DISP-2003_L01B_20Feb03.ppt

FOURIER ANALYSIS
PART 1: Fourier Series
Maria Elena Angoletta,
AB/BDI
DISP 2003, 20 February 2003

M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 2 / 24
TOPICS
1. Frequency analysis: a powerful tool
2. A tour of Fourier Transforms
3. Continuous Fourier Series (FS)
4. Discrete Fourier Series (DFS)
5. Example: DFS by DDCs & DSP

Frequency analysis: why?
 Fast & efficient insight on signal’s building blocks.
 Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE).
 Powerful & complementary to time domain analysis techniques.
 Several transforms in DSPing: Fourier, Laplace, z, etc.
time, t frequency, f
F
s(t) S(f) = F[s(t)]
analysis
synthesis
s(t), S(f) :
Transform Pair
General Transform as
problem-solving tool

Fourier analysis - applications
Applications wide ranging and ever present in modern life
• Telecomms - GSM/cellular phones,
• Electronics/IT - most DSP-based applications,
• Entertainment - music, audio, multimedia,
• Accelerator control (tune measurement for beam steering/control),
• Imaging, image processing,
• Industry/research - X-ray spectrometry, chemical analysis (FT
spectrometry), PDE solution, radar design,
• Medical - (PET scanner, CAT scans & MRI interpretation for sleep
disorder & heart malfunction diagnosis,
• Speech analysis (voice activated “devices”, biometry, …).

Fourier analysis - tools
Input Time Signal Frequency spectrum






1
N
0
n
N
n
k
π
2
j
e
s[n]
N
1
k
c
~
Discrete
Discrete
DFS
Periodic
(period T)
Continuous
DTFT
Aperiodic
Discrete
DFT
n
f
π
2
j
e
n
s[n]
S(f) 





 
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
time, tk
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8
time, tk






1
N
0
n
N
n
k
π
2
j
e
s[n]
N
1
k
c
~
**
**
Calculated via FFT
**
dt
t
f
π
j2
e
s(t)
S(f)






 
dt
T
0
t
ω
k
j
e
s(t)
T
1
k
c 




Periodic
(period T)
Discrete
Continuous
FT
Aperiodic
FS
Continuous
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8
time, t
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
time, t
Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples

A little history
 Astronomic predictions by Babylonians/Egyptians likely via trigonometric sums.
 1669: Newton stumbles upon light spectra (specter = ghost) but fails to
recognise “frequency” concept (corpuscular theory of light, & no waves).
 18th century: two outstanding problems
 celestial bodies orbits: Lagrange, Euler & Clairaut approximate observation data
with linear combination of periodic functions; Clairaut,1754(!) first DFT formula.
 vibrating strings: Euler describes vibrating string motion by sinusoids (wave
equation). BUT peers’ consensus is that sum of sinusoids only represents smooth
curves. Big blow to utility of such sums for all but Fourier ...
 1807: Fourier presents his work on heat conduction  Fourier analysis born.
 Diffusion equation  series (infinite) of sines & cosines. Strong criticism by peers
blocks publication. Work published, 1822 (“Theorie Analytique de la chaleur”).

A little history -2
 19th / 20th century: two paths for Fourier analysis - Continuous & Discrete.
CONTINUOUS
 Fourier extends the analysis to arbitrary function (Fourier Transform).
 Dirichlet, Poisson, Riemann, Lebesgue address FS convergence.
 Other FT variants born from varied needs (ex.: Short Time FT - speech analysis).
DISCRETE: Fast calculation methods (FFT)
 1805 - Gauss, first usage of FFT (manuscript in Latin went unnoticed!!!
Published 1866).
 1965 - IBM’s Cooley & Tukey “rediscover” FFT algorithm (“An algorithm for
the machine calculation of complex Fourier series”).
 Other DFT variants for different applications (ex.: Warped DFT - filter design &
signal compression).
 FFT algorithm refined & modified for most computer platforms.

Fourier Series (FS)
* see next slide
A periodic function s(t) satisfying Dirichlet’s conditions * can be expressed
as a Fourier series, with harmonically related sine/cosine terms.
 









1
k
t)
ω
(k
sin
k
b
t)
ω
(k
cos
k
a
0
a
s(t)
a0, ak, bk : Fourier coefficients.
k: harmonic number,
T: period,  = 2/T
For all t but discontinuities
Note: {cos(kωt), sin(kωt) }k
form orthogonal base of
function space.



T
0
s(t)dt
T
1
0
a
 


T
0
dt
t)
ω
sin(k
s(t)
T
2
k
b
-
 


T
0
dt
t)
ω
cos(k
s(t)
T
2
k
a
(signal average over a period, i.e. DC term &
zero-frequency component.)

