FOURIER ANALYSIS PART 1: Fourier Series

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FOURIER ANALYSIS PART 1: Fourier Series. Maria Elena Angoletta, AB/BDI DISP 2003, 20 February 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 &amp; DSP. analysis.

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### FOURIER ANALYSISPART 1: Fourier Series

Maria Elena Angoletta,

AB/BDI

DISP 2003, 20 February 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

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 2 / 24

analysis

General Transform as problem-solving tool

s(t), S(f) : Transform Pair

time, tfrequency, f

F

s(t) S(f) = F[s(t)]

synthesis

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.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 3 / 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, …).

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 4 / 24

Periodic (period T)

FS

Discrete

Continuous

Aperiodic

FT

Continuous

**

Periodic (period T)

DFS

Discrete

Discrete

DTFT

Continuous

Aperiodic

**

DFT

Discrete

**

Calculated via FFT

Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples

Fourier analysis - tools

Input Time Signal Frequency spectrum

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 5 / 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”).

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 6 / 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.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 7 / 24

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

(signal average over a period, i.e. DC term & zero-frequency component.)

* see next slide

Fourier Series (FS)

synthesis

analysis

Note: {cos(kωt), sin(kωt) }k form orthogonal base of function space.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 8 / 24

Dirichlet conditions

(a)

s(t) piecewise-continuous;

s(t) piecewise-monotonic;

s(t) absolutely integrable ,

(b)

In any period:

(c)

Rate of convergence

Example: square wave

if s(t) discontinuous then |ak|<M/k for large k (M>0)

(a)

(b)

(c)

FS convergence

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 9 / 24

FS of odd* function: square wave.

(zero average)

(odd function)

*Even & Odd functions

Even :

s(-x) = s(x)

Odd :

s(-x) = -s(x)

FS analysis - 1

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 10 / 24

Fourier spectrum representations

Rectangular

Polar

vk = akcos(k t) - bksin(k t)

vk =rk cos (k t + k)

rK = amplitude, K= phase

fk=k /2

Fourier spectrum

of square-wave.

FS analysis - 2

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 11 / 24

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.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 12 / 24

First observed by Michelson, 1898. Explained by Gibbs.

• 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.
Gibbs phenomenon

Overshoot exist @ each discontinuity

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 13 / 24

(zero average)

amplitude

(even function)

phase

FS time shifting

FS of even function:

Note: amplitudes unchanged BUTphases advance by k/2.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 14 / 24

Euler’s notation:

e-jt = (ejt)* = cos(t) - j·sin(t) “phasor”

Complex form of FS (Laplace 1782). Harmonics ck separated by f = 1/T on frequency plot.

Complex FS

analysis

synthesis

Note: c-k = (ck)*

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 15 / 24

Homogeneity a·s(t) a·S(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)

*Explained in next week’s lecture

FS properties

Time Frequency

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 16 / 24

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

Negative frequencies & time reversal

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.

Careful: phases important when combining several signals!

FS - “oddities”

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 17 / 24

Average power W :

Parseval’s Theorem

Example

Pulse train, duty cycle  = 2 t / T

Wk = 2 W0sync2(k )

bk = 0

a0 =  sMAX

W0 = ( sMAX)2

sync(u) = sin( u)/( u)

ak = 2sMAXsync(k )

FS - power
• FS convergence ~1/k
•  lower frequency terms
• Wk = |ck|2 carry most power.
• Wk vs. ωk: Power density spectrum.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 18 / 24

FS of main waveforms

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 19 / 24

Band-limited signal s[n], period = N.

DFS defined as:

Orthogonality in DFS:

~

~

Note: ck+N = ck same period N

i.e. time periodicity propagates to frequencies!

Kronecker’s delta

Synthesis: finite sum  band-limited s[n]

Discrete Fourier Series (DFS)

DFS generate periodic ck with same signal period

analysis

synthesis

N consecutive samples of s[n] completely describe s in time or frequency domains.

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 20 / 24

s[n]: period N, duty factor L/N

amplitude

phase

DFS analysis

DFS of periodic discrete

1-Volt square-wave

Discrete signals periodic frequency spectra.

Compare to continuous rectangular function (slide # 10, “FS analysis - 1”)

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 21 / 24

Homogeneitya·s[n] a·S(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

DFS properties

Time Frequency

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 22 / 24

(1)

(2)

(3)

(2)

(1)

tn = n/fS , n = 1, 2 .. NS, NS= No. samples

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).

DFS analysis: DDC + ...

s(t) periodic with period TREV (ex: particle bunch in “racetrack” accelerator)

M. E. Angoletta - DISP2003 - Fourier analysis - Part1: Fourier Series 23 / 24

Example: Real-life DDC

Fourier coefficients a k*, b k*

harmonic k* = LO/REV

... + DSP

DDCs with different fLO yield more DFS components

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