Overviewliacs.leidenuniv.nl/~bakkerem2/SR2009/SR2009_FFT.pdf · 2009. 3. 5. · SR2009 Fourier...

SR2009

Fourier Transforms 1

SHORT TIME FOURIER TRANSFORMS

Erwin M. BakkerLML LIACS

SR 2009, March 5th 2009

- SR2009 - Fourier Transforms 2 / 54

OverviewOverview

• Analog and Digital Signals• Fourier Transforms

Material slightly adapted from lectures byDr M.E. Angoletta at DISP2003, a DSP course given by CERN and University of Lausanne

(UNIL) http://humanresources.web.cern.ch/humanresources/external/training/tech/spe

cial/DISP2003.asp


Analog and Digital SignalsAnalog and Digital Signals

1. From Analog to Digital Signal

2. Sampling & Aliasing


Analog and Digital SignalsAnalog and Digital Signals

Continuous functionContinuous function F of a continuouscontinuous variable t (t can be time, space etc) : F(t)

Analog SignalsDiscrete functionDiscrete function Fk of a discretediscrete (sampling) variable tk, with k an integer: Fk = F(tk).

Digital Signals

Uniform (periodic) sampling with sampling frequency fS = 1/ tS, e.g., ts = 0.001 sec => fs = 1000Hz

SR2009



Digital System ImplementationDigital System Implementation

• Sampling rate.

• Pass / stop bands.

KEY DECISION POINTS:KEY DECISION POINTS:Analysis bandwidth, Dynamic range

• No. of bits. Parameters.

Digital Processing

A/D

AntialiasingFilter

ANALOG INPUTANALOG INPUT

DIGITAL OUTPUTDIGITAL OUTPUT

• Digital format.


SamplingSamplingHow fast must we sample a continuous signal to preserve its info content?

Ex: wheels in a movie.

25 frames (=samples) per second.

Frequency misidentification due to low sampling frequency.

Train starts wheels ‘go’ clockwise.

Train accelerates wheels ‘go’ counter-clockwise.

1

Why?Why?

* Sampling: independent variable (ex: time) continuous → discrete.Quantisation: dependent variable (ex: voltage) continuous → discrete.Here we’ll talk about uniform sampling.

**


Sampling - 2Sampling - 2

__ s(t) = sin(2πf0t)

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

s(t) @ fSf0 = 1 Hz, fS = 3 Hz

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

__ s1(t) = sin(8πf0t)

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

__ s2(t) = sin(14πf0t)-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

sk (t) = sin( 2π (f0 + k fS) t ) , k ∈

s(t) @ fS represents exactly all sine-waves sk(t) defined by:

1


The sampling theoremThe sampling theoremA signal s(t) with maximum frequency fMAX can be recovered if sampled at frequency fS > 2 fMAX .

Condition on fS?

fS > 300 Hz

t)cos(100πt)πsin(30010t)πcos(503s(t) −⋅+⋅=

F1=25 Hz, F2 = 150 Hz, F3 = 50 Hz

F1 F2 F3

fMAX

Example

1

Theorem

* Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel’nikov.

Nyquist frequency (rate) fN = 2 fMAX

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Frequency DomainFrequency Domain

•• Time & frequencyTime & frequency: two complementary signal descriptions. Signals seen as “projected’ onto time or frequency domains.

1

•• BandwidthBandwidth: indicates rate of change of a signal. High bandwidth signal changes fast.

EarEar + brain act as frequency analyser: audio spectrum split into many narrow bands low-power sounds detected out of loud background.

Example


Sampling low-pass signalsSampling low-pass signals

-B 0 B f

Continuous spectrum (a) Band-limited signal: frequencies in [-B, B] (fMAX = B).

(a)

-B 0 B fS/2 f

Discrete spectrumNo aliasing (b) Time sampling frequency

repetition.

fS > 2 B no aliasing.

(b)

1

0 fS/2 f

Discrete spectrum Aliasing & corruption (c)

(c) fS 2 B aliasing !aliasing !

