fourier analisys ii

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FOURIER ANALYSIS

PART 2:

Cont inuous & Disc rete Four ier

Transforms

Maria Elena Angoletta

AB/BDI

DISP 2003, 27 February 2003



M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 2 / 26

TOPICSTOPICS

1. Infinite Fourier Transform (FT)

2. FT & generalised impulse

3. Uncertainty principle

4. Discrete Time Fourier Transform (DTFT)

5. Discrete Fourier Transform (DFT)

6. Comparing signal by DFS, DTFT & DFT

7. DFT leakage & coherent sampling




Fourier analysis - toolsFourier analysis - toolsInput Time Signal Frequency spectrum

∑−

=

−⋅=

1N

0n

N

nkp2 j

es[n]N

1kc~

Discrete

DiscreteDFSDFSPeriodic(per iod T)

ContinuousDTFT Aperiodic

DiscreteDFT DFT

nf p2 je

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

∑−

=

−⋅=

1N

0n

N

nkp2 j

es[n]N

1kc~

dttf p j2

es(t)S(f)−∞+

∞− ⋅= ∫

dt

T

0

t?k jes(t)T

1kc ∫ −⋅⋅=Periodic

(per iod T) Discrete

ContinuousFT FT Aperiodic

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 = 2 π /T , s [n ]=s (t n ), N = No . o f samp les




Fourier Integral (FI)Fourier Integral (FI)Fourier analysis tools

for aperiodic signals.

{ }∫ ∞+

⋅+⋅=0

d?t)sin(?)B(?t)cos(?) A(?s(t)

Any aperiodic signal s(t) can be expressed as aFourier integral if s(t) piecewise smooth(1) in anyfinite interval (-L,L) and absolute integrable(2) .

Fourier Integral TheoremFourier Integral Theorem

(3)

+∞<∞+

∞∫ dt-

s(t)(2)

s(t) continuous,

s’(t) monotonic(1)

∫ ∞+

∞−

⋅= dtt)cos(?s(t)p

1) A(? ∫

∞+

∞−

⋅= dtt)sin(?s(t)p

1)B(?(3)

FourierFourier

TransformTransform

(Pair)(Pair) -- FTFT

dt

t? j

es(t))S(?

−∞+∞− ⋅= ∫ a n a l y

s i s

a n a l y s i s

?dt? j

e)?S(p2

1s(t) ∫

∞+∞−

⋅⋅= s y

n t h e

s i s

s y n t h e

s i s

Complex formComplex form

Real Real --toto--complex link complex link

[ ])B(? j) A(?p)S(? ⋅−⋅=




Let’s summarise a littleLet’s summarise a little

FS

Signal →

Time

Frequency

FI

ak, bk A(ω), B(ω)

Periodic Aperiodic

ck C(ω)

Domain↓

real

complex

FT




From FS to FTFrom FS to FTFS moves to FT as period T

increases: continuous spectrum

2 τ

0 50 100 150 200

f

|S(f)|

FT

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

k f

|ak|

T = 0.05

0

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200

k f

|ak|

T = 0.1

0

0.05

0.10.15

0.2

0.25

0 50 100 150 200

k f

|ak|

T = 0.2

Pulse train, width 2 τ = 0.025

T

2 τ

t

s t

Note: |ak |→2 a0 as k →0 ⇒ 2 a0 is plotted at k=0

Frequency spacingFrequency spacing→0 ! 0 !




Getting FT from FSGetting FT from FS

∫ −

⋅⋅=T/2

T/2

dtt?k j-es(t)T

1kc

∑∞

−∞=⋅=

k

t?k jekcs(t)

∆f = ∆ω/(2π) = 1/Tfrequency spacingfrequency spacing

As ∆f →0 , replace ∆f , ωK ,

by df, 2πf,

∑∞

−∞=k?

