ECEN 5696 Fourier Optics
Professor Kelvin WagnerDept ECE, UCB 425, ECEE 232, x24661
What you will learn by completing this specialization
• Fourier transforms in time and space. 1D, 2D, 3D and 4D
• From Maxwell’s equations to diffraction and imaging
•Numerical techniques in wave opticsAberations and Beam Propagation
•Holography and Optical Information Processing
• Spatio-Temporal Fourier Optics and multiple wavelengths
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 1
Fourier Optics Learning Objectives
• Review Fourier transforms and develop deep intuitive understanding
• Generalize the Fourier transform to 2-D images and fields
• Construct arbitrary solutions to Maxwell’s Eqn as a superposition of plane waves
• Understand how waves propagate through space and are focused by lenses
• Develop a clear intuition for the propagation of plane waves and Gaussian beams
• Compare, contrast, and analyze coherent and incoherent imaging systems
• Formulate a wave theory of aberations and visualize them
• Develop numerical techniques for optical beam propagation as one line of code
• Discover the use of optical correlations for pattern recognition
• Invent holography to record and transform optical fields
• Extend the ideas of holography to computer generated and digital holography
• Further generalize the Fourier approach to the case of broadband fields
• Utilize the Fourier decomposition to invent and evaluate novel optical systems
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 2
Suggested References for AdditionalReading
Texts and suggested references:J. Goodman , Introduction to Fourier Optics, 3rd EdJ. Shamir, Optical Systems & ProcessesJ. Gaskill, Linear Systems, Fourier Transforms, and OpticsT. Cathey, Optical Information Processing and HolographyB. Saleh, Fundamentals of Photonics Chapter 4D. Brady, Optical Imaging and Spectroscopy, 2009D. Voelz, Computational Fourier Optics: A MATLAB Tutorial, 2011J. Schmidt Numerical Simulation of Optical Wave Propagation, 2011N. George Fourier Optics, 2012 on-line short manuscriptR.K. Tyson, Principles and Applications of Fourier Optics, 2014Kedar Khare, Fourier Optics and Computational Imaging , 2016
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 3
Course Outline
Linear Systems and Fourier TransformsSampling Theory and the Fast Fourier Transform (FFT)
2-D Systems and Transforms, OperatorsWave Propagation, momentum spaceDiffraction Theory
Beam Propagation MethodFranhoffer and Fresnel DiffractionCoherent Optical ImagingIncoherent ImagingWave theory of aberrationsHolography
Computer Generated HolographyDigital Holography
Optical Information ProcessingSynthetic Aperture Radar (SAR) and TomographyVolume Holography: 3-D Fourier TransformsSpatial and Temporal Fourier Optics: 4-D Fourier transformsVector Effects, Subwavelength Structures
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 4
Review of 1-D Fourier TransformsLearning Objectives and Outcomes
•Remember integral definition of Fourier transform– define operator representation
•Recognize equivalence of 1-D spatial FT and temporal FT
•Review some FT pairs of compact and singular functions
•Visually identify 1-D Fourier transform pair plots
• Summarize the properties of 1-D Fourier transforms
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 5
4 types of Fourier TransformsR.A. Roberts, C.T. Mullis, Digital Signal processing, Addison Wesley 1987
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 6
The 1-D Temporal Fourier Transform:Definitions Valid for finite support or
periodic signalsForward temporal Fourier transform (Hz)
G(f) =
ˆ
g(t)e−i2πftdt = Fg(t)
Inverse transformg(t) =
ˆ
G(f)ei2πftdf = F−1G(f)
Alternate definition using angular radian frequency ω = 2πf
Forward temporal Fourier transform (rad/sec)
G(ω) =
ˆ
g(t)e−iωtdt = Fg(t) ω = 2πf
Inverse transform
g(t) =1
2π
ˆ
G(ω)eiωtdω = F−1G(ω) ≡ F−1G(ω) dω = 2πdf
Note that these FT functions are scaled versions of each other G(ω) = G(ω/2π)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 7
Exactly Analogous 1-D Spatial FourierTransform: Definitions
Can similarly define FT in space using spatial frequency, u = fx [lines/mm], analogousto temporal frequency f [Hz], or use wavevector, kx [rad/mm], analogous to angularfrequency ω [rad/sec].
Forward 1-D spatial Fourier transform
G(u) =
ˆ
g(x)e−i2πuxdx = Fxg(x)
Inverse 1-D spatial Fourier transform
g(x) =
ˆ
G(u)ei2πuxdu = F−1x G(u)
or in terms of wavevector kx
G(kx) =
ˆ
g(x)e−ikxxdx = Fg(x) ≡ Fxg(x)
g(x) =1
2π
ˆ
G(kx)eikxxdkx = F−1G(kx) ≡ F−1x G(kx) ≡ F−1kx
G(kx)
Note that these FT functions are scaled versions of each other G(kx) = G(kx/2π)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 8
1-dimensional Fourier transforms
Well defined continuous functionsrect(t) = Π(t) sinc(f)
rect( tT ) = Π(tT
) T sinc(Tf)
sinc(t) rect(f)
tri(t) = Λ(t) sinc2(f)
sinc2(t) tri(f)
e−πt2 e−πf
2
e±iπt2 e±iπ/4e∓iπf
2
e−|t| 21+(2πf)2
e−|t|H(t) 11+i2πf = 1−i2πf
1+(2πf)2
sechπt sechπf
jinc(t) = J1(πt)2t
√1− (2f)2Π(f)
1|t|1/2 1
|f |1/2
Singular functions in t or f
δ(t) 1(f)
δ(ta
)= |a|δ(t) |a|1(f)δ(t− t0) e−i2πt0f
1iπt sgn(−f)
u(t) = H(t) 12δ(f) + 1
i2πf
ei2πf0t δ(f − f0)cos(2πf0t) 1
2 [δ(f−f0)+δ(f+f0)]
sin(2πf0t) 12i [δ(f−f0)−δ(f+f0)]
comb(t) comb(f)
comb(tT
) |T |comb(Tf)
tk (−1i2π
)kδ(k)(f)
(1i2π
)kδ(k)(t) fk
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 9
Visual Fourier Transform DictionaryDiscontinuous Functions
rect(t)=P(t)
-1 0 1 2time
sinc(t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 10
Visual Fourier Transform DictionaryExponentially Decaying Functions
gausssian(t)
-2 0 2 4time
gaussian(f)
-1 0 1 2freq
exp(t)
-2 0 2 4time
e-|t|
-2 0 2 4time
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 11
Visual Fourier Transform DictionaryImpulsive Functions
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 12
Visual Fourier Transform DictionarySingular Functions
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 13
The Temporal Fourier Transform:Properties
Linearityag(t) + bh(t) aG(f) + bH(f)
Conjugationg∗(±t) G∗(∓f)
Scale
g(αt)1
|α|G(f
α
)
Shiftg(t− t0) e−i2πft0G(f)
Modulationei2πf0tg(t) G(f − f0)
Derivative and Integration
dn
dxng(t) (i2πf)nG(f)
ˆ t
−∞g(τ )dτ
1
i2πfG(f) +
G(0)
2δ(f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 14
The 1-D Fourier Transform: Properties 2
Parseval’s Theoremˆ ∞
−∞|g(t)|2dt =
ˆ ∞
−∞|G(f)|2df
Convolutionˆ
g(t′)h(t− t′)dt′ G(f)H(f)