Aliasing: signal ambiguity Aliasing: signal ambiguity in frequency domainin frequency domain

sk (t) = sin( 2π (f0 + k fS) t ) , k ∈ N

Note: s(t) at fS represents all sine-waves sk(t) defined by:k = 1 k=0 k = 1


Antialiasing filterAntialiasing filter

-B 0 B f

Signal of interest

Out of band noise Out of band

noise

-B 0 B fS/2 f

(a),(b) Out-of-band noise can aliaseinto band of interest. Filter it before!Filter it before!

(a)

(b)

-B 0 B f

Antialiasing filter Passband

frequency

(c)

Passband: depends on bandwidth of interest.

(c) AntialiasingAntialiasing filterfilter

1


Under-samplingUnder-sampling1

Using spectral replications to reduce Using spectral replications to reduce sampling frequency sampling frequency ffSS requirements.requirements.

mBCf2

Sf1mBCf2 −⋅

≤≤+

+⋅

m∈ , selected so that fS > 2B

B

0 fC

Bandpass signalcentered on fC

-fS 0 fS 2fS fC

AdvantagesAdvantages

Slower ADCs / electronics Slower ADCs / electronics needed.needed.

Simpler Simpler antialiasingantialiasing filters.filters.

fC = 20 MHz, B = 5MHz

Without under-sampling fS > 40 MHz.

With under-sampling fS = 22.5 MHz (m=1);

= 17.5 MHz (m=2); = 11.66 MHz (m=3).

ExampleExample

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Over-samplingOver-sampling1

fOS = over-sampling frequency,w = additional bits required.fOS = 4w · fS

Each additional bit implies/requires overEach additional bit implies/requires over--sampling by a factor of four. sampling by a factor of four.

Oversampling : sampling at frequencies fS >> 2 fMAX .

Over- sampling & averaging may improve ADC resolution( i.e. SNR, see )2


(Some) ADC parameters(Some) ADC parameters1. Number of bits N (~resolution)

2. Data throughput (~speed)

3. Signal-to-noise ratio (SNR)

4. Signal-to-noise-&-distortion rate (SINAD)

5. Effective Number of Bits (ENOB)

6. …

NB: Definitions may be slightly manufacturerNB: Definitions may be slightly manufacturer--dependent!dependent!

Different applications Different applications have different needs.have different needs.

2

Static distortionStatic distortion

Imaging / video

Communication

Radar systems


ADC - Number of bits NADC - Number of bits NContinuous input signal digitized into 2N levels.

-4

-3

-2

-1

0

1

2

3

-4 -3 -2 -1 0 1 2 3 4

000

001

111

010

V

VFSR

Uniform, bipolar transfer function (N=3)Uniform, bipolar transfer function (N=3)

Quantisation stepQuantisation step q =V max

2N

Ex: Vmax = 1V , N = 12 q = 244.1 µV

Voltage ( = q)

Scale factor (= 1 / 2N )

Percentage (= 100 / 2N )

-1

-0.5

0

0.5

1

-4 -3 -2 -1 0 1 2 3 4

- q / 2

q / 2

Quantisation errorQuantisation error

2


ADC - Quantisation errorADC - Quantisation error2

• Quantisation Error eq in[-0.5 q, +0.5 q].

• eq limits ability to resolve small signal.

• Higher resolution means lower eq.

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

time [ms]

Volta

ge [V

]

QE forN = 12VFS = 1

0 2 4 6 8 10

Sampling time, tk

|e q

| [V

]

10-4

2 10-4

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SNR of ideal ADCSNR of ideal ADC2

( )

⋅=

)qRMS(einputRMS

10log20idealSNR (1)

Also called SQNR(signal-to-quantisation-noise ratio)

(RMS = root mean square)

Ideal ADC: only quantisation error eeqq(p(e)p(e) constant, no stuck bits…)

eeqq uncorrelated with signal.

ADC performance constant in time.