∫ ∞+

∞−

FSFS defined defined

∫ −

⋅=≡T/2

T/2

dtt? j-es(t)

?f

?c?G kk

k

∑∞

−∞=⋅=

k

kk

?

?f t? je?Gs(t)

2

kk ?c

T/2

T/2

dtt? j-es(t)?f kc ≡

−

⋅⋅= ∫

∑∞

−∞=⋅=

k

kk

?

t? je?cs(t)1 dtt? jes(t)?G

0? f limS(f)

k

−⋅∞+

∞−

=→

= ∫

∫ ∞+

∞−

⋅= df ft j2peS(f)s(t)

FTFT defined defined




FT & Dirac’s DeltaFT & Dirac’s Delta

The FT of the generalised impulseThe FT of the generalised impulse δδ

((DiracDirac) is a complex exponential) is a complex exponential

<<

=−⋅∫ otherwise,0

b0taif ,)0y(t

dt)0t(td

b

a

y(t)

=

≠=−

0ttundefined,

0ttif ,0

)0t(td

DiracDirac’s’s δδ defineddefined

)0y(tdt)0t(td

-

y(t) =−∞+

∞

⋅∫ Hence

FT of an infinite train ofFT of an infinite train of δδ::

t

T

f

1/T ∑∑∑

∞+

−∞=

∞+

−∞=

∞+

−∞= ⋅−==

− mm

T?m j

km)T

p2

(?dT

2p

ekT)(tdFT

a.k.a. Sampling function, Shah(T) = ? (T) or “comb”a.k.a. Sampling function, Shah(T) = ? (T) or “comb”

Not e Not e :: δδ && ? = “generalised “functions? = “generalised “functions

{ } 0tf p2 je)0t(tdFT −=−

FT ofFT of DiracDirac’s’s δδpropertyproperty

)a(?d2peFT ta j −=




FT propertiesFT properties

Linearity a·s(t) + b·u(t) a·S(f)+b·U(f)

Multiplication s(t)·u(t)

Convolution S(f)·U(f)

Time shifting

Frequency shifting

Time reversal s(-t) S(-f)

Differentiation j2πf S(f)

Parseval’s identity ∫ h(t) g*(t) dt = ∫ H(f) G*(f) df

Integration S(f)/(j2πf )

Energy & Parseval’s(E is t-to-f invariant)

Time FrequencyTime Frequency

f d)f U()f S(f ∫ ∞+

∞−

⋅−

td)t

-

u()ts(t∫ ∞+

∞

⋅−

S(f)tf 2p je ⋅−

s(t)f p2 je ⋅+

)ts(t −

∫ ∫ ∞+

∞

∞+

∞

==

-

df 2

S(f)

-

dt2

s(t)E

dt

ds(t)

∫ ∞

t

-

dus(u)

)f -S(f




FT - uncertainty principleFT - uncertainty principle

Fourier uncertaintyprinciple

⇒ ∆t•∆f ≥ 1/4π

ImplicationsImplications• Limited accuracy on simultaneous observation of s(t) & S(f).

• Good time resolution (small ∆t) requires large bandwidth ∆f & vice-versa.

For effective duration∆

t & bandwidth∆

f∃ γ > 0 ∆t•∆f ≥ γ uncertainty productuncertainty product

Bandwidt h Theorem Bandwidt h Theorem

For Energy Signals:

E=∫ |s(t)|2dt = ∫ |S(f)|2df < ∞

dts(t)tE

1t

22 ∫ +∞

∞−

⋅⋅= df S(f)f E

1f

22 ∫ +∞

∞−

⋅⋅=

Define mean valuesDefine mean values

dts(t))t(tE

1?t

22 ⋅−⋅= ∫ +∞

∞−

df S(f))f (f E

1? f

22 ⋅−⋅= ∫ +∞

∞−

Define std. dev.Define std. dev.