g(t)h(t)
ˆ
G(t′)H(t− t′)dt′
Correlationˆ
g(t′)h∗(t′ − t)dt′ G(f)H∗(f)ˆ
g(t′)g∗(t′ − t)dt′ |G(f)|2
Fourier Integral
FtF−1t g(t) = F−1t Ftg(t) = g(t)
FtFtg(t) = g(−t)Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 15
Operation x(t) = F−1f X(f) = F−1X (ω) Ff⇐⇒ X(f) =
ˆ ∞
−∞x(t)e−i2πftdt
F⇐⇒ X (ω) =ˆ ∞
−∞x(t)e−iωtdt
Linearity Ax(t) +By(t)Ff⇐⇒ AX(f) +BY (f)
F⇐⇒ AX (ω) +BY(ω)Conjugation x∗(t)
Ff⇐⇒ X∗(−f) F⇐⇒ X ∗(−ω)Scale x(at)
Ff⇐⇒ 1|a|X
(fa
) F⇐⇒ 1|a|X
(ωa
)
Mirror x(−t) Ff⇐⇒ X(−f) F⇐⇒ 2πX (−ω)Duality X(t) or X (t) Ff⇐⇒ x(−f) F⇐⇒ x(−ω)
Time shift x(t− T ) Ff⇐⇒ e−i2πfTX(f)F⇐⇒ e−iωTX (ω)
Frequency shift e−i2πf0tx(t) = e−iω0tx(t)Ff⇐⇒ X(f − f0) F⇐⇒ X (ω − ωs)
Time differentiation dx(t)dt
Ff⇐⇒ i2πfX(f)F⇐⇒ iωX (ω)
d2x(t)dt2
Ff⇐⇒ (i2πf)2X(f)F⇐⇒ iω)2X (ω)
dnx(t)dtn
Ff⇐⇒ (i2πf)nX(f)F⇐⇒ (iω)nX (ω)
frequency differentiation (−i)ntnx(t) Ff⇐⇒ 1(2π)n
dnX(f)dfn
F⇐⇒ dnX (ω)dωn
Time Integrationˆ t
−∞x(τ)dτ = u ∗ x Ff⇐⇒ X(f)
i2πf+ 1
2X(0)δ(f)
F⇐⇒ X (ω)iω
+ πX (0)δ(ω)
frequency integration x(t)t
Ff⇐⇒ˆ ∞
f
X(f ′)df ′ F⇐⇒ˆ ∞
ω
X (ω′)dω′
Convolution x(t) ∗ y(t) =ˆ ∞
−∞x(τ)y(t− τ)dτ Ff⇐⇒ X(f)Y (f)
F⇐⇒ X (ω)Y(ω)
Correlation x(t) ⋆ y(t) =
ˆ ∞
−∞x(τ)y∗(τ − t)dτ Ff⇐⇒ X(f)Y ∗(f)
F⇐⇒ X (ω)Y∗(ω)
Frequency Convolution x(t)y(t)Ff⇐⇒ X(f) ∗ Y (f)
F⇐⇒ 12πX (ω) ∗ Y(ω)
Time periodicity x(t)=combT (t)∗p(t)Ff⇐⇒ 1
Tcomb1/T (f)P (f)
F⇐⇒ 1Tcombω0(ω)P(ω)
Real & Even x(t)=Eℜx(t)
Ff⇐⇒ Real&Even F⇐⇒ Real&Even
Real & Odd x(t)=Oℜx(t)
Ff⇐⇒ Imagnary&Odd F⇐⇒ Imagnary&Odd
Decomposition of Real Ex(t) Ff⇐⇒ ℜX(f) F⇐⇒ ℜX (ω)Signals into Even & Odd Ox(t) Ff⇐⇒ iℑX(f) F⇐⇒ iℑX (ω)
Name x(t) = F−1f X(f) = F−1X (ω) Ff⇐⇒ X(f) =
ˆ ∞
−∞x(t)e−i2πftdt
F⇐⇒ X (ω) =ˆ ∞
−∞x(t)e−iωtdt
rect Π(t) = rect(t)Ff⇐⇒ sinc(f)
F⇐⇒ sinc(
ω2π
)
sinc sinc(t)= sinπtπt
or snc(t)= sin tt
Ff⇐⇒ Π(f) = rect(f) or πrect(πf)F⇐⇒ rect
(ω2π
)or πrect
(ω2
)
Causal exponential e−αtu(t)Ff⇐⇒ 1
α+i2πf
F⇐⇒ 1α+iω
tent e−α|t| Ff⇐⇒ 2αα2+(2πf)2
F⇐⇒ 2αα+ω2
Triangle = tri(t) Λ(t) = Π(t) ∗ Π(t) =
1−|t| |t|<10 |t|>1
Ff⇐⇒ sinc2fF⇐⇒ sinc2
(ω2π
)
Gaussian e−πt2 e−12(t/σ)2 Ff⇐⇒ e−πf2
σ√2πe−2π2f2σ2 F⇐⇒ e−ω2/4π σ
√2πe−ω2σ2/2
Delta δ(t/T ) = |T |δ(f) Ff⇐⇒ |T |1(f) F⇐⇒ |T |1(ω)sign sgn(t) = t
|t| = −1 t<0
1 t>0
Ff⇐⇒ 1iπf
F⇐⇒ 2iω
step u(t) =
0 t<01 t>0
Ff⇐⇒ 1i2πf
+ 12δ(f)
F⇐⇒ 1iω
+ πδ(ω)
Hilbert kernel iπt
Ff⇐⇒ sgn(f)F⇐⇒ sgn(ω)
Constant KFf⇐⇒ Kδ(f)
F⇐⇒ K2πδ(ω)
Cosine cos(2πf0t) = cos(ω0t)Ff⇐⇒ 1
2[δ(f + f0) + δ(f − f0)] F⇐⇒ π[δ(ω + ω0) + δ(ω − ω0)]
Sine sin(2πf0t) = sin(ω0t)Ff⇐⇒ 1
2i[δ(f − f0)− δ(f + f0)]
F⇐⇒ iπ[δ(ω + ω0)− δ(ω − ω0)]
Complex exponential ei2πf0t = eiω0tFf⇐⇒ δ(f − f0) F⇐⇒ 2πδ(ω − ω0)
Causal Cosine cos(2πf0t)u(t) = cos(ω0t)u(t)Ff⇐⇒ 1
2[δ(f+f0)+δ(f−f0)] + if
2π(f20−f2)
F⇐⇒ π[δ(ω + ω0) + δ(ω − ω0)] +iω
ω20−ω2
Causal Decaying Sine e−at sin(ω0t)u(t)Ff⇐⇒ 2πf0
(2πf0)2+(a+i2πf)2)
F⇐⇒ ω0
ω20+(a+iω)2
Chirp eiat2= eiπbt
2 Ff⇐⇒√
ibe−iπf2/b F⇐⇒
√iπae−iω2/4a
CombT combT (t) =∑
n
δ(t− nT ) Ff⇐⇒ 1T
∑
m
δ(f − m
T
)= 1
Tcomb 1
T(f)
F⇐⇒ 2πT
∑
m
δ(ω − m2π
T
)= 2π
Tcomb 2π
T(f)
ramp tu(t)Ff⇐⇒ i
4πδ′(f)− 1
4π2f2
F⇐⇒ iπδ′(ω)− 1ω2
power tnFf⇐⇒
(i2π
)nδ(n)(f)
F⇐⇒ in(2π)δ(n)(ω)
Bessel J0(t)Ff⇐⇒ 2√
1−(2πf)2
F⇐⇒ 2√1−ω2 |ω| < 1 , 0 |ω| > 1
Jinc jinc(t) = J1(t)/2tFf⇐⇒
√1− (2πf)2
F⇐⇒√1− ω2 |ω| < 1 , 0 |ω| > 1
Periodic wave x(t) =∑
n
p(t− nT ) Ff⇐⇒ 1T
∑
n
P(nT
)δ(f − n/T ) F⇐⇒ 2π
T
∑
n
P
(2πn
T
)δ(ω − n2π/T )
x(t) = x(t + T ) =∑
n
Xnei2πnt/T Ff⇐⇒ =
∑
n
Xnδ(f − n/T ) F⇐⇒ = 2π∑
n
Xnδ(ω − n2π/T )
Fourier Transform Pairs:Sin,Rect,SincLearning Objectives and Outcomes
• Practice manipulation of simple Fourier integralsComplex exponentialsSine and CosineDelta functions and the sifting property
•Convergence factorsEvaluate diverging integrals as limits
•Rect and sinc and bandlimited superpositions
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 18
FT of complex exponentials
The most basic and fundamental FT pairs are the complex exponentials
Fei2πf0t
=
ˆ ∞
−∞ei2πf0te−i2πftdt =
ˆ ∞
−∞e−i2π(f−f0)tdt = δ(f − f0)
and
Fe−i2πf0t
=
ˆ ∞
−∞e−i2πf0te−i2πftdt =
ˆ ∞
−∞e−i2π(f+f0)tdt = δ(f + f0)
which produce singular FT spectra since when f 6= f0 the integrand is oscillatory andintegrates to zero, but when matched, f = f0 the integral is unbounded.
The inverse transform is of a singular function, but we can just use sifting property
F−1δ(f − f0) =ˆ ∞
−∞δ(f − f0)ei2πftdt = ei2πf0t
Using linearity of the FT to combine these results we get
Fcos(2πf0t) = F12[e
i2πf0t + e−i2πf0t]= 1
2[δ(f − f0) + δ(f + f0)]
and
Fsin(2πf0t) = F
12i[ei2πf0t − e−i2πf0t]
= 1
2i[δ(f − f0)− δ(f + f0)]
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 19
Fourier Transform of Impulse Pair: Cosineand Sine Duality gives FT of symetric
temporal impulse pairP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 20
Fourier Transform of complex phase factortimes cosine eiπ/4 cos(ωot) and eiπ/8 cos(ωot+π/4)
eiπ/4 cos(ωot)F⇐⇒ eiπ/4
2 [δ(ω − ωo) + δ(ω + ωo)]
eiπ/8 cos(ωo + π/4)tF⇐⇒ ei(π/8+π/4)
2 δ(ω − ωo) + ei(π/8−π/4)2 δ(ω + ωo)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 21
FT of complex exponential
Or we can use equivalent ω-transform notation
Feiω0t
=
ˆ ∞
−∞eiω0te−iωtdt =
ˆ ∞
−∞ei(ω0−ω)tdt = 2πδ(ω0 − ω)
and scaled inverse transform
F−1δ(ω − ω0) =1
2π
ˆ ∞
−∞δ(ω − ω0)e
iωtdt =1
2πeiω0t
FT of the delta impulseDuality or direct integration can be used to evaluate
Fδ(t) =ˆ ∞
−∞δ(t)e−iωtdt = 1
and for a shifted impulse
Fδ(t− T ) =ˆ ∞
−∞δ(t− T )e−iωtdt = e−iωT
which also comes out directly from time shifting property
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 22
Phase shifts of complex exponentials
Positive and Negative Linear Phase Factorseiωot
F⇐⇒ δ(ω − ωo) e−iωot F⇐⇒ δ(ω + ωo)
Phase Shifting Complex Exponential by multiplying by eiφ
iei2ωotF⇐⇒ iδ(ω − 2ωo) −iei2ωot F⇐⇒ −iδ(ω−2ωo)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 23
FT of causal decreasing exponential
Fe−atu(t)
=
ˆ ∞
−∞u(t)e−ate−i2πftdt=
ˆ ∞
0
e(−a−i2πf)tdt=e(−a−i2πf)t
−a− i2πf
∣∣∣∣∞
0
= 0−1−a−i2πf =
1
a + i2πf
As long as e−a∞ → 0 which requires a > 0 and a convergent causal exponential
This could be performed with the ω transfom definition to obtain the equivalent result
Fe−atu(t)
=
ˆ ∞
−∞e−ate−iωtdt =
ˆ ∞
0
e(−a−iω)tdt =e(−a−iω)t
−a− iω
∣∣∣∣∞
0
=0− 1
−a− iω =1
a + iω
Which should be obvious from just the substitution ω = 2πf
Note that a causal sinusoid does not FT to a delta, and there is spectral spreading. Sincethe integral is not convergent, use the modulation theorem and known FT of u(t)
Fe−i2πf0tu(t)
=
ˆ ∞
0
e−i2π(f+f0)tdt =e−i2π(f+f0)t
−i2π(f + f0)
∣∣∣∣∞
0
=?