AssumptionsAssumptions

( ) ( )22

FSRVT

0dt

2ωtsin

2FSRV

T1inputRMS =

⋅⋅= ∫ Input(t) = ½ VInput(t) = ½ VFSRFSR sin(sin(ωω t).t).

( )12N2

FSRV12q

q/2

q/2-qdeqep2

qe)qRMS(e⋅

==⋅= ∫

eeqq Error value

pp((ee)) quantisation error probability density

1q

q2

q2

(sampling frequency fS = 2 fMAX)


SNR of ideal ADC - 2SNR of ideal ADC - 2

[dB]1.76N6.02SNRideal +⋅= (2)Substituting in (1) =>

One additional bit SNR increased by 6 dBOne additional bit SNR increased by 6 dB

2

Actually (2) needs correction factor depending on ratio between sampling freq & Nyquist freq. Processing gain due to oversampling.

- Real signals have noise.

- Forcing input to full scale unwise.

- Real ADCs have additional noise (aperture jitter, non-linearities etc).

Real SNR lower because:Real SNR lower because:


ADC selection dilemmaADC selection dilemmaSpeed & resolution:Speed & resolution:

a tradeoff.a tradeoff.Effective Number of Bits (ENOB)

2

High resolution (bit #)- Higher cost & dissipation.

- Tailored onto DSP word width.

High speed- Large amount of data to store/analyse.

- Lower accuracy & input impedance.

**

** DIFFICULTDIFFICULT area moves down & right every year. Rule of thumb: 1 bit improvement every 3 years.

may increase SNR.2OversamplingOversampling & averaging& averaging (see ).

DitheringDithering ( = adding small random noise before quantisation).

Sound cards: x


Finite word-length effects Finite word-length effects

Dynamic Dynamic rangerangedBdB = 20 log10largest value

smallest value

Fixed point ~ 180 dB

Floating point ~1500 dB

High dynamic range wide data set representation with no overflow.

NB: Different applications have different needs. Ex: telecomms: 50 dB; HiFi audio: 90 dB.

OverflowOverflow : arises when arithmetic operation result has one too many bits to be represented in a certain format.

SR2009



Fourier TransformsFourier Transforms

1. Frequency analysis: a powerful tool

2. A tour of Fourier Transforms

3. Continuous Fourier Series (FS)

4. Discrete Fourier Series (DFS)


Frequency AnalysisFrequency Analysis• 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: Fourier, Discrete Cosine, Laplace, z, Wavelet, etc.

time, t frequency, fF

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

analysisanalysis

synthesissynthesis

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

General Transform as General Transform as problemproblem--solving toolsolving tool


Fourier Analysis - ApplicationsFourier Analysis - ApplicationsWide range of applications:Wide range of applications:

•• TelecommsTelecomms - GSM/cellular phones, codec's,

•• Electronics/ITElectronics/IT - most DSP-based applications

•• Imaging, image processing, Compression techniquesImaging, image processing, Compression techniques

•• MMultimedia: sound, image, audio, video

• Remote sensing: SAR

•• Industry/researchIndustry/research - X-ray spectrometry, chemical analysis (FT spectrometry), PDE solution, radar design,

•• MedicalMedical - PET scanner, CAT scans & MRI interpretation for sleep disorder & heart malfunction diagnosis,

•• Speech analysisSpeech analysis: speech recognition, speech synthesis,


Fourier Analysis - Tools Fourier Analysis - Tools Input Time Signal Frequency spectrum

∑−

=

−⋅=

1N

0n

Nnkπ2j

k es[n]N1c~

Discrete

DiscreteDFSDFSPeriodic (period T)

ContinuousDTFTAperiodic

DiscreteDFTDFT

nfπ2jen

s[n]S(f) −⋅∞+

−∞== ∑

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12time, tk

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8

time, tk

∑−

=

−⋅=

1N

0nN

nkπ2jes[n]

N1

kc~

**

**

Calculated via FFT**

dttfπj2

es(t)S(f)−∞+

∞−⋅= ∫

dtT

0

tωkjes(t)T1

kc ∫ −⋅⋅=Periodic (period T)

Discrete

ContinuousFTFTAperiodic

FSFSContinuous

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, ω = 2π/T, s[n]=s(tn), N = No. of samples

SR2009



History Fourier TransformHistory Fourier Transform16691669: Newton stumbles upon light spectra (specter = ghost) but fails to recognise “frequency” concept (corpuscularcorpuscular theory of light, & no waves).

