FROM PTOLEMY TO MP3 DIGITAL MUSIC:
2000 Years of Applications of Harmonic Analysis
Yuri S. Ledyaev
Department of Mathematics
Western Michigan University
℡ (269) 387-4557
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 1/13
Ptolemaic Geocentric System
Ancient Greek Astronomer Claudius Ptolemaeus (c.90-169) (inEnglish, Ptolem y)
Famous astronomical treatiseAlma gest ("The Great Treatise")It was preserved (as many AncientGreek texts in Arabic manuscriprs)Translated into Latinin 12th century by Gerard of Cremona
Geocentric model of solarsystem
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Ancient Greek Astronomer Claudius Ptolemaeus (c.90-169) (inEnglish, Ptolem y)
Moon
Mercury
Venus
Sun
Mars
Jupiter
Saturn
Fixed StarsIdeal motions are circular (Aristotle, Plato)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Ancient Greek Astronomer Claudius Ptolemaeus (c.90-169) (inEnglish, Ptolem y)
Moon
Mercury
Venus
Sun
Mars
Jupiter
Saturn
Fixed StarsGeocentric system: planets moving around spherical EarthDeferent-and-epicycle model
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Geocentric system: planets moving around spherical EarthDeferent-and-epicycle model
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Geocentric system: planets moving around spherical EarthDeferent-and-epicycle model
planets moving around spherical Earth
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Heliocentric solar system of Copernicus, Galileo and Kepler:All planets (including Earth) moving around Sun
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Heliocentric solar system of Copernicus, Galileo and Kepler:All planets (including Earth) moving around Sun
QUESTION: Why geocentric system gave satisfactoryapproximation of observations?
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Heliocentric solar system of Copernicus, Galileo and Kepler:All planets (including Earth) moving around Sun
QUESTION: Why geocentric system gave satisfactoryapproximation of observations?
ANSWER: For the same reason that we can listen digitalmusic now
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Ancient Greek Astronomer Claudius Ptolemaeus (c.90-169)
Portable media players designed by Apple iPod (c.2001 - )
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Ptolemaic Geocentric System
Ancient Greek Astronomer Claudius Ptolemaeus (c.90-169)
Portable media players designed by Apple iPod (c.2001 - )
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 2/13
Harmonic Analysis during last 2000 years
Harmonic analysis : approximation of functions by using finiteseries of simple periodic functions
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Harmonic Analysis during last 2000 years
Harmonic analysis : approximation of functions by using finiteseries of simple periodic functions
sin(t) and cos(t)
Consider function f : [−T, T ] → R
f(t) v a0+a1 cos(πt/T )+b1 sin(πt/T )+a2 cos(2πt/T )+b2 sin(2πt/T )) . . .
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Harmonic Analysis during last 2000 years
Harmonic analysis : approximation of functions by using finiteseries of simple periodic functions
sin(t) and cos(t)
Consider function f : [−T, T ] → R
f(t) v c0+c1 cos(πt/T )+s1 sin(πt/T )+c2 cos(2πt/T )+s2 sin(2πt/T )) . . .
f(t) v∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Harmonic Analysis during last 2000 years
Consider function f : [−T, T ] → R
f(t) v c0+c1 cos(πt/T )+s1 sin(πt/T )+c2 cos(2πt/T )+s2 sin(2πt/T )) . . .
