1. 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

2. 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

3. 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???

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

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

6. 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=

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

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

9. 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.

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

11. 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

12. 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.

13. Time-Frequency Resolution

14. 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.

15. STFT Example
300 Hz Hz Hz 50Hz

16. STFT Example

17. STFT Example
a=0.01

18. STFT Example
a=0.001

19. STFT Example
a=0.0001

20. STFT Example a=

21. Discrete Time Stft