FS convergence
s(t) piecewise-continuous;
s(t) piecewise-monotonic;
s(t) absolutely integrable , 


T
0
dt
s(t)
(a)
(b)
(c)
Dirichlet conditions
In any period:
Example:
square wave
T
(a) (b)
T
s(t)
(c)
if s(t) discontinuous then
|ak|<M/k for large k (M>0)
Rate of convergence

FS analysis - 1
* Even & Odd functions
Odd :
s(-x) = -s(x)
x
s(x)
s(x)
x
Even :
s(-x) = s(x)
FS of odd* function: square wave.
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 t
square
signal,
sw(t)

0
π
0
2π
π
1)dt
(
dt
2π
1
0
a 













  
0
π
0
2π
π
dt
kt
cos
dt
kt
cos
π
1
k
a 












  
 

















   kπ
cos
1
π
k
2
...
π
0
2π
π
dt
kt
sin
dt
kt
sin
π
1
k
b
-









even
k
,
0
odd
k
,
π
k
4
1
ω
2π
T 


(zero average)
(odd function)
...
t
5
sin
π
5
4
t
3
sin
π
3
4
t
sin
π
4
sw(t) 











FS analysis - 2
-π/2
f1 3f1 5f1 f
f1 3f1 5f1 f
rk
θk
4/π
4/3π
fk=k /2
rK = amplitude,
K = phase
rk
k
ak
bk
k = arctan(bk/ak)
rk = ak
2
+ bk
2
zk = (rk , k)
vk = rk cos (k t + k)
Polar
Fourier spectrum
representations




0
k
(t)
k
v
s(t)
Rectangular
vk = akcos(k t) - bksin(k t)
4/π
f1 2f1 3f1 4f1 5f1 6f1 f
f1 2f1 3f1 4f1 5f1 6f1 f
4/3π
ak
-bk
Fourier spectrum
of square-wave.

 




7
1
k
sin(kt)
k
b
-
(t)
7
sw
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
 




5
1
k
sin(kt)
k
b
-
(t)
5
sw  




3
1
k
sin(kt)
k
b
-
(t)
3
sw
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
 




1
1
k
sin(kt)
k
b
-
(t)
1
sw
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
 




9
1
k
sin(kt)
k
b
-
(t)
9
sw
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
 




11
1
k
sin(kt)
k
b
-
(t)
11
sw
FS synthesis
Square wave reconstruction
from spectral terms
Convergence may be slow (~1/k) - ideally need infinite terms.
Practically, series truncated when remainder below computer tolerance
( error). BUT … Gibbs’ Phenomenon.

Gibbs phenomenon
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
t
square
signal,
sw(t)
 




79
1
k
k
79 sin(kt)
b
-
(t)
sw
Overshoot exist @
each discontinuity
• Max overshoot pk-to-pk = 8.95% of discontinuity magnitude.
Just a minor annoyance.
• FS converges to (-1+1)/2 = 0 @ discontinuities, in this case.
• First observed by Michelson, 1898. Explained by Gibbs.

FS time shifting
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 t
square
signal,
sw(t)

FS of even function:
/2-advanced square-wave
π
f1 3f1 5f1 7f1 f
f1 3f1 5f1 7f1 f
rk
θk
4/π
4/3π
0
0
a 
0

k
b
-















even.
k
,
0
11...
7,
3,
k
odd,
k
,
π
k
4
9...
5,
1,
k
odd,
k
,
π
k
4
k
a
(even function)
(zero average)
Note: amplitudes unchanged BUT
phases advance by k/2.

Complex FS
Complex form of FS (Laplace 1782). Harmonics
ck separated by f = 1/T on frequency plot.
r

a
b
 = arctan(b/a)
r = a2
+ b2
z = r e
j
2
e
e
cos(t)
jt
jt 


j
2
e
e
sin(t)
jt
jt




Euler’s notation:
e-jt = (ejt)* = cos(t) - j·sin(t) “phasor”
Note: c-k = (ck)*
   
k
b
j
k
a
2
1
k
b
j
k
a
2
1
k
c 









0
a
0
c 
Link to FS real coeffs.






k
t
ω
k
j
e
k
c
s(t)
 


T
0
dt
t
ω
k
j
-
e
s(t)
T
1
k
c

FS properties
Time Frequency
* Explained in next week’s lecture
Homogeneity a·s(t) a·S(k)
Additivity s(t) + u(t) S(k)+U(k)
Linearity a·s(t) + b·u(t) a·S(k)+b·U(k)
Time reversal s(-t) S(-k)
Multiplication * s(t)·u(t)
Convolution * S(k)·U(k)
Time shifting
Frequency shifting S(k - m)





m
m)U(m)
S(k
t
d
)
t
T
0
u(
)
t
s(t
T
1
 


S(k)
e T
t
k
2π
j



s(t)
T
t
m
2π
j
e 

)
t
s(t 

FS - “oddities”
k = - , … -2,-1,0,1,2, …+ , ωk = kω, k = ωkt, phasor turns anti-clockwise.
Negative k  phasor turns clockwise (negative phase k ), equivalent to negative time t,
 time reversal.
Negative frequencies & time reversal
Fourier components {uk} form orthonormal base of signal space:
uk = (1/T) exp(jkωt) (|k| = 0,1 2, …+) Def.: Internal product :
uk  um = δk,m (1 if k = m, 0 otherwise). (Remember (ejt)* = e-jt )
Then ck = (1/T) s(t)  uk i.e. (1/T) times projection of signal s(t) on component uk
Orthonormal base
 


T
o
*
m
k
m
k dt
u
u
u
u
Careful: phases important when combining several signals!

