FOURIER ANALYSIS PART 2: Continuous & Discrete Fourier Transforms Dr.M.A.Kashem Asst. Professor CSE,DUET

FOURIER ANALYSISPART 2:

Continuous & Discrete Fourier Transforms

Dr.M.A.Kashem

Asst. Professor CSE,DUET

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 - tools Fourier analysis - tools Input Time Signal Frequency

spectrum

1N

0n

Nnkπ2

jes[n]

N

1kc~

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 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

Nnkπ2

jes[n]

N

1kc~

dttfπj2

es(t)S(f)

dtT

0

tωkjes(t)T

1kc 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, = 2/T, s[n]=s(tn), N = No. of samples


Fourier Integral (FI)Fourier Integral (FI)Fourier analysis tools for aperiodic signals.

0

dωt)sin(ω)B(ωt)cos(ω)A(ωs(t)

Any aperiodic signal s(t) can be expressed as a Fourier integral if s(t) piecewise smooth(1) in any finite 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(ωs(t)

π

1)A(ω

dtt)sin(ωs(t)

π

1)B(ω(3)

Fourier Fourier Transform Transform (Pair) - FT(Pair) - FT

dttωj

es(t))S(ω

analysis

analysis

ωdtωj

e)ωS(π2

1s(t)

synth

esis

synthesis

Complex formComplex form

Real-to-complex linkReal-to-complex link

)B(ωj)A(ωπ)S(ω


Let’s summarise a little Let’s summarise a little

FS

Signal

Time

Frequency

FI

ak, bk A(), B()

Periodic

Aperiodic

ck C()

Domain

realcomplex

FT


From FS to FTFrom FS to FTFS moves to FT as period T increases: continuous spectrum

2

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

0.1

0.15

0.2

0.25

0 50 100 150 200

k f

|ak|

T = 0.2

Pulse train, width 2 = 0.025

T

2

t

s(t)

Note: |ak|2 a0 as k0 2 a0 is plotted at k=0

Frequency spacing Frequency spacing 0 !0 !


Getting FT from FSGetting FT from FS

T/2

T/2

dttωkj-es(t)T

1kc

k

tωkjekcs(t)

f = /(2) = 1/T frequency spacingfrequency spacing

As f 0 , replace f , K ,

by df, 2f,

kω

FS FS defineddefined

T/2

T/2

dttωj-es(t)Δf

ωcωΓ kk

k

k

kk

ωΔftωjeωΓs(t)

2

kk ωc

T/2

T/2

dttωj-es(t)Δfkc

k

kk

ω

tωjeωcs(t)1 dttωjes(t)ωΓ

0ΔflimS(f)

k

dfftj2πeS(f)s(t)

FT FT defineddefined


FT & Dirac’s DeltaFT & Dirac’s DeltaThe FT of the generalised The FT of the generalised impulse impulse (Dirac) is a complex (Dirac) is a complex exponentialexponential

otherwise,0

b0taif,)0y(t

dt)0t(tδb

a

y(t)

0ttundefined,

0ttif,0

)0t(tδ

Dirac’s Dirac’s defined defined

)0y(tdt)0t(tδ

-

y(t)

Hence

FT of an infinite train of FT of an infinite train of ::

t

T

f

1/ T

mm

Tωmj

k

m)T

π2(ωδ

T

2πekT)(tδFT

a.k.a. Sampling function, Shah(T) = Щ(T) or a.k.a. Sampling function, Shah(T) = Щ(T) or “comb”“comb”NoteNote: : & Щ = “generalised “functions & Щ = “generalised “functions

0tfπ2je)0t(tδFT

FT of Dirac’sFT of Dirac’s propertyproperty

)α(ωδ2πeFT tαj


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 j2f S(f)

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

Integration S(f)/(j2f )

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

Time FrequencyTime Frequency

fd)fU()fS(f

td)t

-

u()ts(t

S(f)tf2πje

s(t)fπ2je

)ts(t

-

df2S(f)

-

dt2s(t) E

dt

ds(t)

t

-

dus(u)