f(t) v∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
where c0 =1
2T
∫ T
−Tf(t)dt and for k ≥ 1
ck =1
T
∫ T
−Tf(t) cos(
πk
Tt)dt, sk =
1
T
∫ T
−Tf(t) sin(
πk
Tt)dt
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Harmonic Analysis during last 2000 years
Fourier series (Fourier 1807 , Analytical Theory of Heat )
f(t) v∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
where c0 =1
2T
∫ T
−Tf(t)dt and for k ≥ 1
ck =1
T
∫ T
−Tf(t) cos(
πk
Tt)dt, sk =
1
T
∫ T
−Tf(t) sin(
πk
Tt)dt
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Harmonic Analysis during last 2000 years
Fourier series (Fourier 1807 , Analyt-ical Theory of Heat )For large class of functions f
f(t)! =∞∑
k=0
ck cos(πk
Tt) + sk sin(
2πk
Tt)
Good approximating properties
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 3/13
Examples: Approximation by Fourier Trig. Polynomials
f(t) = t(1 − t). t ∈ [0, 1]
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 4/13
Examples: Approximation by Fourier Trig. Polynomials
0.1666666667 − 0.1013211836 cos (6.283185308 t) −4.156265120 × 10−11 sin (6.283185308 t)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 4/13
Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 4/13
Examples: Approximation by Fourier Trig. Polynomials
0.1666666667 − 0.1013211836 cos (6.283185308 t) −4.156265120 × 10−11 sin (6.283185308 t) − 0.0253303 cos (12.5663 t) −0.01125790930 cos (18.84955592 t) MATH 5710, Advanced Calculus II, March 23, 2020 – p. 4/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
f(t) = 1 − |t − 1|, t ∈ [0, 2]
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
0.5000000000 − 0.4052847344 cos (3.141592654 t) −0.0000000001677970458 sin (3.141592654 t)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
0.5000000000 − 0.4052847344 cos (3.141592654 t) −0.0000000001677970458 sin (3.141592654 t) +
3.469446952 × 10−17 cos (6.283185308 t) +
3.199817670 × 10−12 sin (6.283185308 t)MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 5/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3
x
f(t) = t2, t ∈ [0, 1]
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 6/13
Examples: Approximation by Fourier Trig. Polynomials
0.3333333333 + 0.1013211839 cos (6.283185308 t) −0.3183098860 sin (6.283185308 t)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 6/13
Examples: Approximation by Fourier Trig. Polynomials
0.3333333333 + 0.1013211839 cos (6.283185308 t) −0.3183098860 sin (6.283185308 t) + 0.02533029678 cos (12.56637062 t)−0.1591549429 sin (12.56637062 t)
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 6/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
f(t) =
{1, t ∈ [0, 1]
0, t ∈ [1, 2)MATH 5710, Advanced Calculus II, March 23, 2020 – p. 7/13
Examples: Approximation by Fourier Trig. Polynomials
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 7/13
Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Examples: Approximation by Fourier Trig. Polynomials
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Fourier Series: Approximation and Ptolemaic Geocentric System
Fourier series (Fourier 1807 , Analyt-ical Theory of Heat )For large class of functions f
f(t)! =∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
Good approximating properties – Ptole-maic Geocentric System gave satisfac-tory approximation of observation data
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 8/13
Fourier Series: Approximation and Ptolemaic Geocentric System
Good approximating properties – Ptolemaic Geocentric Systemgave satisfactory approximation of observation data
Geocentric system: planets moving around spherical EarthDeferent-and-epicycle model
planets moving around spherical Earth
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 8/13
Fourier Series: Approximation and Ptolemaic Geocentric System
Good approximating properties – Ptolemaic Geocentric Systemgave satisfactory approximation of observation dataDescription of rotation with period T1
r1(t) = Ω1(t)r1(0)
where Ω1(t) is the rotation matrix
Ω1(t) =
[cos(2π
T1t) sin(2π
T1t)
− cos(2πT1
t) cos(2πT1
t)
]
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 8/13
Fourier Series: Approximation and Ptolemaic Geocentric System
Why motion of planet is approximated by the function r(t)
r(t) =
[cos(2π
T1t) sin(2π
T1t)
− cos(2πT1
t) cos(2πT1
t)
]
r1(0) +
[cos(2π
T2t) sin(2π
T2t)
− cos(2πT2
t) cos(2πT2
t)
]
r2(0)
planets moving around spherical Earth
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 8/13
Fourier series, Cosine and Sine Series, Fourier Transform
Fourier series (Fourier 1807 , Analyt-ical Theory of Heat )For large class of functions f
f(t)! =∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
Good approximating properties – Ptole-maic Geocentric System gave satisfac-tory approximation of observation data
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Why in
f(t)=∞∑
k=0
ck cos(πk
Tt) + sk sin(
πk
Tt)
we have
ck =1
T
∫ T
−Tf(t) cos(
πk
Tt)dt, sk =
1
T
∫ T
−Tf(t) sin(
πk
Tt)dt
Consider R3 with orthogonal basis: vec-tors −→e1 ,−→e2 ,−→e3 ( −→ei ∙ −→ej = 0). Vector
−→f :
−→f = c1
−→e1 + c2−→e2 + c3
−→e3
−→e1 ∙−→f = c1
−→e1 ∙ −→e1 c1 =−→e1 ∙
−→f
−→e1 ∙ −→e1MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Cosine series for f : [0, T ] → R
f(t)=∞∑
k=0
ck cos(πk
Tt)
Sine series for f : [0, T ] → R
f(t)=∞∑
k=1
sk sin(πk
Tt)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Cosine series for f : [0, T ] → R
f(t)=∞∑
k=0
ck cos(πk
Tt)
Sine series for f : [0, T ] → R
f(t)=∞∑
k=1
sk sin(πk
Tt)
Integral representation for f : (−∞, +∞) → R
f(t) =
∫ +∞
0C(ω) cos ωt dω +
∫ +∞
0S(ω) sin ωt dω
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Integral representation for f : (−∞, +∞) → R
f(t) =
∫ +∞
0C(ω) cos ωt dω +
∫ +∞
0S(ω) sin ωt dω
where Fourier transform
C(ω) =1
π
∫ +∞
−∞f(t) cos ωt dt, S(ω) =
1
π
∫ +∞
−∞f(t) sin ωt dt
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Integral representation for f : (−∞, +∞) → R
f(t) =
∫ +∞
0C(ω) cos ωt dω +
∫ +∞
0S(ω) sin ωt dω
where Fourier transform
C(ω) =1
π
∫ +∞
−∞f(t) cos ωt dt, S(ω) =
1
π
∫ +∞
−∞f(t) sin ωt dt
Hundreds applications : Electric and Control engineering, Signalprocessing, Optics and Spectroscopy, X-ray Crystallography(Protein structure, DNA,. . . ), Computerized Tomography,Radioastronomy, . . .