= δ(f + f0) ∗[12δ(f) +
1
i2πf
]= 1
2δ(f + f0) +1
i2π(f + f0)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 24
Example Fourier Transform of rect
rect
(t
2T
)= Π
(t
2T
)=
0 t > T1 t < T
X(iω)=
ˆ ∞
−∞Π
(t
2T
)e−iωtdt =
ˆ T
−Te−iωtdt =
1
−iωe−iωt∣∣∣∣T
−T=
1
−iω(e−iωT − e+iωT
)
=2
ω
1
2i
(eiωT − e−iωT
)=
2
ωsinωT =
2
2πf
T
Tsin 2πft = 2T
sinπ2Tf
π2Tf= 2T sinc2Tf
where sinc(f) = sin πfπf
or using alternative sncω = sinωω
we get X(ω) = 2T snc (ωT )
Π
(t
1
) 1 · sinc(1 · f)
Π
(t
T
) T sinc(Tf)
1T
1T
1T
-
.636
.128
-.212t f
1.0
T
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 25
Sinc and its width
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 26
Pictorial Fourier Transform of a symmetricrect
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 27
Pictorial Inverse Fourier Transformproducing a temporal rect in the limit of
infinite bandwidthP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 28
Duality Pictorial Inverse Fourier Transformof a rect in frequency
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 29
Time lapse movie of phasor evolution ofeach Fourier component of rect in
frequencyP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 30
FT of sinc
Fsinc(t)=ˆ ∞
−∞sinc(t)e−i2πftdt=
ˆ ∞
−∞
sin πt
πtcos(2πft) dt− i
0 =´
oddˆ ∞
−∞
sinπt
πtsin(2πft) dt
=
ˆ ∞
−∞
sin(πt + 2πft)
2πt+sin(πt− 2πft)
2πtdt
=
ˆ ∞
−∞
sin[π(1 + 2f)t]
(1 + 2f)t
(1 + 2f)
2+sin[π(1− 2f)t]
(1− 2f)t
(1− 2f)
2dt
=1 + 2f
2
ˆ ∞
−∞
sin[π(1 + 2f)t]
(1 + 2f)tdt +
1− 2f
2
ˆ ∞
−∞
sin[π(1− 2f)t]
(1− 2f)tdt
=1 + 2f
2|1 + 2f | +1− 2f
2|1− 2f | = Π(t) = rect (t)
Where we used ˆ ∞
−∞sinc(at) dt =
1
|a|1+2f
2|1+2f | =12sgn(f + 1
2)1−2f
2|1−2f | =12sgn(−(f − 1
2))
Or directly by duality since
f|f|
1+2f
|1+2f|1-2f
|1-2f|
-2
23
1 2-1-2
12
-1
3
1 2-1-2
12
-1
3
-1-2 1 2
1
-1
-1-2 1 2
1
-1-1-2 1 2
1
-1-1-2 1 2
1
-1-1-2 1 2
1
-1
-1-2 1 2
1
-1
-1-2 1 2
1
-1
Π(t) sinc(f) ⇒ sinc(t) Π(−f) = Π(f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 31
Fourier Transform Pairs:Step,GaussianLearning Objectives and Outcomes
• Practice manipulation of simple 1-D Fourier integralsSign and Step functionsGaussian function needed for lasersComplex Gaussian as generalization of real Gaussian
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 32
FT of sign function
The sign function is defined as
sgn(t) = 2u(t)− 1 = u(t)− u(−t) =−1 t < 01 t > 0
and Fsgn(t) not absolutely convergent since´∞−∞ |sgn(t)|dt→∞
Must define FT in terms of series of functions that in the limit become sgn(t)
sgn(t)= lima→∞
sa(t) where sa(t)=
−et/a t < 0e−t/a t > 0
For finite a we have a well defined FT
Fsa(t)=ˆ ∞
0
e−t/ae−i2πftdt−ˆ 0
−∞et/ae−i2πftdt=
ˆ ∞
0
e(−1/a−i2πf)tdt−ˆ 0
−∞e(1/a−i2πf)tdt
=e(−1/a−i2πf)t
−1/a− i2πf
∣∣∣∣∞
0
− e(1/a−i2πf)t
1/a− i2πf
∣∣∣∣0
−∞=
0− 1
−1/a− i2πf −1− 0
1/a− i2πf =−i4πf
(1/a)2 + (2πf)2
Giving the FT of sgn in the limit a→∞
Fsgn(t) = Flima→∞
sa(t)=
−i4πf(0)2 + (2πf)2
=1
iπf
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 33
FT of sign function
Note that an easier route is to use the derivative and integration properties
d
dt(sgn(t) +K) = 2δ(t)
Fd
dt(sgn(t) +K)
= iωFsgn(t) +K = 2
Thus
iωFsgn(t) + iω2πKδ(ω) = 2 ⇒ Fsgn(t) = 2
iω− 2πKδ(ω)
The constant K must be zero since sgn(t) + sgn(−t) = 0
2
iω− 2πKδ(ω) +
2
−iω − 2πKδ(ω) = 0 ⇒ K = 0
Fsgn(t) = 2
iω
And from this we can get the FT of u(t) = 12 +
12sgn(t)
Fu(t) = πδ(ω) +1
iω
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 34
FT of Heavyside step function
´∞−∞ |u(t)|dt→∞ not convergent so Laplace transform 1/s not defined along iω axis.
Must instead define in terms of series of functions that in the limit become u(t)
u(t) = lima→∞
ua(t) where ua(t) =
0 t < 0
e−t/a t > 0
Or we can use that u(t) = 12 +
12sgn(t) with known FT and linearity of FT
Fu(t) = F12+ 1
2sgn(t)
= 1
2δ(f) +
1
i2πf
Now using the scaling of a delta to write δ(f) = δ(ω2π
)= 2πδ(ω)
Fu(t) = πδ(ω) +1
iω
Duality can be invoked to evaluate1
iπt sgn(−f)
12δ(t) +
1
i2πt u(−f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 35
FT of 1/iπt
F
1
iπt
=
ˆ ∞
−∞
1
iπte−i2πftdt =
0 =´
evenoddˆ ∞
−∞
1
iπtcos(2πft)dt− i
ˆ ∞
−∞
1
iπtsin(2πft)dt
=−i2fi
ˆ ∞
−∞
sin(2πft)
2πftdt = −2f
ˆ ∞
−∞sinc(2ft)dt =
−2f|2f | = sgn(−f)
where the infinite integral (eg area) of a scaled sinc of width 1/2f is just given by thewidth 1/2f , but when f is negative this area of the even sinc is −1/2f , thus
ˆ ∞
−∞sinc(2ft)dt =
1
2|f |
or duality can be invoked from the known FT Fsgn(t) = 1/iπt to evaluate
1
iπt sgn(−f)
But in this case the FT of 1/iπt is actually the convergent one, so duality should beinvoked instead to find FT of sgn(t).