1818thth centurycentury: two outstanding problemstwo 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). BUTBUT peers’ consensus is that sum of sinusoids onlyonly represents smooth curves. Big blow to utility of such sums for all but Fourier ...

18071807: Fourier presents his work on heat conduction Fourier presents his work on heat conduction ⇒⇒ Fourier analysis born.Fourier analysis born.→ Diffusion equation ⇔ series (infinite) of sines & cosines. Strong criticism by peers

blocks publication. Work published, 1822 (“Theorie Analytique de la chaleur”).


History Fourier Transform -2History Fourier Transform -21919thth / 20/ 20thth centurycentury: two paths for Fourier analysis two paths for Fourier analysis -- Continuous & Discrete.Continuous & Discrete.

CONTINUOUSCONTINUOUS

→ 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)DISCRETE: Fast calculation methods (FFT)

→ 18051805 - Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866).

→ 19651965 - 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.


Bases of Vector Spaces Bases of Vector Spaces

v = xe1+ye2+ze3

e1

e2

e3

v is a linear combination of the

basis vectors ei (i = 1,2,3)

eit

e2it

e3it

f(t) = k1eit+ k2e2it+k3e3it

f is a linear combination of the

basis functions eit,e2it,e3it


Fourier CoefficientsFourier Coefficients

w

v

cw

v-cw

Let <.,.> an in-product for our vector space V. Then we calculate

the Fourier coefficient of v in V with respect to (basis) vector w by:

cw is the component of v along the direction of w.><><

=wwwvc

,,

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Another Space, Another BaseAnother Space, Another Base

F(t) = sin(2π.t)


Another Space, Another BaseAnother Space, Another Base

F(t) = sin(2π.2t)F(t) = sin(2π.t) F(t) = sin(2π.3t)

F(t) = cos(2π.t) F(t) = cos(2π.2t) F(t) = cos(2π.3t)

{ cos(2π.kt), sin(2 π.kt) }k forms a orthogonal basis


Linear Combination of FunctionsLinear Combination of Functions

F(t) = sin(2π.2t)

F(t) = sin(2π.t)

F(t) = sin(2π.t) + sin(2π.2t)



F(t) = sin(2π.2t)F(t) = sin(2π.3t)

F(t) = sin(2π.t) + sin(2π.2t) + sin(2π.3t)

F(t) = sin(2π.t) + sin(2π.2t)

SR2009





F(t) = sin(2π.t) + sin(2π.2t) F(t) = sin(2π.t) + sin(2π.2t) + sin(2π.3t)




F(t) = sin(2π.t)

F(t) = cos(2π.t) – 2.sin(2π.2t)

Phase Shift:


Low Band Pass FiltersLow Band Pass Filters

F(t) = sin(2π.t) + sin(2π.2t) F(t) = sin(2π.t) + sin(2π.2t) + <- low freq.

sin(2π.30t) + sin(2π.120t) <- high freq.


Fourier TransformsFourier Transforms

The values h of some process can be described in the time domain:• h(t) values as a function of time t (-∞< t <∞)

The same process can be described as amplitudes and phases (complex values) H in the frequency domain:

• H(f) values as a function of frequency f (-∞< f <∞)

One can transform the representation in the time domain to the representation in the frequency domain by using the Fourier Transform equation:

∫∞

∞−= dtethfH iftπ2).()(

∫∞

∞−

−= dfefHth iftπ2).()(And back, using the FT-equation:

SR2009



Fourier Series (FS)Fourier Series (FS)

** see next slidesee next slide

A A periodicperiodic function s(t) satisfying function s(t) satisfying Dirichlet’sDirichlet’s conditions conditions ** can be expressed can be expressed as a Fourier series, with harmonically related sine/cosine termsas a Fourier series, with harmonically related sine/cosine terms..

