1. Lecture on Continuous and Discrete Fourier Transforms

2. TOPICS 1. Infinite Fourier Transform (FT) 2. FT & generalised impulse
3. Uncertainty principle
4. Discrete Time Fourier Transform (DTFT)
5. Discrete Fourier Transform (DFT)
6. Comparing signal by DFS, DTFT & DFT
7. DFT leakage & coherent sampling

3. Fourier analysis - tools
Input Time Signal Frequency spectrum
Periodic (period T)
Discrete
Continuous FT
Aperiodic FS
Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples
Discrete
DFS
Periodic (period T)
Continuous
DTFT
Aperiodic
DFT

4. Fourier Integral (FI) Fourier Integral Theorem
Any aperiodic signal s(t) can be expressed as a Fourier integral if s(t) piecewise smooth(1) in any finite interval (-L,L) and absolute integrable(2).
Fourier Integral Theorem
(3)
Fourier analysis tools for aperiodic signals.
(2)
s(t) continuous, s’(t) monotonic
(1)
(3)
Fourier Transform (Pair) - FT
analysis
synthesis
Complex form
Real-to-complex link

5. Let’s summarise a little
Signal 
Periodic
Aperiodic
Domain  FS FI
Time
real
ak, bk
A(), B()
Frequency FT
complex ck
C()

6. Note: |ak|2 a0 as k0  2 a0 is plotted at k=0
From FS to FT
FS moves to FT as period T increases: continuous spectrum
T = 0.05
Pulse train, width 2 t = 0.025
T = 0.1
Frequency spacing 0 !
T = 0.2 FT
Note: |ak|2 a0 as k0  2 a0 is plotted at k=0

7. f = /(2) = 1/T frequency spacing
Getting FT from FS 2
FS defined
f = /(2) = 1/T frequency spacing
FT defined
As f 0 , replace f , K ,
by df, 2f, 1

8. FT & Dirac’s Delta Dirac’s  defined
The FT of the generalised impulse  (Dirac) is a complex exponential
Hence
FT of Dirac’s 
property
FT of an infinite train of :
a.k.a. Sampling function, Shah(T) = Щ(T) or “comb”
Note:  & Щ = “generalised “functions

9. FT properties Time Frequency Linearity a·s(t) + b·u(t) a·S(f)+b·U(f)
Multiplication s(t)·u(t)
Convolution S(f)·U(f)
Time shifting
Frequency shifting
Time reversal s(-t) S(-f)
Differentiation j2f S(f)
Parseval’s identity  h(t) g*(t) dt =  H(f) G*(f) df
Integration S(f)/(j2f )
Energy & Parseval’s
(E is t-to-f invariant)

10. FT - uncertainty principle
For effective duration t & bandwidth f
  > 0 t•f  
uncertainty product
Bandwidth Theorem
Fourier uncertainty principle
For Energy Signals:
E= |s(t)|2dt = |S(f)|2df < 
Define mean values
Define std. dev.
 t•f  1/4
Implications
• Limited accuracy on simultaneous observation of s(t) & S(f).
• Good time resolution (small t) requires large bandwidth f & vice-versa.

11. FT - example FT of 2-wide square window Fourier uncertainty Choose
S(f) = 2 sMAX sync(2f)
FT of 2-wide square window
Choose
t = |s(t)/s(0) dt| = 2,
f = |S(f)/S(0) df|=1/(2) = half distance btwn first 2 zeroes (f1,-1 = 1/2) of S(f)
then: t · f = 1
Fourier uncertainty
Power Spectral Density (PSD) vs. frequency f plot. Note: Phases unimportant!