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 36
FT Gaussian and Complex Gaussian
g(t) = e−πt2 |
´∞−∞ e
−πt2dt = 1 n(t) = e−(t/τ)2
G(f) =
ˆ
e−πt2e−i2πftdt =
ˆ
e−π(t2+i2ft)dt
note (t + if)2 = t2 + 2ift− f 2
= e−πf2ˆ
e−π(t+if)2dt = e−πf
2ˆ
e−πt′2dt′
︸ ︷︷ ︸= e−πf
2
t′ = t + if dt′ = dt 1= area of normalized unit GaussianScaled Gaussian
g(t) = e−π(t/τ)2
τe−π(τf)2
Complex Gaussian (Chirp with Gaussian amplitude modulation)1/τ 2 = a + ib when a = 0 this is a chirp
e−π(a+ib)t2
1√a + ib
e−πf2/(a+ib)
e−iπbt2
1√ibe+iπf
2/b
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 37
Fourier Transform PropertiesLearning Objectives and Outcomes
•Recognize and use Linearity in evaluating FT
•Understand Shift and Scale theorems
•Gain insight from Duality
•Understand the Convolution Theorem
•Apply FT Symmetries
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 38
Linearity
The Fourier integral is clearly linear, so that
Fax(t) + by(t) = aFx(t) + bFy(t)and
F−1aX(f) + bY (f) = aF−1X(f) + bF−1Y (f)
Linearity is an implicit assumption in the decomposition of functions into even andodd parts. Similarly implicit in decomposition of signals or transforms into real andimaginary parts
Example
Fe−|t|
= F
e−tu(t) + etu(−t)
=
1
1 + j2πf+
1
1− j2πf =1−
j2πf + (1 +j2πf)
1 + (2πf)2
=2
1 + (2πf)2
Note this follows from the Laplace transform evaluated at s = 2πf
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 39
Shift Theorem
x(t− T ) e−i2πfTX(f)
ˆ ∞
−∞e−i2πfTX(f)ei2πftdf =
ˆ ∞
−∞X(f)ei2πf(
t′︷ ︸︸ ︷t− T )df = x(t′) = x(t− T )
Π(t) sinc(f)
Π(t− T ) sinc(f)e−i2πfT
Π(t− 1/2) sinc(f)e−iπf
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 40
Fourier Transform of Shifted FunctionsP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 41
Conjugation
x∗(t) X∗(−ω)Proof
X∗(ω) =
(ˆ ∞
−∞x(t)e−iωtdt
)∗=
ˆ ∞
−∞x∗(t)e+iωtdt
let ω′ = −ωX∗(−ω′) =
ˆ ∞
−∞x∗(t)e−iω
′tdt
So when x(t) is real x(t) = x∗(t)
X∗(−ω) =ˆ ∞
−∞x(t)e−iωtdt = X(ω)
So we can write X(ω) in terms of real and imaginary parts
X(ω) = ℜX(ω) + iℑX(ω) = Xr(ω) + iXi(ω)
Xr(ω) + iXi(ω) = Xr(−ω)− iXi(−ω)So we can conclude for real x(t) we have Hermition X(ω)
Xr(ω) = Xr(−ω) eveniXi(ω) = −iXi(−ω) odd
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 42
Cosine and Sine parts of Fourier TransformP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
inverse Fourier transform of unitheight rect pulse
Note the even character of cosωt
Sum along ridge does not cancel
sine carrier surface
Odd symetry of sinωt
Symmetric sums along ω cancelfor all t giving 0 real part of f(t)for even ℑF (jω)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 43
Scaling
x(at)1
|a|X(f
a
)
ˆ ∞
−∞x(at)e−i2πftdt =
ˆ ∞
−∞x(τ )e−i2πfτ/adτ/a τ = at dτ = adt a > 0
=
ˆ −∞
+∞x(τ )e−i2πfτ/adτ/a =
1
|a|
ˆ ∞
−∞x(τ )e−i2πfτ/adτ a < 0
=1
|a|
ˆ ∞
−∞x(τ )e
−i2π(fa
)τdτ =
1
|a|X(f
a
)
special case for mirroring in time
x(−t) X(−f)
Normalized time width T form of the scaling thm
x
(t
T
) |T |X(fT )
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 44
Fourier Transform of Scaled FunctionsP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 45
Shifted and scaled functions
Find FT of shifted and scaled rectangle by breaking it down into steps
x(t) = 25Π
(t− 4
10
)
apply scale theorem
Π
(t
10
)F⇐⇒10sinc(10f )
apply shift theorem to that result
Π
(t− 4
10
)F⇐⇒10sinc(10f ) e−i2π4f
apply linearity
25Π
(t− 4
10
)F⇐⇒250sinc(10f ) e−i2π4f
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 46
Shifted and scaled functions
Find FT of composite function containing shifted, scaled,phase shifted, and weighted functions.
sinc
[t− 7
2
]ei2π11t + 4Π[3t + 15]e−i2π13t
Looks complicated. Don’t panic, just break it down into in-dividual parts.
First note the interesting duality
sinc(t) + Π(t)F⇐⇒Π(f ) + sinc(f )
This works for even functions
For general functions there is a more complete duality
x(t) +X(t) + x(−t) +X(−t) F⇐⇒X(f ) + x(−f ) +X(−f ) + x(f )Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 47
linearity combined with shifted and scaledfunctions
Find FT of composite function
sinc
[t− 7
2
]ei2π11t + 4Π[3t + 15]e−i2π13t
Build up the complicated functions from its parts
sinc
(t
2
)F⇐⇒2Π(2f)
Π(3t)F⇐⇒ 1
3sinc
(f
3
)
sinc
(t− 7
2
)F⇐⇒2Π(2f)e−i2π7f
Π[3(t + 5)]F⇐⇒ 1
3sinc
(f
3
)ei2π5f
sinc
(t− 7
2
)ei2π11t
F⇐⇒2Π[2(f − 11)]e−i2π7(f−11)
Π[3(t + 5)]e−i2π13tF⇐⇒ 1
3sinc
(f + 13
3
)ei2π5(f+13)
sinc[t−72
]ei2π11t+4Π[3t+15]e−i2π13t
F⇐⇒2Π[2(f−11)
]e−i2π7(f−11)+ 4
3sinc(f+133
)ei2π5(f+13)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 48
Duality and the Transform of a Transform
Suppose h(t) H(f) but H() is just a function (eg sinc) and that we want to knowthe Fourier transform of that function of timelet g(t) = H(t)
FH(t) = Fg(t) = G(f) =
ˆ ∞
−∞g(t)e−i2πftdt =
ˆ ∞
−∞H(t)e−i2πftdt
=
ˆ ∞
−∞H(t′)ei2π(−f)t
′dt′ = h(−f)
This means thatFFh(t) = h(−t)
Duality suggests that property of time domain has dual when applied to frequencydomain
dX(f)
df=d
df
ˆ ∞
−∞x(t)e−i2πftdt =
ˆ ∞
−∞x(t)(−i2πt)e−i2πftdt
−i2πtx(t) dX(f)
dfDifferentiation in frequency
e+i2πf0tx(t) X(f − f0) Modulation theorem
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 49
DualityP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 50
Duality: Shifting in time and Shifting inFrequency
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 51
Convolution Theorem
y(t) =
ˆ ∞
−∞x(τ )h(t− τ )dτ =
ˆ ∞
−∞h(τ ′)x(t− τ ′)dτ ′
Fy(t)=Y (f)=X(f)H(f)=H(f)X(f)
Transform domain shift by τ
Y (iω) =
ˆ
[ˆ ∞
−∞x(τ )h(t− τ )dτ
]e−iωtdt =
ˆ
x(τ )
[ˆ
e−iωt︷ ︸︸ ︷h(t− τ ) dt
]dτ
=
ˆ
x(τ )[H(ω)e−iωτ
]dτ = H(ω)
ˆ
x(τ )e−iωτdτ = H(ω)X(ω)
y(t) = h(t) ∗ x(t) F⇐⇒ Y(ω) = H(ω)X(ω)
Y (f) = H(f)X(f)
Modulation Theorem : Duality - convolution in frequency domain
r(t) = s(t)p(t)F⇐⇒ 1
2π
ˆ
S(µ)P(ω − µ)dµ =1
2πS(ω) ∗ P (ω)
ˆ
S(f ′)P (f − f ′)df ′ = S(f) ∗ P (f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 52
Convolution Theorem Examples
Tri(t) = Λ(t) = Π(t) ∗ Π(t) = rect (t) ∗ rect (t)
FΛ(t) = FΠ(t)FΠ(t) = sinc2(f)
Simple proofs of convolution properitesCommutative
x(t) ∗ y(t) = y(t) ∗ x(t) since X(f)Y (f) = Y (f)X(f)
Associative
x(t)∗[y(t)∗z(t)
]=[x(t)∗y(t)
]∗z(t) since X(f)
[Y (f)Z(f)
]=[X(f)Y (f)
]Z(f)
Distributive
x(t) ∗[y(t) + z(t)
]=x(t) ∗ y(t) + x(t) ∗ z(t) since X
[Y + Z
]=XY +XZ
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 53
Convolution Theorem Examples
Find the FT of the convolution of x(t) = 10 sin 2πf0t with y(t) = 2δ(t + 4)
Do convolution first then FT
10 sin(2πf0t) ∗ 2δ(t + 4) = 20 sin[2πf0(t + 4)]F⇐⇒20
1
2i(δ(f − f0)− δ(f + f0))e
i2π4f
Do FT first to avoid having to do convolution
10 sin(2πf0t) ∗ 2δ(t + 4)F⇐⇒10
1
2i(δ(f − f0)− δ(f + f0)) · 2ei2π4f
where we used10 sin(2πf0t)
F⇐⇒101
2i(δ(f − f0)− δ(f + f0))
2δ(t + 4)F⇐⇒ei2π4f
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 54
Convolution Theorem Example: Timeshifting as convolution with shifted impulse
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 55
Correlation Theorem
The correlation function measures the similarity between functions
x(t) ⋆ y(t) =
ˆ ∞
−∞x(τ )y∗(τ − t)dτ =
ˆ ∞
−∞x(τ ′ + t)y∗(τ ′)dτ ′
