1. FOURIER ANALYSIS PART 1: Fourier Series
DISP-2003
FOURIER ANALYSIS PART 1: Fourier Series
Maria Elena Angoletta,
AB/BDI
DISP 2003, 20 February 2003
Introduction to Digital Signal Processing

2. TOPICS 2. A tour of Fourier Transforms
DISP-2003
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
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Introduction to Digital Signal Processing

3. 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
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4. 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, …).
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5. 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 **
Calculated via FFT
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6. 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”).
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7. 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)
Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866).
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.
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8. For all t but discontinuities
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.
a0, ak, bk : Fourier coefficients.
k: harmonic number,
T: period,  = 2/T
For all t but discontinuities
synthesis
analysis
(signal average over a period, i.e. DC term & zero-frequency component.)
Note: {cos(kωt), sin(kωt) }k form orthogonal base of function space.
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9. if s(t) discontinuous then |ak|<M/k for large k (M>0)
FS convergence
s(t) piecewise-continuous;
s(t) piecewise-monotonic;
s(t) absolutely integrable ,
(a)
(b)
(c)
Dirichlet conditions
In any period:
if s(t) discontinuous then |ak|<M/k for large k (M>0)
Rate of convergence
Example: square wave
(a)
(b)
(c)
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10. FS of odd* function: square wave.
FS analysis - 1
FS of odd* function: square wave.
(zero average)
(odd function)
* Even & Odd functions
Odd :
s(-x) = -s(x)
Even :
s(-x) = s(x)
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11. FS analysis - 2 Fourier spectrum representations fk=k /2
rK = amplitude, K = phase
vk = rk cos (k t + k)
Polar
Rectangular
vk = akcos(k t) - bksin(k t)
Fourier spectrum
of square-wave.
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12. Square wave reconstruction from spectral terms
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.
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13. Overshoot exist @ each discontinuity
Gibbs phenomenon
Overshoot each discontinuity
Max overshoot pk-to-pk = 8.95% of discontinuity magnitude.
Just a minor annoyance.
FS converges to (-1+1)/2 = discontinuities, in this case.
First observed by Michelson, Explained by Gibbs.
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14. /2-advanced square-wave
FS time shifting
FS of even function:
/2-advanced square-wave
(even function)
(zero average)
phase
amplitude
Note: amplitudes unchanged BUT phases advance by k/2.
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15. Complex FS Euler’s notation:
e-jt = (ejt)* = cos(t) - j·sin(t) “phasor”
analysis
Complex form of FS (Laplace 1782). Harmonics ck separated by f = 1/T on frequency plot.
synthesis
Note: c-k = (ck)*
Link to FS real coeffs.
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16. * Explained in next week’s lecture
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)
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17. FS - “oddities” Orthonormal base
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
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
Careful: phases important when combining several signals!
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18. FS - power Average power W : Example Parseval’s Theorem
FS convergence ~1/k
 lower frequency terms
Wk = |ck|2 carry most power.
Wk vs. ωk: Power density spectrum.
Example
Wk = 2 W0 sync2(k )
W0 = ( sMAX)2
sync(u) = sin( u)/( u)
Pulse train, duty cycle  = 2 t / T
bk = 0
a0 =  sMAX
ak = 2sMAX sync(k )
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19. FS of main waveforms
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20. Discrete Fourier Series (DFS)
Band-limited signal s[n], period = N.
DFS generate periodic ck with same signal period
Note: ck+N = ck  same period N
i.e. time periodicity propagates to frequencies!
DFS defined as: ~
Kronecker’s delta
Orthogonality in DFS:
analysis
Synthesis: finite sum  band-limited s[n]
synthesis
N consecutive samples of s[n] completely describe s in time or frequency domains.
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21. DFS of periodic discrete
DFS analysis
s[n]: period N, duty factor L/N
DFS of periodic discrete
1-Volt square-wave
amplitude
phase
Discrete signals  periodic frequency spectra.
Compare to continuous rectangular function (slide # 10, “FS analysis - 1”)
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22. * Explained in next week’s lecture
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)
* Explained in next week’s lecture
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23. DFS analysis: DDC + ... (1) (2) (3)
s(t) periodic with period TREV (ex: particle bunch in “racetrack” accelerator)
tn = n/fS , n = 1, 2 .. NS , NS = No. samples
(1)
(2)
I[tn ]+j Q[tn ] = s[tn ] e -jLOtn
(3)
I[tp ]+j Q[tp ] p = 1, 2 .. NT , Ns / NT = decimation. (Down-converted to baseband).
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24. Example: Real-life DDC
... + DSP
Fourier coefficients a k*, b k*
harmonic k* = LO/REV
DDCs with different fLO yield more DFS components
Example: Real-life DDC
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