)f-S(f


FT - uncertainty principle FT - uncertainty principle

Fourier uncertainty principle

t•f 1/4

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

> 0 t•f uncertainty uncertainty productproduct

Bandwidth TheoremBandwidth Theorem

For Energy Signals:E= |s(t)|2dt = |S(f)|2df <

dts(t)tE

1t 22

dfS(f)fE

1f 22

Define mean valuesDefine mean values

dts(t))t(tE

1Δt 22

dfS(f))f(f

E

1Δf 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)

= 0.1

- 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

= 0.2

- t

s(t) 1

= 0.4

- 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 2-wide square window

Chooset = |s(t)/s(0) dt| = 2,f = |S(f)/S(0) df|=1/(2) = half distance btwn first 2 zeroes (f1,-1 = 1/2) of S(f)

then: t · f = 1

Fourier uncertaintyFourier uncertainty

Power Spectral Density (PSD) vs. frequency f plot. Note: Note: Phases unimportant!

S(f) = 2 sMAX sync(2f)


FT - power spectrumFT - power spectrum

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

Phone signals PSD Phone signals PSD masksmasks

US = Upstream DS = 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

Nnkπ2

jes[n]

N

1kc~

n

s[n] 1 period

n

s[n]

2π

0

nfπ2j dfS(f)e2π

1s[n]synthesis

synthesis

nfπ2j

n

es[n]S(f)

analysis

analysis

Obtained from DFS as N

DTFT defined as:DTFT defined as:


DTFT - convolution DTFT - convolutionDigital 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)

DI GI TAL LTI SYSTEM

h[n]

x[n] y[n]

h[t] = impulse response

DI GI TAL LTI

SYSTEM 0 n

[n] 1

0 n

h[n]

0 f

DTFT([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 = Щ(t)


Discrete FT (DFT)Discrete FT (DFT)

1N

0k

Nnk2π

jekcs[n] ~synth

esis

synthesis

DFT defined as:DFT defined as:

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

1N

0n

Nnk2π

jes[n]

N

1kc~analysis

analysis

Applies to discrete time and frequency signals.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 @ fm

Frequency resolution

Analysis frequencies fAnalysis frequencies fmm

1N...20,m,N

fmf Sm


DFT - pulse & sinewaveDFT - pulse & sinewave

ck = (1/N) e-jk(N-1)/N sin(k)/ sin(k/N) ~~

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

r[n] =1 , if 0nN-1

0 , otherwise

b) real sinewave, frequency fb) real sinewave, frequency f00 = L/N = L/N

cs[n] = cos(j2f0n)

ck = (1/N) ej{(Nf0-k)-(Nf0 -k)/N} (½) sin{(Nf0-k)}/ sin{(Nf0-k)/N)} +

(1/N) ej{(Nf0+k)-(Nf0+k)/N} (½) sin{(Nf0+k)}/ sin{(Nf0+k)/N)}

~~

i.e. L complete cycles in N sampled points

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

s[n] 1

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

1 1

ck ~


DFT Examples DFT Examples

DFT plots are sampled version of windowed 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 Tmk2π

j

s[n]Tth2π

je

DFT propertiesDFT propertiesTime FrequencyTime Frequency


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

t 0 T/ 2 T 2T f

s[n] S(f)

f

~ cK

t

s”[n] DFT I DFT

(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 – leakage DFT – leakage

Spectral components belonging to frequencies Spectral components belonging to frequencies between two successive frequency bins propagate between two successive frequency bins propagate to all bins.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 non-integer number of Leakage caused by sampling for a non-integer number of periodsperiods

s[n] · u[n] S(f) * U(f) (Convolution)


2. Rectangular window4. Sampling function1. Cosine wave

1. Cosine wave3. Windowed cos wave

5. Sampled windowed wave

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

}

Coherent sampling: NC input cycles exactly into NS = NC (fS/fIN) 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 frequency components of same signal (ex: vibration studies).

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

LeakageLeakage

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


COFFEE BREAKCOFFEE BREAK

Be back in ~15 minutesBe back in ~15 minutes

Coffee in room #13Coffee in room #13

Documents

FOURIER ANALYSIS PART 2: Continuous & Discrete Fourier Transforms Dr.M.A.Kashem Asst. Professor CSE,DUET