introducing the impulse response h(t) = y∗(−t) we can write this as a convolution
x(t)∗h(t)=ˆ ∞
−∞x(τ )h(t−τ )dτ =
ˆ ∞
−∞x(τ )y∗(−[t−τ ])dτ =
ˆ ∞
−∞x(τ )y∗(τ−t)dτ =x(t)⋆y(t)
Now we can use the convolution theorem and the relation y∗(−t) ⇐⇒ Y ∗(ω) to writethe correlation theorem
x(t) ⋆ y(t) = x(t) ∗ h(t) ⇐⇒ X(ω)H(ω) = X(ω)Y ∗(ω) X(f)Y ∗(f)
when x(t) = y(t− T ) and for a flat bandwidth B code (|Y (f)|2 = Π[fB
]) we get
y(t−T )⋆y(t) = F−1Y (f)e−i2πfT · Y (f)∗
= F−1
|Y (f)|2e−i2πfT
= sinc[B(t−T )]
Autocorrelation Theorem
x(t) ⋆ x(t) =
ˆ ∞
−∞x∗(τ )x(τ + t)dτ ⇐⇒ |X(f)|2
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 56
Modulation Theorem
Feiω0tf(t)
= F (ω − ω0)
Now using standard identities cosω0t =12[e
iω0t + e−iω0t] we get
Fcos(ω0t)f(t) =1
2
[F (ω − ω0) + F (ω + ω0)
]
And using sinω0t =12i[eiω0t − e−iω0t] we get
Fsin(ω0t)f(t) =1
2i
[F (ω − ω0)− F (ω + ω0)
]
1T
1T
1T
-
.636
.128
-.212t f
1.0
T
-f0 f0t f
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 57
Differentiation and Integration
Differentiationdx(t)
dt=d
dt
ˆ ∞
−∞X(f)ei2πftdt =
ˆ ∞
−∞
d
dt
X(f)ei2πft
dt =
ˆ ∞
−∞X(f)
d
dt
ei2πft
dt
=
ˆ ∞
−∞[X(f)(i2πf)]ei2πftdt
dx(t)
dt i2πfX(f)
dx(t)
dt iωX(ω)
Integration
y(t) =
ˆ t
−∞x(τ )dτ = x(t) ∗ u(t) = u(t) ∗ x(t)
=
ˆ ∞
−∞u(τ )x(t− τ )dτ =
ˆ ∞
0
x(t− τ )dτ =
ˆ ∞
−∞x(τ )u(t− τ )dτ =
ˆ t
−∞x(τ )dτ
Now using the convolution theorem
Y (f) = X(f)U (f) = X(f)
[1
−i2πf + 12δ(f)
]=
X(f)
−i2πf +X(0)
2δ(f)
ˆ t
−∞x(τ )dτ
X(f)
−i2πf +X(0)
2δ(f)
X(ω)
−iω +X(0)πδ(ω)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 58
FT of Triangle Function and its DerivativeP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 59
Pictorial FT of IntegrationP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 60
Differentiation in Frequency
Duality in its f and ω form gives
X(t)F⇐⇒x(−f) X(t)
F⇐⇒2πx(−ω)
Differentiation in timedx(t)
dt
F⇐⇒ i2πfX(f)dx(t)
dt
F⇐⇒ iωX(ω)
Dual is differentiation in frequency
−i2πtx(t) F⇐⇒ dX(f)
df− itx(t) F⇐⇒ dX(ω)
dω
Now we can use this to evaluate FT of causal ramp, remember
u(t)F⇐⇒πδ(ω) +
1
iω
Thus
tu(t)F⇐⇒ i
dU (ω)
dω= i
d
dω
[πδ(ω) +
1
iω
]= iπδ′(ω)− ω−2
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 61
Parseval’s Theorem(Rayleigh’s or Plancherel’s Thm)
Power P (t) = v2(t)R
= i2(t)R so Energy delivered to R = 1Ω load isW =
´∞−∞P (t)dt=
´∞−∞f
2(t)dt=´∞−∞|x(t)|2dt where f(t) is either real v(t) or i(t)
Total Energy can be calculated by integrating energy per unit time, |x(t)|2, across timeOr by integrating energy per unit frequency, |X(f)|2, across all frequencies
ˆ ∞
−∞|x(t)|2dt =
ˆ ∞
−∞|X(f)|2df
Proof
E =
ˆ ∞
−∞x(t)x∗(t)dt =
ˆ ∞
−∞x(t)x∗(t)e−i2πf
′tdt
∣∣∣∣f ′=0
= X(f) ∗X∗(−f)∣∣∣f ′=0
=
ˆ ∞
−∞X(f)X∗(−(f ′ − f))df
∣∣∣∣f ′=0
=
ˆ ∞
−∞X(f)X∗(f − f ′)df
∣∣∣∣f ′=0
=
ˆ ∞
−∞X(f)X∗(f)df
For periodic x(t) with period T0 = 1/f0
1
T0
ˆ t0+T0
t0
|x(t)|2dt =∞∑
n=−∞|Xn|2 where X(f) =
∞∑
n=−∞Xnδ(f − nf0)
The Xn are the Fourier series coefficients
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 62
Power Theoremˆ ∞
−∞x(t)y∗(t)dt =
ˆ ∞
−∞X(f)Y ∗(f)df
Compact Proofˆ ∞
−∞x(t)y∗(t)dt = Fx(t)y∗(t)
∣∣∣0= X(f) ∗ Y ∗(−f)
∣∣∣0=
ˆ ∞
−∞X(f)Y ∗(f)df
For real x and y, FT are Hermitian (evenXr(f) = Xr(−f) and oddXi(f) = −Xi(−f))ˆ ∞
−∞x(t)y(t)dt =
ˆ ∞
−∞Xr(f)Y
∗r (f) +Xi(f)Y
∗i (f)df
This tells us something about the geometry of the Fourier Transform function spaceThe inner product between signals in function space,
´∞−∞ x(t)y
∗(t)dt, is preserved bythe FT, thus FT is a rotation operator in∞-dimensional function space since rotationspreserve angles between vectors.Since FFx(t) = x(−t) we have
FFFFx(t) = x(t)
And the FT rotation is a 90 rotation in function space
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 63
The Fourier Transform of Even and OddFunctions
Even(real) Even(real)12[g(t) + g(−t)] 1
2[G(f) +G(−f)]
Odd(real) Odd(imaginary)12[g(t)− g(−t)] 1
2[G(f)−G(−f)])
real Hermitian
g(t) G(f) such that G(−f) = G∗(f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 64
Fourier Transform of Odd Function isImaginary Spectrum
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 65
Real and Imaginary parts of Even functionFourier Transform
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 66
Real and Imaginary parts of Odd functionFourier Transform
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 67
Even and Odd Function Properties andFourier Transform Relations
• g(t) = e(t) + o(t)
•´ T
−T e(t)dt = 2´ T
0 e(t)dt
•∑N−N e[n] = e[0]+2
∑N1 e[n]
•´ T
−T o(t)dt = 0
•∑N−N o[n] = 0
• o + o = o
• e + e = e
• e× e = e
• o× o = e
• o× e = o
• ddto = e
• ddte = o
• e ∗ e = e
• o ∗ o = e
• e ∗ o = o
Even Even
Odd Odd
Real and even Real and even
Real and odd imaginary and odd
Imaginary and even Imaginary and even
Imaginary and odd Real and odd
Complex and even Complex and even
Complex and odd Complex and odd
Real even plus Imaginary odd Real
Real odd plus Imaginary even Imaginary
Real & asymmetrical Complex & Hermitian
Imaginary & asymmetrical Complex & anti-Hermitian
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 68
Fourier Transform Properties of Even andOdd Function
R.N. Bracewell, The Fourier Transform and its Applications, Mcgraw-Hill, 2000
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 69
Causal function Fourier TransformP. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 70
Real and Imaginary parts of Causalfunction Fourier Transform
P. Kraniauskas, Transforms in Signals and Systems, Addison-Wesley, 1992
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 71
Delta and other singularity functions
Delta function is the limiting form of any compact function as itis scaled narower and taller
δ(t) = limA→∞
AΠ [At] = limw→0
1
wΠ
[t
w
]= lim
T→0
1
Te−π(t/T )
2
Or the delta function can be defined by its integral representationˆ
e−i2π(t−t0)fdf = δ(t− t0) δ(t) =1
2π
ˆ ∞
−∞eiωtdω
Or a delta can be defined as the derivative of a step function
δ(t) =d
dtu(t) = 1
2
d
dtsgn(t)
Or a delta function can be defined by its properties
δ(t− t0) = 0 t 6= t0 δ(t)→∞ at t→ 0
sifting :
1
1
u(t)
δ(t)=u’(t) 1
1
Nature abhors a naked singularity, δ() functions should only appear under an integralˆ
f(t)δ(t− t0)dt = f(t0)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 72
Delta function properties, continued
scaling: How do you stretch or compress something that is infinitely narrow?
δ(−t) = δ(t) An even function
δ
(t
b
)= |b|δ(t)
δ
(t− t0b
)= |b|δ(t− t0)
δ(at− t0) =1
|a|δ(t− t0/a)
delta of functions with simple roots ti (gives above scaling relation for g(t) = at)
δ(g(t)) =∑
i
|g′(ti)|−1δ(t− ti)
Properties in products
f(t)δ(t− t0) = f(t0)δ(t− t0)ˆ
δ(τ − t0)δ(t− τ )dτ = δ(t− t0)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 73
Scaling: How do you stretch (b>1,a<1) orcompress (b<1,a>1) something that is
infinitely narrow?δ(tb
)= |b|δ(t) δ(at) = 1
|a|δ(t)
Proof: first consider b > 0 T ′=bT = |b|Tδ(t) = lim
T→0
1T e−π( tT )
2
⇒ δ(tb
)= lim
T→0
1
Te−π(t/bT
)2= lim
T ′→0
b
T ′e−π(
tT ′)
2
= bδ(t)
Now consider b < 0, since delta is even this must give the same result: δ(bt) = δ(|b|t).Or introduce T ′ = −bT = |b|T inside the Gaussian exponential without changing it.