0 50 100 150 200
k f
Wk/W0
10-3
10-2
10-1
1
2
Wk = 2 W0 sync2(k )
W0 = ( sMAX)2









 



 
1
k 0
W
k
W
1
0
W
W
FS - power
• FS convergence ~1/k
 lower frequency terms
Wk = |ck|2 carry most power.
• Wk vs. ωk: Power density spectrum.
Example
sync(u) = sin( u)/( u)
Pulse train, duty cycle  = 2 t/ T
T
2 t
t
s(t)
bk = 0 a0 =  sMAX
ak = 2sMAX sync(k )
Average power W : s(t)
s(t)
T
o
dt
2
s(t)
T
1
W 

 









 






1
k
2
k
b
2
k
a
2
1
2
0
a
k
2
k
c
W
Parseval’s Theorem

FS of main waveforms

Discrete Fourier Series (DFS)
N consecutive samples of s[n]
completely describe s in time
or frequency domains.
DFS generate periodic ck
with same signal period





1
N
0
k
N
n
k
2π
j
e
k
c
s[n] ~
Synthesis: finite sum  band-limited s[n]
Band-limited signal s[n], period = N.
m
k,
δ
1
N
0
n
N
-m)
n(k
2π
j
e
N
1




Kronecker’s delta
Orthogonality in DFS:






1
N
0
n
N
n
k
2π
j
e
s[n]
N
1
k
c
~
Note: ck+N = ck  same period N
i.e. time periodicity propagates to frequencies!
DFS defined as:
~
~

DFS analysis
DFS of periodic discrete
1-Volt square-wave






























otherwise
,
N
k
π
sin
N
kL
π
sin
N
N
1)
(L
k
π
j
e
2N,...
N,
0,
k
,
N
L
k
c
~
0.6
0 1 2 3 4 5 6 7 8 9 10 k
1
0 2 4 5 6 7 8 9 10 n
k
-0.4
0.2
0.24 0.24
0.6 0.6
0.24
1
0.24
-0.2
0.4
0.2
-0.4
-0.2
0.4
0.2
0.6 ck
~
Discrete signals  periodic frequency spectra.
Compare to continuous rectangular function
(slide # 10, “FS analysis - 1”)
-5 0 1 2 3 4 5 6 7 8 9 10 n
0 L N
s[n]
1
s[n]: period N, duty factor L/N

DFS properties
Time Frequency
Homogeneity a·s[n] a·S(k)
Additivity s[n] + u[n] S(k)+U(k)
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)




1
N
0
h
h)
-
S(h)U(k
N
1
* Explained in next week’s lecture





1
N
0
m
m]
u[n
s[m]
S(k)
e T
m
k
2π
j



s[n]
T
t
h
2π
j
e 


Q
Q[
[t
tp
p]
]:
: “
“Q
Qu
ua
ad
dr
ra
at
tu
ur
re
e”
”
s[tn]
s(t)
cos[LO tn]
I
I[
[t
tn
n]
]
-sin[LO tn]
Q
Q[
[t
tn
n]
]
ADC
(fS)
I
I[
[t
tp
p]
]:
: “
“I
In
n-
-p
ph
ha
as
se
e”
”
LPF
&
DECIMATION
N = NS/NT
TO DSP
(next slide)
D
DI
IG
GI
IT
TA
AL
L D
DO
OW
WN
N
C
CO
ON
NV
VE
ER
RT
TE
ER
R
DFS analysis: DDC + ...
s(t) periodic with period TREV (ex: particle bunch in “racetrack” accelerator)
tn = n/fS , n = 1, 2 .. NS , NS = No. samples
(1)
(1)
(2)
(2) I[tn ]+j Q[tn ] = s[tn ] e -jLOtn
(3)
(3) I[tp ]+j Q[tp ] p = 1, 2 .. NT , Ns / NT = decimation. (Down-converted to baseband).
0 f
B
B
B
B
B
B
fLO f

... + DSP
DDCs with different fLO
yield more DFS components
Fourier coefficients a k*, b k*
harmonic k* = LO/REV




T
N
1
p
p
I
T
N
1
*
k
a 




T
N
1
p
p
Q
T
N
1
*
k
b
Parallel
digital
input
from ADC
TUNABLE LOCAL
OSCILLATOR
(DIRECT DIGITAL SYNTHESIZER)
Clock
from
ADC
sin cos
COMPLEX MIXER
fS fS fS/N
Decimation factor N
I
Q
Central
frequency
LOW PASS
FILTER
(DECIMATION)
Example: Real-life DDC

DISP-2003_L01B_20Feb03.ppt

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