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Fourier series, Cosine and Sine Series, Fourier Transform
Integral representation for f : (−∞, +∞) → R
f(t) =
∫ +∞
0C(ω) cos ωt dω +
∫ +∞
0S(ω) sin ωt dω
where Fourier transform
C(ω) =1
π
∫ +∞
−∞f(t) cos ωt dt, S(ω) =
1
π
∫ +∞
−∞f(t) sin ωt dt
Hundreds applications : Electric and Control engineering, Signalprocessing, Optics and Spectroscopy, X-ray Crystallography(Protein structure, DNA,. . . ),At WMU : Math 5710 - Analysis (Fourier series), Math 5740Advanced Differential Equations (basics of Harmonic Analysis,applications to Partial Differential Equations)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 9/13
Harmonic Analysis and Signal Processing
Assume that we have an audio signal given by a continuousfunction f(t).How to reproduce such function? Fourier series?
f(t)=∞∑
k=0
ck cos(2πk
Tt) + sk sin(
2πk
Tt)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Assume that we have an audio signal given by a continuousfunction f(t).How to reproduce such function? Fourier series?
f(t)=∞∑
k=0
ck cos(2πk
Tt) + sk sin(
2πk
Tt)
But
ck =2
T
∫ T
0f(t) cos(
2πk
Tt)dt, sk =
2
T
∫ T
0f(t) sin(
2πk
Tt)dt
and f(t) should be T−periodic and it cannot be transmitted in realtime
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Practical approach based on the Sampling Theorem :
any practical audio- (or video-) signal has bounded bandwidth
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Practical approach based on the Sampling Theorem :
any practical audio- (or video-) signal has bounded bandwidth(namely, human ear can hear frequencies bounded fromabove by 20 KHz)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Practical approach based on the Sampling Theorem :
any practical audio- (or video-) signal has bounded bandwidth(namely, human ear can hear frequencies bounded fromabove by 20 KHz)
We can assume that there exists a frequency ωm such that
C(ω) = 0, S(ω) = 0 ∀ ω ≥ ωm
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Practical approach based on the Sampling Theorem :
any practical audio- (or video-) signal has bounded bandwidth(namely, human ear can hear frequencies bounded fromabove by 20 KHz)
We can assume that there exists a frequency ωm such that
C(ω) = 0, S(ω) = 0 ∀ ω ≥ ωm
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Practical approach based on the Sampling Theorem :
any practical audio- (or video-) signal has bounded bandwidth(namely, human ear can hear frequencies bounded fromabove by 20 KHz)
We can assume that there exists a frequency ωm such that
C(ω) = 0, S(ω) = 0 ∀ ω ≥ ωm
Thus, the signal f(t)
f(t) =
∫ ωm
0C(ω) cos ωt dω +
∫ ωm
0S(ω) sin ωt dω
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
Sampling TheoremTHEOREM: Let f(t) have bounded bandwidth ωm andT > 0 satisfy
ω0 :=2π
T> 2 ωm
Then
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
THEOREM: Let f(t) have bounded bandwidth ωm andT > 0 satisfy
ω0 :=2π
T> 2 ωm
Then
f(t) =+∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
CONCLUSION: To reconstruct the entire signal f(t) it is enoughto know values of this signal at sampling moments kT , T shouldbe "small"
f(0), f(T ), f(2T ), f(3T ), . . . , f (kT ), . . .
