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7/24/2019 Fourier Analisys II
http://slidepdf.com/reader/full/fourier-analisys-ii 1/26
FOURIER ANALYSIS
PART 2:
Cont inuous & Disc rete Four ier
Transforms
Maria Elena Angoletta
AB/BDI
DISP 2003, 27 February 2003
7/24/2019 Fourier Analisys II
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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
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 3 / 26
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
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 4 / 26
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(? ⋅−⋅=
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 5 / 26
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
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 6 / 26
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 !
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 7 / 26
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
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 8 / 26
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 −=
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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
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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.
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 11 / 26
-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)
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M . E. Angoletta - DISP2003 - Fourier analysis - Part 2.1 12 / 26
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
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FT of main waveformsFT of main waveforms
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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:
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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
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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)
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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 −=
⋅=
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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 ~
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DFT ExamplesDFT Examples
DFT plots are
sampled version ofwindowed DTFT
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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
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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).
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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
*
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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)
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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
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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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COFFEE BREAKCOFFEE BREAK
Be back in ~15 minutesBe back in ~15 minutes
Coffee in room #13Coffee in room #13