12. Phone signals PSD masks
FT - power spectrum
POTS = Voice/Fax/modem Phone HPNA = Home Phone Network
Phone signals PSD masks
US = Upstream DS = Downstream
From power spectrum we can deduce if signals coexist without interfering!
Power Spectral Density, PSD(f) = dE/df = |S(f)|2

13. FT of main waveforms

14. Discrete Time FT (DTFT)
DTFT defined as:
Note: continuous frequency domain! (frequency density function)
analysis
Holds for aperiodic signals
synthesis
Obtained from DFS as N  

15. DTFT - convolution
Digital Linear Time Invariant system: obeys superposition principle.
h[t] = impulse response
Convolution
x[n] h[n]
X(f) H(f) Y(f) = X(f) · H(f)

16. DTFT - Sampling/convolution
Time Frequency
s[n] * u[n]  S(f) · U(f) , s[n] · u[n]  S(f) * U(f)
(From FT properties)
Sampling s(t)
Multiply s(t) by Shah = Щ(t)

17. Discrete FT (DFT) DFT defined as: ~ Frequency resolution
synthesis
analysis
Frequency resolution
Analysis frequencies fm
DFT bins analysis frequencies fm
DFT ~ bandpass filters fm
Applies to discrete time and frequency signals.
Same form of DFS but for aperiodic signals:
signal treated as periodic for computational purpose only.
Note: ck+N = ck  spectrum has period N ~

18. DFT - pulse & sinewave a) rectangular pulse, width N ~
ck = (1/N) e-jk(N-1)/N sin(k)/ sin(k/N) ~
a) rectangular pulse, width N
r[n] =
1 , if 0nN-1
0 , otherwise
b) real sinewave, frequency f0 = L/N
cs[n] = cos(j2f0n)
ck = (1/N) ej{(Nf0-k)-(Nf0 -k)/N} (½) sin{(Nf0-k)}/ sin{(Nf0-k)/N)} + (1/N) ej{(Nf0+k)-(Nf0+k)/N} (½) sin{(Nf0+k)}/ sin{(Nf0+k)/N)} ~
i.e. L complete cycles in N sampled points

19. DFT Examples DFT plots are sampled version of windowed DTFT

20. DFT properties Time Frequency Linearity a·s[n] + b·u[n] a·S(k)+b·U(k)
Multiplication s[n] ·u[n]
Convolution S(k)·U(k)
Time shifting s[n - m]
Frequency shifting S(k - h)

21. DTFT vs. DFT vs. DFS (b) (a) (d) (c) (f) (e) DTFT transform magnitude.
Aperiodic discrete signal.
(d)
DFS coefficients = samples of (b).
(c)
Periodic version of (a).
(f)
DFT estimates a single period of (d).
(e)
Inverse DFT estimates a single period of s[n]

22. DFT – leakage * * N·Magnitude Leakage (a) (b) (c)
Spectral components belonging to frequencies between two successive frequency bins propagate to all bins.
Leakage
Ex: 32-bins DFT of 1 VP sinusoid 32kHz. 1 kHz frequency resolution.
(a)
8 kHz sinusoid
* N·Magnitude *
(b)
8.5 kHz sinusoid
(c)
8.75 kHz sinusoid

23. DFT - leakage example s(t) FT{s(t)} 0.25 Hz Cosine wave
5. Sampled windowed wave
1. Cosine wave
3. Windowed cos wave
s[n] · u[n]  S(f) * U(f) (Convolution)
4. Sampling function
2. Rectangular window
1. Cosine wave
Leakage caused by sampling for a non-integer number of periods

24. DFT - coherent sampling
s(t)
FT{s(t)}
0.2 Hz Cosine wave
5. Sampled windowed wave
3. Windowed cos wave
1. Cosine wave
s[n] ·u[n]  S(f) * U(f) (Convolution)
2. Rectangular window
1. Cosine wave
4. Sampling function
Coherent sampling: NC input cycles exactly into NS = NC (fS/fIN) sampled points.

25. DFT – leakage notes Leakage
Affects Real & Imaginary DFT parts magnitude & phase.
Has same effect on harmonics as on fundamental frequency.
Affects differently harmonically un-related frequency components of same signal (ex: vibration studies).
Leakage depends on the form of the window (so far only rectangular window).
After coffee we’ll see how to take advantage of different windows.