1. Discrete Fourier Transform (DFT)
Jyh-Shing Roger Jang (張智星)
MIR Lab (多媒體資訊檢索實驗室)
CSIE, NTU (台灣大學 資訊工程系)

2. Discrete Fourier Transform
Goal
Decompose a given vector of discrete signals into sinusoidal components whose amplitudes can be view as energy distribution over a range of frequencies
Applications
Speech recognition
Digital filtering
Many many more...
Due to sampling

3. Decomposition
How to decompose xn=x(n/fs), n=0~N-1 into a linear combination of sinusoidal functions?
Magnitude spectrum 
Phase spectrum 
Remaining problems
To determine no. of terms and their frequencies

4. How to Select Frequencies
Signals of N points with sample rate fs
Total duration d=N/fs (sec)
Periods = d, d/2, d/3, …
Freq. of sinusoids (bin frequencies)
Total count = N/2+1

5. No. of Terms for Decomposition
Curve fitting for a parabola
# of unknowns = # of equations (= # of points)
Curve fitting view for sinusoidal decomp.
How many terms do we need if there are N points?  N/2+1!

6. Example When N=4

7. Example When N=8

8. Sinusoidal Basis Functions (1/2)
Can be converted into
sin & cos
Phase is necessary

9. Sinusoidal Basis Functions (2/2)

10. Decompose x(t) into Basis Functions
n=0,1,2,…, N-1
# of terms = # of bin freq = N/2+1
# of fitting parameters = N

11. Decomposition into Basis Functions
Overall expression:
No. of parameters is 1+2*(N/2-1)+1=N, which is equal to the no. of data points  Exact solution is likely to exist.
Amplitude
Freq=
Phase

12. Parameter Identification
How to identify coefficients of the basis functions:
Solving simultaneous linear equations
Integration (which take advantage of the orthogonality of the basis functions)
Fast Fourier transform (FFT)
A fast algorithm with a complexity of O(N log N)

13. From Time-domain to Freq-domain
Many animation over the web
3D animation
Please post on FB if you find more.

14. Frequencies of Basis Functions
Due to Euler identity, we can express the k-th term shown on the right.
Quiz!

15. Frequencies of Basis Functions
Plug in the simplified k-th term:
Include the first and last terms:

16. Representations of DFT: Two-side
Characteristics
General representation
If x is complex, then c is not conjugate symmetric
MATLAB command: c=fft(x)
Usually complex numbers

17. Representations of DFT: One-side
Characteristics
When x is real, c is conjugate symmetric and we only have to look at one-side of FFT result.
MATLAB command: magSpec=fftOneside(x)
Real
Real
In SAP toolbox
Complex
conjugate

18. Representations of DFT: One-side
Formulas

20. Example: Conjugate Symmetric
Conjugate symmetric of DFT for real x
fftSym01.m

21. Example: Two-side FFT
Two-side FFT of a pure sinusoid at one of the bin freqency
fft01.m

22. Example: Two-side FFT
Two-side FFT of a pure sinusoid NOT at one of the bin frequencies
fft02.m

23. Example: One-side FFT
One-side FFT of a pure sinusoid NOT at one of the bin frequencies
fft03.m

24. Example: One-side FFT
One-side FFT of a frame of audio signals
fft04.m

25. Example: FFT for Data Compression
Use partial coefficients to reconstruct the original signals
fftApproximate01.m
High-frequency components are not so important

26. Example: FFT for Data Compression
Use low-freq components to reconstruct a frame
fftApproximate01.m

27. Example: FFT for Periodic Signals
Quiz!
FFT on periodic signals
fftRepeat01.m
Useful for pitch tracking
Interpretation of harmonics from viewpoints of
Approximation by basis functions
Integration for obtaining the coefficients

28. Why Harmonic Structures?

29. Example: Zero-padding for FFT
Purpose
N’=N+a=2^n, easier for FFT computation
Interpolation to have better resolution
Example
fftZeroPadding01.m
Quiz!

30. Example: Down-sampling
High-frequency components are missing
fftResample01.m

31. Harmonics Why do we have harmonic structures in a power spectrum?
Since the original frame is quasi-periodic…
If we want to use the spectrum for pitch tracking, we need to enhance the harmonics.  How?

32. How to Enhance Harmonics?
Goal
Enhance harmonics for pitch tracking
Approach
Take an integer number of fundamental periods Hard!
Use windowing
Use zero-padding for better resolution

33. Example: Harmonics Enhancement (1)
fftHarmonics01.m

34. Example: Harmonics Enhancement (2)
fftHarmonics02.m

35. Example: Harmonics Enhancement (3)
fftHarmonics03.m