-20 -10 0 10 20

f/Hz

|S(f)|2

10-5

10-4

10-1

-0.1

0

0.1

0.2

-20 -10 0 10 20

f/Hz

S(f)

τ = 0.1

-τ τ t

s(t)1

-0.1

0

0.1

0.2

0.3

0.4

-20 -10 0 10 20

f/Hz

S(f)

-20 -10 0 10 20

f/Hz

|S(f)|2

10-5

10-4

10-1

10-3

10-2

1

τ = 0.2

-τ τ t

s(t)1

τ = 0.4

-τ τ t

s(t)1

-0.2

0

0.2

0.4

0.6

0.8

-20 -10 0 10 20

f/Hz

S(f)

-20 -10 0 10 20

f/Hz

|S(f)|2

10-5

10-4

10-1

10-3

10-2

1

FT - exampleFT - exampleFT of 2τ-wide

square window

Choose

∆t = |∫ s(t)/s(0) dt| = 2τ,∆f = |∫ S(f)/S(0) df|=1/(2τ) = half distance btwn

first 2 zeroes (f 1,-1 = ±1/2τ) of S(f)then: ∆t · ∆f = 1

Fourier uncertaintyFourier uncertainty

Power Spectral Density

(PSD) vs. frequency f plot.

Note:Note: Phases unimportant!

S(f) = 2τ sMAX sync(2f τ)




FT - power spectrumFT - power spectrum

POTS = Voice/Fax/modem PhoneHPNA = Home Phone Network

Phone signals PSD masksPhone signals PSD masks

US = UpstreamDS = Downstream

From power spectrum we can deduce if signals coexist without interfering!

Power Spectral Density,

PSD(f) = dE/df = |S(f)|2




FT of main waveformsFT of main waveforms




Discrete Time FT (DTFT)Discrete Time FT (DTFT)NoteNote: continuous frequency domain!

(frequency density function)

Holds for aperiodic signals

∑−

=

−⋅=

1N

0n

N

nkp2 j

es[n]N

1kc

~

n

s[n] 1 period

n

s[n]

∫ ⋅=2p

0

nf p2 j df S(f)e2p

1s[n] s y

n t h e s i s

s y n t h e s

i s

nf p2 j

n

es[n]S(f) −+∞

−∞=⋅= ∑ a n

a l y s i s

a n a l y s i s

Obtained from DFS as N → ∞

DTFT defined as:DTFT defined as:




DTFT - convolutionDTFT - convolution

Digital Linear Time Invariant systemDigital Linear Time Invariant system: obeys superposition principle.

∑∞=

⋅−=∗=0m

h[m]m]x[nh[n]x[n]y[n]x[n] h[n]

ConvolutionConvolution

X(f) H(f) Y(f) = X(f) · H(f)

DIGITAL LTISYSTEM

h[n]

x[n] y[n]

h[t] = impulse response

DIGITALLTI

SYSTEM0 n

δ[n]1

0 n

h[n]

0 f

DTFT(δ[n])

1




DTFT - Sampling/convolutionDTFT - Sampling/convolution

s[n] * u[n] ⇔ S(f) · U(f) ,

s[n] · u[n] ⇔ S(f) * U(f)

(From FT properties)

Time Frequency

t f

s t S f

t f

ts fs u(t) U f)

n f

s” n S” f

Sampling s(t)

Multiply s(t) by

Shah = ? (t)




Discrete FT (DFT)Discrete FT (DFT)

∑−

=⋅=

1N

0k

N

nk2p j

ekcs[n] ~ s y n t h e

s i s

s y n t h e

s i s

DFT defined as:DFT defined as:

Note: Note: ck+N = ck ⇔ spectrum has period N~~~~

∑−

=

−⋅=

1N

0n

N

nk2p j

es[n]N

1kc

~ a n a l y s i s

a n a l y s i s

Ø Applies to discrete time and frequency signals.Ø Same form of DFS but for aperiodic signals:

signal treated as periodic for computational purpose only.