δ(tb
)= |b|δ(t)
Alternative proof using defining integral property: First consider a > 0 and intro-duce t′ = at = |a|t into scaled delta sifting integral so t = t′/a and dt = dt′/a
ˆ ∞
−∞g(t)δ(at)dt =
ˆ ∞
−∞g(t′a
)δ(t′)dt
′a=
1
ag(0)
Now for a < 0 we should use t′ = −at = |a|t to avoid flipping g(), but this changessign of limits, however sign of 1/a changes them back givingˆ ∞
−∞g(t)δ(at)dt =
1
|a|g(0) =1
|a|
ˆ ∞
−∞g(t)δ(t)dt ⇒ δ(at) =
1
|a|δ(t)
which is the sifting behavior of an unscaled delta, with amplitude normalization by 1|a|.
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 74
Derivatives and integrals of delta functions
δ(k)(t) =dkδ(t)
dtk=dk+1u(t)
dtk+1
where the unit step, u(t) = δ(−1)(t) = H(t) =
0 t < 012 t = 01 t > 0
The derivatives are hard to visualize so instead defined in terms of integral propertiesConsider a function f(t) with derivatives defined up to at least kth order defined (con-tinuous at t0)
f (k)(t) =dkf(t)
dtk
Then the defining properties of derivatives of δ(t)
δ(k)(t− t0) = 0 t 6= t0
ˆ ∞
−∞δ(k)(t)dt = 0
ˆ t2
t1
f(t)δ(k)(t− t0)dt = (−1)kf (k)(t0) t1 < t0 < t2
unlike the delta function, the integrals are identically zero for k 6= 0
δ(t) = limb→0
1
bΠ
[t
b
]δ(1)(t) = lim
b→0
1
b[δ(t + b/2)− δ(t− b/2)]
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 75
FT of delta functionssifting property of delta allows easy calculation of FT
Fδ(t) =ˆ ∞
−∞δ(t)e−i2πftdt = e−i2πf0 = 1(f)
FT of doublet, acts like a differentiator of the integrand at the point it fires
Fδ(1)(t)
=
ˆ ∞
−∞δ(1)(t)e−i2πftdt =
d
dte−i2πft|t=0 = (−i2πf)e−i2πft|t=0 = (−i2πf)
Higher order doublets
Fδ(n)(t)
=
ˆ ∞
−∞δ(n)(t)e−i2πftdt =
dn
dtne−i2πft|t=0 = (−i2πf)ne−i2πft|t=0 = (−i2πf)n
And the function whose derivative is a delta function behaves in the opposite ways
F12sgn(t)
=
1
i2πf
whileFu(t) = F
12 +
12sgn(t)
= 1
2δ(f) +1
i2πf
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 76
Comb function(Shah III(t))
comb(t) =
∞∑
n=−∞δ(t− n)
scaled and shifted
comb
(t− t0T
)= |T |
∞∑
n=−∞δ(t− t0 − nT )
comb(tT
)=
∞∑
n=−∞δ(tT− n
)= |T |
∞∑
n=−∞δ(t− nT )
combT (t) =1
|T |comb(t
T
)=
∞∑
n=−∞δ(t− nT )
Finite number of comb teeth
Π
(t
T
)comb∆(t)
Sampling
f(t)
[1
|T |comb(t− t0T
)]=
∞∑
n=−∞f(t0+nT )δ(t−t0−nT )
1
0 1 2 3-1
T
to-T to to+T
big T
small T
density remains constant
T
f(t)
T0 2T
1
0 2∆-∆ ∆-2∆ Τ/2−Τ/2
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 77
Fourier Transform of the comb
comb (t) =
∞∑
n=−∞δ(t− n · 1) comb (f) =
∞∑
k=−∞δ(f − k · 1)
comb is limit of weighted envelope under which there are periodically placed narrowbumps
c(t) = τ−1e−πτ2t2
∞∑
n=−∞e−π(t−n·1)
2/τ2 limτ→0
c(t) = comb (t)
envelope timewidth 1/τ
τ
t f
envelope frequencywidth 1/τ
FT of the product of envelope of width 1τ
with∞ array of Gaussian bumps of width τconvolution of Gaussian of width τ in frequency with∞ sum of FT of shifted Gaussian
e−π(t/τ−1)2 τ−1e−π(τ
−1f)2
e−π(t−n1τ )
2
e−i2πn1fτe−π(fτ)2
∞∑
n=−∞e−π(
t−n1τ )
2
τe−π(fτ)2∞∑
n=−∞e−i2πn1f
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 78
comb FT
this can instead be expanded as a Fourier series
τ−1∞∑
n=−∞e−π(
t−n1τ )
2
=∞∑
m=−∞e−πτ
2m2cos 2πmt
c(t) = e−πτ2t2
∞∑
m=−∞e−πτ
2m2cos 2πmt =
∞∑
m=−∞e−πτ
2t2e−πτ2m2e−i2πmt
⇒ C(f) =∞∑
m=−∞e−πτ
2m2τ−1e
−π(f−m/1
τ
)2
Thus
comb (f) = limτ→0
C(f) =∞∑
m=−∞δ(f −m1)
So, in the limit the temporal comb function, comb (t), Fourier transforms to the fre-quency comb, comb (f). True even though this function not only has singularities likea δ(t), it has an infinite number of such singularities!
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 79
FT of periodic signals
Arbitrary periodic waveform with period T can be produced by a superposition ofharmonic tones weighted appropriately
X(f) =∑
k
akδ (f − k/T )a1
a5a3-1 T
1T 3
T
5T
FT FSak
⇒ x(t) =
ˆ ∞
−∞X(f)ei2πftdf =
∑
k
ak
ˆ ∞
−∞δ (f − k/T ) ei2πftdf =
∑
k
akei2π kT t
Can synthesize a time domain periodic waveform by convolving a comb with element*
*
=
=
Now use convolution thm
p(t) ∗∑
δ(t− nT ) F⇐⇒P (f) · comb (fT )
a1
a5a3-1 T
1T 3
T
5T
FT FSak
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 80
Array Theorem
*
comb
window
element
# zeroes =N-1 # subpeaks =N-2To represent diffraction gratings and other finite width periodic structuresConsider a finite width periodic structure
g(x) = w(x)·[f(x) ∗ 1
Xcomb
( xX
)] G(u) = W (u)∗[F (u)·comb (uX)]
2Dg(x, y) = w(x, y) ·
[f(x, y) ∗ ∗ 1
XYcomb
( xX,y
Y
)]
G(u, v) =W (u, v) ∗ [F (u, v) · comb (uX, vY )]
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 81
Fourier Transform WidthsLearning Objectives and Outcomes
• Learn about relation between moments and widths offunctions
• Study variance as a measure of width of common func-tions
•Recognize uncertainty principle as a Fourier theorem
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 82
Moments of the Fourier transform
Moments m0 =
ˆ
g(t)dt = G(0)
m1 =
ˆ
tg(t)dt =−12πi
G′(0)
mn =
ˆ
tnf(t)dt =
(−12πi
)ndnG(f)
dfn
∣∣∣∣f=0
Centroid center of gravity〈g〉 =
´∞−∞ tg(t)dt´∞−∞ g(t)dt
=−G′(0)2πiG(0)
=m1
m0
Moment of Inertia 2nd moment proved with derivative thm´∞−∞(−i2πt)2g(t)e−i2πftdt = G”(f)
m2 =
ˆ ∞
−∞t2g(t)dt = − 1
4π2G”(0) curvature at origin in f
Mean Square
〈t2〉 =´∞−∞ t
2g(t)dt´∞−∞ g(t)dt
=m2
m0= − 1
4π2G”(0)
G(0)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 83
Variance and Equivalent Widths offunctions and transforms
Instead of using area of function (which can be 0)Define Variance
σ2g = 〈(t−〈t〉)2〉= 〈t2〉−〈t〉2=´∞−∞(t− 〈t〉)2g(t)dt´∞−∞ g(t)dt
= − G”(0)
4π2G(0)+
[G′(0)]2
4π2[G(0)]2=m2
m0−m
11
m20
Example
Gaus(t/τ )F⇐⇒τGaus(fτ ) m0 = τ, m1 = 0, t = m1
m0= 0, m2 =
G”(0)
−4π2 , σ2 =
τ 2
2π
To have finite 2nd moment, must die out more rapidly than x−2
sinc2(t) Λ(f) cusp has∞ 2nd derivative ⇒ m2 →∞
These measures breakdown for oscillatory functions when the area is 0.To cope with this case consider the centroid and variance of energy density
σ|g|2 = (∆x)2 =
´∞−∞ x
2|g(x)|2dx´∞−∞ |g(x)|2dx
−[´∞−∞ x|g(x)|2dx´∞−∞ |g(x)|2dx
]
This behaves more like our intuition and will be utilized in the uncertainty relationcalculation
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 84
Table of various widths
〈t〉 = m1m0
〈t2〉 = m2m0
G”(0) σ|g|2 = (∆t)2
Π(t) 0 1/12 −π2/3 1/12
Λ(t) 0 1/6 −2π2/3 1/4sinc(t) 0 osc 0 ∞sinc2(t) 0 ∞ ∞ .030...
