cy

Time Frequency Signal Expansion and STFT Synthesis

f

tfjx fddetgfSTFTtx

~

~2)( ~

)()~

,()(

Synthesis window

Basis functions

Coefficients (weights)

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.

1)(*)( dtttg

300 Hz 200 Hz 100Hz 50Hz

STFT Example

2/2

)( atet

STFT Example

a=0.01

STFT Example

a=0.001

STFT Example

a=0.0001

STFT Example

a=0.00001

STFT Example

Discrete Time Stft

n k

kFtjx enTtgkFnTSTFTtx 2)( )(),()(

dtenTttxkFnTSTFT kFtj

tx

2*)( )()(),(