[ ]∑∞+

=⋅−⋅+=

1kt)ω(ksinkbt)ω(kcoska0as(t)

a0, ak, bk : Fourier coefficients.

k: harmonic number,

T: period, ω = 2π/TFor all t but discontinuitiesFor all t but discontinuities

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

∫⋅=T

0s(t)dt

T1

0a

∫ ⋅⋅=T

0dtt)ωsin(ks(t)

T2

kb-

∫ ⋅⋅=T

0dtt)ωcos(ks(t)

T2

ka

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

analysis

analysis

synthesis

synthesis


Fourier Series Convergence Fourier Series Convergence

s(t) piecewise-continuous;

s(t) piecewise-monotonic;

s(t) absolutely integrable , ∞<∫T

0dts(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 convergenceRate of convergence


Fourier Series Analysis - 1Fourier Series 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

squa

re s

igna

l, sw

(t)

2 π

0π

0

2π

π1)dt(dt

2π1

0a =

−+⋅= ∫ ∫

0π

0

2π

πdtktcosdtktcos

π1

ka =

−⋅= ∫ ∫

{ }=−⋅⋅

==

−⋅= ∫ ∫ kπcos1πk

2...π

0

2π

πdtktsindtktsin

π1

kb-

⋅

=evenk,0

oddk,πk

4

1ω2πT =⇒=

(zero average)(zero average)

(odd function)(odd function)

...t5sinπ5

4t3sinπ3

4tsinπ4sw(t) +⋅⋅

⋅+⋅⋅

⋅+⋅=


Fourier Series Analysis - 2Fourier Series Analysis - 2

-π/2

f1 3f1 5f1 f

f1 3f1 5f1 f

rk

θk

4/π

4/3π

fk=k ω/2π

rK = amplitude, θK = phase

rk

θk

ak

bkθk = arctan(bk/ak)rk = ak

2 + bk2

zk = (rk , θk)

vk = rk cos (ωk t + θk)

Polar

Fourier spectrum Fourier spectrum representationsrepresentations

∑∞

==

0k(t)kvs(t)

Rectangularvk = akcos(ωk t) - bksin(ωk t)

4/π

f1 2f1 3f1 4f1 5f1 6f1 f

f1 2f1 3f1 4f1 5f1 6f1 f

4/3π

ak

-bk

Fourier spectrum Fourier spectrum of squareof square--wave.wave.

SR2009



[ ]∑=

⋅=7

1ksin(kt)kb-(t)7sw

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

[ ]∑=

⋅=5

1ksin(kt)kb-(t)5sw [ ]∑

=⋅=

3

1ksin(kt)kb-(t)3sw

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

[ ]∑=

⋅=1

1ksin(kt)kb-(t)1sw

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

[ ]∑=

⋅=9

1ksin(kt)kb-(t)9sw

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

[ ]∑=

⋅=11

1ksin(kt)kb-(t)11sw

Fourier Series SynthesisFourier Series SynthesisSquare wave reconstruction Square wave reconstruction

from spectral termsfrom spectral terms

Convergence may be slow (~1/k) - ideally need infinite terms.Practically, series truncated when remainder below computer tolerance(⇒ errorerror). BUTBUT … Gibbs’ Phenomenon.


Gibbs PhenomenonGibbs Phenomenon

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10t

squa

re s

igna

l, sw

(t)

[ ]∑=

⋅=79

1kk79 sin(kt)b-(t)sw

Overshoot exist @ Overshoot exist @ each discontinuityeach 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 casein this case.

• First observed by Michelson, 1898. Explained by Gibbs.


