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

2. 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

3. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 3 / 24
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

4. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 4 / 24
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, …).

5. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 5 / 24
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

6. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 6 / 24
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”).

7. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 7 / 24
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.

8. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 8 / 24
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.)

9. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 9 / 24
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

10. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 10 / 24
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) 











11. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 11 / 24
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.

12. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 12 / 24
 




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.

13. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 13 / 24
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.

14. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 14 / 24
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.

15. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 15 / 24
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

16. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 16 / 24
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 

17. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 17 / 24
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!

18. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 18 / 24
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

19. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 19 / 24
FS of main waveforms

20. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 20 / 24
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:
~
~

21. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 21 / 24
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

22. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 22 / 24
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 


23. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 23 / 24
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

24. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 24 / 24
... + 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