DFT bins located @ analysis frequencies fm

DFT ~ bandpass filters centred @ f m

Frequency resolution

Analysis frequencies f Analysis frequencies f mm

1N...20,m,N

f mf Sm −=

⋅=




DFT - pulse & sinewaveDFT - pulse & sinewave

ck = (1/N) e-jπk(N-1)/N

sin(πk)/ sin(πk/N)

~~

a) rectangular pulse, width Na) rectangular pulse, width N

r[n] =

1 , if 0≤n≤N-1

0 , otherwise

b) realb) real sinewavesinewave, frequency f, frequency f00 = L/N= L/N

cs[n] = cos(j2πf 0n)

ck = (1/N) e jπ{(Nf 0-k)-(Nf 0 -k)/N} (½) sin{π(Nf 0-k)}/ sin{π(Nf 0-k)/N)} +

(1/N) e jπ{(Nf 0+k)-(Nf 0+k)/N} (½) sin{π(Nf 0+k)}/ sin{π(Nf 0+k)/N)}

~~

i.e. L complete cycles in N sampled points

-5 0 1 2 3 4 5 6 7 8 9 10 n0 N

s[n]1

0 1 2 3 4 5 6 7 8 9 10 k

1 1

ck ~




DFT ExamplesDFT Examples

DFT plots are

sampled version ofwindowed DTFT




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

0h

h)-S(h)U(kN

1

∑−

=−⋅

1N

0m

m]u[ns[m]

S(k)e T

mk2p j

⋅⋅

−

s[n]T

th2p j

e ⋅

+

DFT propertiesDFT propertiesTime FrequencyTime Frequency




DTFT vs. DFT vs. DFSDTFT vs. DFT vs. DFS

t0 T/2 T 2T f

s[n] S(f)

f

~

cK

t

s”[n] DFTIDFT

(a)

(a) Aperiodic discrete signal.

(b)

(b) DTFT transform magnitude.

(c)

(c) Periodic version of (a).

(d)

(d) DFS coefficients = samples of (b).

(e)

(e) Inverse DFT estimates a single period of s[n]

(f)

(f) DFT estimates a single period of (d).




DFT – leakageDFT – leakage

Spectral components belonging to frequencies betweenSpectral components belonging to frequencies between

two successive frequency bins propagate to all bins.two successive frequency bins propagate to all bins.LeakageLeakage

Ex: 32-bins DFT of 1 VP sinusoid sampled @ 32kHz. 1 kHz frequency resolution.

(b)(b) 8.5 kHz sinusoid

(c)(c) 8.75 kHz sinusoid

(a)(a) 8 kHz sinusoid

* N·Magnitude

*




1. Cosine wave

DFT - leakage exampleDFT - leakage examples(t) FT{s(t)}

2. Rectangular window4. Sampling function1. Cosine wave

0.25 Hz Cosine wave

3. Windowed cos wave5. Sampled windowed

wave

Leakage caused by sampling for a nonLeakage caused by sampling for a non--integer number of periodsinteger number of periods

s[n] · u[n] ⇔ S(f) * U(f)(Convolution)




2. Rectangular window4. Sampling function1. Cosine wave

1. Cosine wave3. Windowed cos wave5. Sampled windowedwave

DFT - coherent samplingDFT - coherent samplings(t) FT{s(t)}

Coherent sampling: NC input cycles exactly into NS = NC (f S/f IN) sampled points.

s[n] ·u[n] ⇔ S(f) * U(f)(Convolution)

0.2 Hz Cosine wave




DFT – leakage notesDFT – leakage notes

1. Affects Real & Imaginary DFT parts magnitude & phase.

2. Has same effect on harmonics as on fundamental frequency.

3. Affects differently harmonically un-related frequencycomponents of same signal (ex: vibration studies).

4. Leakage depends on the form of the window (so far onlyrectangular window).

LeakageLeakage

After coffee we’ll see how to takeadvantage of different windows.




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fourier analisys ii

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