e−xu(x) 1 2 −8π2 1/4
e−πx2a 0 1/2πa2 −2π/a3 1/4πa2
11+x2
0 ∞ ∞ ...
sech (πt) 0 14 −π2 1/12
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 85
Uncertainty Principle: Equivalent width ofthe Power in time and Fourier domains
Key limitation on QM, imaging, antennas, radar, etc
∆t∆f ≥ 1
4π∆t∆ω ≥ 1
2
where the variance of the function g(t) = |s(t)|2 in t and transform, G(f) = |S(f)|2, inf are the mean square width about the centroids of power 〈t〉 and 〈f〉
σ2g = (∆t)2 = 〈(t− 〈t〉)2〉 =´∞−∞(t− 〈t〉)2g(t)dt´∞−∞ g(t)dt
= − G”(0)
4π2G(0)+
[G′(0)]2
4π2[G(0)]2
σ2G = (∆f)2 = 〈(f − 〈f〉)2〉 =´∞−∞(f − 〈f〉)2G(f)df´∞−∞G(f)df
Gaussian is minimum uncertainty wavepacket. σ2g is variance of power in t, σ2G in f
s(t) = e−π(at)2= e−
12(t/τ)
2g(t) = |s(t)|2 = e−2πa
2t2= e−(t/τ)2
σ2g=(∆t)2=1
4πa2= 1
2τ2
S(f) =e−π(fa
)2
aG(f)= |S(f)|2= a−2e−2πf
2/a2 σ2G=(∆f)2=a2
4π= 1
2
1
(2π)2τ 2
(∆t)2(∆f)2 =1
4πa2a2
4π=
1
(4π)2⇒ ∆t ·∆f =
1
4πKelvin Wagner, University of Colorado Fourier Optics Fall 2019 86
Examples of Uncertainty Principle
CW signal observed for a finite time T has frequency uncertainty ≈ 1/T
1T
1T
1T
-
.636
.128
-.212t f
1.0
T
-f0 f0t f
Antenna size A gives beamwidth ≈ λ/A
Telescope with diameter D gives angular resolution ≈ λ/D
Atom with lifetime T yields spectral linewidth ≈ 1/T
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 87
Proof of the Uncertainty relationfirst some identities
From the derivative thm dn
dxng(t) (i2πf)nG(f) and conjugation thm g∗(±t) G∗(∓f) we get that
g′(t) =d
dtg(t) (i2πf)G(f) g′∗(t) =
d
dtg∗(t) (−i2πf)G(−f)
Which can be combined with Parsevals Thm´∞−∞ |g(t)|2dt =
´∞−∞ |G(f)|2df to give
ˆ ∞
−∞|g′(t)|2dt = 4π2
ˆ ∞
−∞f 2|G(f)|2df
Schwarz inequality states that |~x|2|~y|2 ≥ |~x · ~y|2 or 4(~x · ~x ~y · ~y) ≥ |~x · ~y+ ~y · ~x|2 whichgives for functions
4
ˆ
g(t)·g∗(t)dtˆ
h(t)·h∗(t)dt ≥∣∣∣∣ˆ
g(t)·h∗(t) + g∗(t)·h(t)dt∣∣∣∣
Integration by parts´
u dv = uv −´
v du for functions which have a well definedwidth and decay to 0 as t→ ±∞ gives (u = t dv = g′dt→ du = dt v = g)
ˆ ∞
−∞tg′(t) dt = tg(t)
∣∣∞−∞ +
ˆ ∞
−∞g(t)dt =
ˆ ∞
−∞g(t)dt
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 88
Uncertainty Relation Proofcalculate variance of power in time and
frequencyConsider the power width in time and Fourier domains p(t)= |s(t)|2 and P (f)= |S(f)|2Transform both functions to be centered at the origin without changing the width
g(t) = p(t)−〈p〉 s(t) = s(t)−〈p〉 G(f) = P (f)−〈P 〉 S(f) = S(f)−〈P 〉Define variance of the powers (∆t)2= σ2p = σ2g=〈t2〉 and (∆f)2= σ2P = σ2G =〈f 2〉 so
(∆t)2(∆f)2 =
´
(t− 〈t〉)2p(t)dt´
p(t)dt
´
(f − 〈f〉)2P (f)df´
P (f)df=
´
t2g(t)dt´
g(t)dt
´
f 2G(f)df´
G(f)df
=
´
t2s(t)s∗(t)dt´
s(t)s∗(t)dt
´
f 2S(f)S∗(f)df´
|S(f)|2df =
´
ts(t) · ts∗(t)dt´
s(t)s∗(t)dt
14π2
´
s′(t)s′∗(t)dt´
|s(t)|2dt
≥14
∣∣´ ts∗(t) · s′(t) + ts(t) · s′∗(t)dt∣∣
4π2(´
s(t)s∗(t)dt)2 =
∣∣´ t ddt(s(t)s∗(t)) dt
∣∣2
16π2(´
s(t)s∗(t)dt)2
=
∣∣´ s(t)s∗(t)dt∣∣2
16π2(´
s(t)s∗(t)dt)2 =
1
16π2
Therefore(∆t)(∆f) = σgσG ≥
1
4π
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 89
Fourier Transform ApodizationLearning Objectives and Outcomes
•Recognize relation between discontinuities and assymp-totic Fourier domain rolloff
•Understand main lobe widening with apodization
•Recognize common apodization windows and Gaussianapodization
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 90
Low pass Filtersconsiderh(t) = e−atu(t)
F⇐⇒ 1
a + jω= H(ω)
magnitude and phase are given by
|H(ω)| = 1√a2 + ω2
∠H(ω) = − tan−1ω
aThe values of magnitude and phase variesfrom
ω |H(ω)| ∠H(ω)0 1 0
a 1/√2 −π/4
∞ 0 −π/2
now consider
h2(t) =0.5
be−b|t|
F⇐⇒ 1
b2 + ω2= H2(ω)
In this case the filter response is phase flat
ω |H2(ω)| ∠H2(ω)0 1 0b 1/2 0∞ 0 0
h2(t) real and even⇐⇒ FT real and even
But as a physial ciruit filter this is not re-alistic since it is non causal, but OK inspace h2(x)
F⇐⇒H2(kx)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 91
Apodization : Lowering the feet (toes)
Fourier domain sidelobes can be lowered by multiplying by smooth weighting functionTypically use raised cosine or Hamming or Hanning window in DSPIn optics we multiply by a truncated gaussian
Discontinuities lead to 20dB/decade assymptotic rolloffDiscontinuities of the derivative lead to 40dB/decade assymptotic rolloffDiscontinuities of the 2nd derivative lead to 60dB/decade assymptotic rolloff
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 92
Unit Width Apodization Functions
• Smoother functions with same support have wider mainlobeBoth 3dB half width (.453,.656,.796) and first zero (1,2,3)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 93
Examples of Apodization
Discontinuitysign(t)
1
iωasymptotics 20db/decade
Π(t) sinc(ω) asymptotics 20db/decade
Discontinuity of slopesign(t) ∗ Π(t)
1
(iω)
sinω/2
ω/2asymptotics 40db/decade
Λ(t) sinc2(ω) asymptotics 40db/decade
Discontinuity of second derivativeΛ(t) ∗ Π(t) sinc3(ω) asymptotics 60db/decade
name Mainlobe width Rolloff rate peak sidelobeΠ(t/T ) rect 2/T -6/oct -20/dec -13.3dBΛ(t/T
2) Bartlett 4/T -12/oct -40/dec -26.5dB
12[1 + cos 2πt
T ] Hanning 4/T -18/oct -60/dec -31.5dB.54 + .46 cos 2πt
T Hamming 4/T -6/oct -20/dec -42.7dB.42 + .5 cos 2πt
T + .08 cos 4πtT Blackman 6/T -18/oct -60/dec -58.1dB
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 94
Truncated Gaussian Impulse Response
w(x) = Π
(x
D− 1
2
)e−4T
2(x/D−.5)2
σ = 12T = ω0
D , truncation ratio T = D2ω0
, 2ω0 = 1/e2 intensity width.
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 95
Effective of Varying the Gaussiantruncation ratio
• Peak intensity drops
•Mainlobe widens
• 1st sidelobe drops
• Assymptotics stays at 20dB/decadeKelvin Wagner, University of Colorado Fourier Optics Fall 2019 96
Routes to computing Fourier Transform
• If possible, use properties and table of known transforms.– Decompose as sum, product, or convolution of known function or deriva-
tive/integral of known function
• If x(t) has finite support its FT exists, so can use integral definition– or can use Laplace transform
• IF Laplace transform X(s) has ROC that contains iω axis, its FT is X(s)∣∣s=iω
• If x(t) is periodic infinite energy but finite power signal, its FT is obtained from FSusing delta functions or equivalently using comb T convolved with a single periodp(t) whose FT can be obtained.
• If x(t) is none of the above, if it has discontinuities (eg u(t) or sgn(t)), or it hasdisconinuities and is not finite energe (eg u(t) cos(ω0t)), then either– add a convergence factor– or take limits of support– or take limits in the Laplace domain.