Fourier Series Time Shifting Fourier Series Time Shifting

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 t

squa

re s

igna

l, sw

(t)

2 πFS of even function:FS of even function:ππ/2/2--advanced sadvanced squarequare--wavewave

π

f1 3f1 5f1 7f1 f

f1 3f1 5f1 7f1 f

rk

θk

4/π

4/3π

00a =

0=kb-

=⋅

−

=⋅

=

even.k,0

11...7,3,kodd,k,πk

4

9...5,1,kodd,k,πk

4

ka

(even function)

(zero average)

phas

eph

ase

ampli

tude

ampli

tude

Note: amplitudes unchanged BUTBUTphases advance by k⋅π/2.


Complex Fourier SeriesComplex Fourier Series

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

rθ

a

bθ = arctan(b/a)r = a2 + b2

z = r ejθ

2eecos(t)

jtjt −+=

j2eesin(t)

jtjt

⋅−

=−

Euler’s notation:

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

NoteNote: c-k = (ck)*

( ) ( )kbjka21

kbjka21

kc −⋅−−⋅=⋅+⋅=

0a0c =Link to FS real Link to FS real coeffscoeffs..

∑∞

−∞=⋅=

k

tωkjekcs(t)

∫ ⋅⋅=T

0dttωkj-es(t)

T1

kcanalysis

analysis

synthesis

synthesis

SR2009



Fourier Series PropertiesFourier Series PropertiesTime FrequencyTime Frequency

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)

∑∞

−∞=−

mm)U(m)S(k

td)tT

0u()ts(t

T1

∫ ⋅−⋅

S(k)e Ttk2πj

⋅⋅

−

s(t)Ttm2πj

e ⋅+

)ts(t −


Fourier Series – ‘Oddities’Fourier Series – ‘Oddities’

k = - ∞, … -2,-1,0,1,2, …+ ∞, ωk = kω, φk = ωkt, phasor turns anti-clockwise.

Negative k ⇒ phasor turns clockwise (negative phase φk ), equivalent to negative time t,

⇒ time reversal.

Negative frequencies & time reversal

Fourier components {Fourier components {uukk} form orthonormal base of signal space:} 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 projectionprojection of signal s(t) on component uk

Orthonormal base

∫ ⋅=⊗T

o

*mkmk dtuuuu


0 50 100 150 200

k f

Wk/W0

10-3

10-2

10-1

12

Wk = 2 W0 sync2(k δ)

W0 = (δ sMAX)2

∞

=+⋅= ∑

1k 0WkW10WW

Fourier Series - Power Fourier Series - Power

•• FS convergence FS convergence ~1/k~1/k

⇒⇒ lower frequency terms

Wk = |ck|2 carry most power.

•• WWkk vs. vs. ωωkk: Power density spectrum: Power density spectrum.

Example

sync(u) = sin(π u)/(π u)

Pulse train, duty cycle Pulse train, duty cycle δδ = 2 = 2 τ τ / T/ T

T

2 τ

t

s(t)

bk = 0 a0 = δ sMAX

ak = 2δsMAX sync(k δ)

Average power WAverage power W : s(t)s(t)T

odt2s(t)

T1W ⊗≡= ∫

∑∑∞

=

++=

∞

−∞==

1k

2kb2

ka212

0ak

2kcW

Parseval’s TheoremParseval’s Theorem


Fourier Series of Some WaveformsFourier Series of Some Waveforms

SR2009



Discrete Fourier Series (DFS)Discrete Fourier Series (DFS)

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

DFS generate periodic ckwith same signal period

∑−

=⋅=

1N

0kN

nk2πjekcs[n] ~

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

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

mk,δ1N

0nN

-m)n(k2πje

N1

=−

=∑

Kronecker’s delta

Orthogonality in DFS:

synthesis

synthesis

∑−

=

−⋅=

1N

0nN

nk2πjes[n]

N1

kc~

Note:Note: ck+N = ck ⇔ same period N i.e. time periodicity propagates to frequencies!i.e. time periodicity propagates to frequencies!