It is best in such cases to use properties and known transforms if possible.
• Last resort: Complex contour integration and residue calculus
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 97
Fourier Transform Properties Summary
• Linearity ax(t) + by(t) aX(f) + bY (f)
• Shift x(t− t0) e−i2πft0X(f)
• Scale x(at) 1|a|X
(fa
)
• Convolutionˆ
g(t′)h(t− t′)dt′ G(f)H(f)
• Correlationˆ
g(t′)h∗(t′ − t)dt′ G(f)H∗(f)
• Modulation g(t)h(t)
ˆ
G(f ′)H(f − f ′)df ′
• Differentiation dn
dtng(t) (i2πf)nG(f)
• Integrationˆ t
−∞g(τ )dτ
1
i2πfG(f) +
G(0)
2δ(f)
• Duality FFg(t) = g(−t)
• Parsevals’ thmˆ ∞
−∞|x(t)|2dt =
ˆ ∞
−∞|X(f)|2df
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 98
FT summary
Symmetries (Duality gives additional symmetries)
Real in t F⇐⇒ Hermitian in f
Even Real in t F⇐⇒ Even Real in f
Odd Real in t F⇐⇒ Odd Imaginary in f
Wide(narrow) in t F⇐⇒ Narrow(wide) in f
Periodic in t F⇐⇒ Sampled in f
Some transforms
rect (t)F⇐⇒ sinc(f)
comb (t)F⇐⇒comb (f)
1(t)F⇐⇒δ(f)
e−αtu(t)F⇐⇒ 1
α+i2πf
e−αt sin(ω0t)u(t)F⇐⇒ ω0
ω20+(α+iω)2= 2πf0
(2πf0)2+(α+i2πf)2
ω transforms are obtained from f transforms by substitution ω = 2πf and vice versa,and scaling any delta functions by 2π
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Fourier Transform Summary
• CTFS is a special case of CTFT
• Frequency spectrum plotted as |X(f)| and ∠X(f) or ℜX(f) and ℑX(f)• Signals and Systems are more usefully described by their f domain properties
– Thinking in both domains yields the clearest insights and checks
• FT of impulse response of LTI system, h(t), yields Transfer function H(f)
Output spectrum Y (f) =X(f)H(f) so in time y(t) =F−1X(f)H(f)• Generalized CTFT allows periodic signals to be FT as frequency impulses
The FT of a periodic signal consist only of impulses regularly spaced in f at kT
• The more a signal is localized in one domain, the less it is localized in the other
• ∗ and × of signals and their transforms are dual operations in t and f
• Signal energy is conserved by the FT, as is signal similarity (dot product)– FT is a 90 rotation in function space and FT are thus unique
• Most FT can be done using tables of transforms and properties of transforms
• Existence of FT unless infinite number of discontinuities or∞ in finite interval
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 100
Linear Space Invariant SystemLearning Objectives and Outcomes
•Recognize shift invariance in space as cousin of LTI
•Understand operation of convolution in 1-D as convolu-tion movie
•Generalize convolution movie to the case of 2-D convolu-tions
•Appreciate imaging and diffraction as examples of 2-Dconvolution
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 101
Linear Time Invariant (LTI) Systems
Linear Circuits and linear fiber optics share fundamental simplifying propertiesLinearity, Causality, Memory
Such time domain linear systems are described by mathematical operators that interre-late the input, x(t), and output, y(t), waveforms as a convolution integral
y(t) = L x(t) = h(t)∗x(t) = x(t)∗h(t) =ˆ ∞
−∞h(t−τ )x(τ )dτ =
ˆ ∞
−∞h(τ )x(t−τ )dτ
Where h(t) is the impulse response and is causal if h(t) = 0 ∀ t < 0
Convolution MovieFor finite support functions the convolution operation can be thought of graphically:
1. plot x(τ ) versus dummy variable of integration τ2. Mirror image the impulse response as h(−τ ) (reflect about origin)3. Slide mirrored impulse far to the left (t≪ 0) to plot h(t− τ )4. Find the area of the product x(τ )h(t− τ ) giving the value of the convolution y(t)
For 0, 1 functions this is the width of the overlap region5. Now slide h(t− τ ) to the right and for every value of t where it overlaps with x(τ )
calculate the area of the product y(t) =´∞−∞ x(τ )h(t− τ )dτ
6. Plot y(t) for all values where the two functions overlap.
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 102
1-D Convolution Movie Sketch of Π( t2
)∗ Π(t)
-.5 .5
-1.5
-1
-.5
0
.5
1
1.5
-1 1
1
1
Π(t)
Π(t/2)
t
t
τ
τ
τ
τ
τ
τ
τ
τ
τ
-1.5 1.5-.5 .5
1
t
Π(t)*Π(t/2)1.5
Note the Π(t/2) is a rect of width 2, NOT a rect of width 1/2, since t has toget twice as large for the value of t/2 to reach ±1/2. Or by unit analysis, t/2is unitless (inside the argument of the rect function), so the width must be 2[s].Π(2t) = Π(t/.5) is a rect of width 1/2.
As the rect of width 1 slides into the rect of width 2, at a shift of t = −1.5 thearea increases linearly until the rect is fully contained in the wider rect starting att = −.5 at which point the full area of 1 remains constant at 1 until t = .5 Thenthe rect slides out and the area decreases linearly back to zero at t=1.5.
The total width of the trapezoid is 3, and each segment is of width 1. Since thetwo inputs are centered at t = 0, the output is also centered at the sum of the twoinputs, so is also centered at t = 0.
b(t) =
ˆ ∞
−∞Π
[t
2
]Π(t− τ )dτ =
ˆ 1
−1Π(t− τ )dτ
=
0 0 < t−1.5´ t+.5
−1 dτ = τ∣∣t+.5−1 = t + 1.5 −1.5 < t < −.5
´ t+.5
t−.5 dτ = 1 −.5 < t < .5´ 1
t−.5 dτ = τ∣∣1t−.5 = 1.5− t .5 < t < 1.5
0 t > 1.5
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 103
1-D Convolution Movie
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 104
1-D Convolution Movie
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CT Table of Convolutions
f(t) h(t) f(t) ∗ h(t) F (ω)H(ω)
f(t) δ(t− T ) f(t− T ) F (ω)e−iωt
Π(t) Π(t) Λ(t) sinc(ω2π
)sinc
(ω2π
)
e−λtu(t) u(t) 1−e−λtλ u(t) 1
iω+λ
(1iω + πδ(ω)
)
u(t) u(t) tu(t)(1iω + πδ(ω)
) (1iω + πδ(ω)
)
e−λtu(t) e−λtu(t) te−λtu(t) 1iω+λ
1iω+λ
e−αtu(t) e−βtu(t) e−αt−e−βtβ−α u(t) 1
iω+α1
iω+β
te−λtu(t) e−λtu(t) 12t2e−λtu(t) 1
(λ+iω)21
iω+λ
tme−λtu(t) tne−λtu(t) m!n!(m+n+1)!
tm+n+1eλtu(t) m!(λ+iω)m
n!(λ+iω)n
e−αt cos(βt + θ)u(t) e−λtu(t) cos(θ−φ)eλt−e−αt cos(βt+θ−φ)√(α+λ)2+β2
u(t) (α+iω) cos θ+β sin θ(α+iω)2+β2
1iω+λ
φ = − tan−1 βα+λ
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 106
Linear Space Invariant (LSI) Systemseg Shift Invariant
Space Variant Linear Transform
g(x) =
ˆ ∞
−∞f(x′)h(x′, x)dx′ = Lf(x) g = H f
impulses at different positions of the input x′ yield different outputs h(x, x′)
When the transfom is space invariant
g(x− a) = Lf(x− a) ⇒ h(x′, x) = h(x− x′)
⇒ g(x) =
ˆ ∞
−∞f(x′)h(x− x′)dx′ g = h ∗ f = F−1x HF
1-D impulse response h(x)
Eigenfunctions of any LSI operator are the complex exponentials with eigenvaluesgiven by the Transfer function H(f) = Ff(x)
Lei2πf0x =ˆ
ei2πf0x′h(x− x′)dx′ =
ˆ
ei2πf0(x−x”)h(x”)dx”
= ei2πf0xˆ
e−i2πf0x”h(x”)dx” = ei2πf0xH(f0)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 107
Auto Convolution Movie:e−tu(t) ∗ e−tu(t) = te−tu(t)
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Auto Convolution Movie:e−2tu(t) ∗ e−2tu(t) = te−2tu(t)
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Cross Convolution Movie:e−2tu(t) ∗ e−tu(t) = e−t−e−2t
2−1 u(t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 110
Cross Convolution Movie:e−4tu(t) ∗ e−tu(t) = e−t−e−4t
4−1 u(t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 111
Cross Convolution Movie:e−8tu(t) ∗ e−tu(t) = e−t−e−8t
8−1 u(t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 112
Convolution Movie:te−2tu(t) ∗ e−2tu(t) = 1
2!t2e−2tu(t)
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Convolution Movie:te−2tu(t) ∗ te−2tu(t) = 1
3!t3e−2tu(t)
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Convolution Movie:t2e−2tu(t) ∗ t2e−2tu(t) = 2!2!
5! t5e−2tu(t)
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Convolution Movie:t2e−4tu(t) ∗ t2e−4tu(t) = 2!2!
5! t5e−4tu(t)
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