DFS defined as:DFS defined as:

~~~~

analysis

analysis


Discrete Fourier Series AnalysisDiscrete Fourier Series AnalysisDFS of periodic discreteDFS of periodic discrete

11--Volt squareVolt square--wavewave

⋅

−−

±+=

=

otherwise,

Nkπsin

NkLπsin

NN

1)(Lkπje

2N,...N,0,k,NL

kc~

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

θk

-0.4π

0.2 0.24 0.24

0.6 0.6

0.24

1

0.24

-0.2π

0.4π

0.2π

-0.4π

-0.2π

0.4π

0.2π

0.6 ck ~

ampli

tude

ampli

tude

phas

eph

ase

Discrete signals Discrete signals ⇒⇒ periodic frequency spectra.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 NN, duty factor L/NL/N


Discrete Fourier Series PropertiesDiscrete Fourier Series PropertiesTime FrequencyTime 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)

∑−

=⋅

1N

0hh)-S(h)U(k

N1

∑−

=−⋅

1N

0mm]u[ns[m]

S(k)e Tmk2πj

⋅⋅−

s[n]Tth2πj

e ⋅+


References-1References-1

1. On bandwidth, David Slepian, IEEE Proceedings, Vol. 64, No 3, pp 291 - 300.

2. The Shannon sampling theorem - Its various extensions and applications: a tutorial review, A. J. Jerri, IEEE Proceedings, Vol. 65, no 11, pp 1565 – 1598.

3. What every computer scientist should know about floating-point arithmetic, David Goldberg.

4. IEEE Standard for radix-independent floating-point arithmetic, ANSI/IEEE Std 854-1987.

PapersPapers

1.Understanding digital signal processing, R. G. Lyons, Addison-Wesley Publishing, 1996.

2.The scientist and engineer’s guide to digital signal processing, S. W. Smith, at http://www.dspguide.com.

3.Discrete-time signal processing, A. V. Oppeheim & R. W. Schafer, Prentice Hall, 1999.

BooksBooks

SR2009



References - 2References - 2

1. Tom, Dick and Mary discover the DFT, J. R. Deller Jr, IEEE Signal Processing Magazine, pg 36 - 50, April 1994.

2. On the use of windows for harmonic analysis with the Discrete Fourier Transform, F. J. Harris, IEEE Proceedings, Vol. 66, No 1, January 1978.

3. Some windows with a very good sidelobe behaviour, A. H. Nuttall, IEEE Trans. on acoustics, speech and signal processing, Vol ASSP-29, no. 1, February 1981.

4. Some novel windows and a concise tutorial comparison of windows families, N. C. Geckinli, D. Yavuz, IEEE Trans. on acoustics, speech and signal processing, VolASSP-26, no. 6, December 1978.

5. Study of the accuracy and computation time requirements of a FFT-based measurement of the frequency, amplitude and phase of betatron oscillations in LEP, H.J. Schmickler, LEP/BI/Note 87-10.

6. Causes et corrections des erreurs dans la mesure des caracteristiques des oscillations betatroniques obtenues a partir d’une transformation de Fourier, E. Asseo, CERN PS 85-9 (LEA).

Papers


References - 3References - 3

1. The Fourier Transform and its applications, R. N. Bracewell, McGraw-Hill, 1986.

2. A History of scientific computing, edited by S. G. Nash, ACM Press, 1990.

3. Introduction to Fourier analysis, N. Morrison, John Wiley & Sons, 1994.

4. The DFT: An owner’s manual for the Discrete Fourier Transform, W. L. Briggs, SIAM, 1995.

5. The FFT: Fundamentals and concepts, R. W. Ramirez, Prentice Hall, 1985.

Books

7. Precise measurements of the betatron tune, R. Bartolini et al., Particle Accel., 1996, vol. 55, pp 247-256.

8. How the FFT gained acceptance, J. W. Cooley, IEEE Signal Processing Magazine, January 1992.

9. A comparative analysis of FFT algorithms, A. Ganapathiraju et al., IEEE Trans.on Signal Processing, December 1997.

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