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Harmonic Analysis and Signal Processing
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 10/13
Sampling Theorem
f(t) =+∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
1
T> 2νm
Sampling rate is greater than the double of bandwidth
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
f(t) =+∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
1
T> 2νm
Sampling rate is greater than the double of bandwidthEXAMPLE: Audio CD (Red Book standard): channel bandwidth22.05 KHz, sampling rate 44.1 KHZHardware: Analog-to-Digital Converter (ADC),to reconstruct signal use Digital-To-Analog Converter (DAC)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
Aliasing
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
Aliasing
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
NO Aliasing if1
T> 2νm
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
Anti-aliasing filters (usually Low-pass filter added to ADC)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
Shannon 1949 Communication inthe presence of noiseVery elegant and short proof using Diracdelta -function
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Sampling Theorem
Kotelnikov 1933 On the transmis-sion capacity of wireless and ca-bles in electrical communicationsShort proof using traditional toolsOther names associated with this result:Whittaker 1915 Interpolation TheoremNyquist 1928Someya circa 1949Weston circa 1949
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 11/13
Proof of the Sampling Theorem ( after Kotelnikov)
Assume that ω0
2 := πT > ωm
STEP1: show that for any signal f(t) with bandwidth bounded byωm
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
for some coefficients Dk
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
Assume that ω0
2 := πT > ωm
STEP1: show that for any signal f(t) with bandwidth bounded byωm
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
for some coefficients Dk
We start with
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
Assume that ω0
2 := πT > ωm
STEP1: show that for any signal f(t) with bandwidth bounded byωm
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
for some coefficients Dk
We start with
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
REMINDER:
C(ω) = 0, S(ω) = 0 for all ω > ωm (Signal with a bounded spectrum)
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
for some coefficients Dk
We start with
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
TRICK: Write Cosine series forC(ω) and Sine series for S(ω)(remember ω0
2 > ωm)
C(ω) =∞∑
k=0
Ak cos(2πk
ω0ω), S(ω) =
∞∑
k=0
Bk sin(2πk
ω0ω)
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
π(t/T − k)
We start with
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
TRICK: Write Cosine series forC(ω) and Sine series for S(ω)(remember ω0
2 > ωm)
C(ω) =∞∑
k=0
Ak cos(2πk
ω0ω), S(ω) =
∞∑
k=0
Bk sin(2πk
ω0ω)
Define Dk := Ak+Bk
2 for k ≥ 0, Dk := A−k−B−k
2 for k < 0
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
C(ω) =∞∑
k=0
Ak cos(2πk
ω0ω), S(ω) =
∞∑
k=0
Bk sin(2πk
ω0ω)
Define Dk := Ak+Bk
2 for k ≥ 0, Dk := A−k−B−k
2 for k < 0. Then
C(ω) =∞∑
k=−∞
Dk cos(2πk
ω0ω), S(ω) =
∞∑
k=−∞
Dk sin(2πk
ω0ω)
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
C(ω) =∞∑
k=0
Ak cos(2πk
ω0ω), S(ω) =
∞∑
k=0
Bk sin(2πk
ω0ω)
Define Dk := Ak+Bk
2 for k ≥ 0, Dk := A−k−B−k
2 for k < 0. Then
C(ω) =∞∑
k=−∞
Dk cos(2πk
ω0ω), S(ω) =
∞∑
k=−∞
Dk sin(2πk
ω0ω)
Plug C(ω) and S(ω) in
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
C(ω) =∞∑
k=−∞
Dk cos(2πk
ω0ω), S(ω) =
∞∑
k=−∞
Dk sin(2πk
ω0ω)
Plug C(ω) and S(ω) in
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
to obtain
f(t) =
∫ ω02
0
∞∑
k=−∞
Dk cos(2πk
ω0ω) cos ωt dω+
∫ ω02
0
∞∑
k=−∞
Dk sin(2πk
ω0ω) sin ωt dω
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
Plug C(ω) and S(ω) in
f(t) =
∫ ω02
0C(ω) cos ωt dω +
∫ ω02
0S(ω) sin ωt dω
to obtain
f(t) =
∫ ω02
0
∞∑
k=−∞
Dk cos(2πk
ω0ω) cos ωt dω+
∫ ω02
0
∞∑
k=−∞
Dk sin(2πk
ω0ω) sin ωt dω
We can rewrite it
f(t) =∞∑
k=−∞
Dk
∫ ω02
0[cos(
2πk
ω0ω) cos ωt + sin(
2πk
ω0ω) sin ωt] dω
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
We can rewrite it
f(t) =∞∑
k=−∞
Dk
∫ ω02
0[cos(
2πk
ω0ω) cos ωt + sin(
2πk
ω0ω) sin ωt] dω
Now we use trig identity
f(t) =∞∑
k=−∞
Dk
∫ ω02
0cos(ωt −
2πk
ω0ω) dω
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
Now we use trig identity
f(t) =∞∑
k=−∞
Dk
∫ ω02
0cos(ωt −
2πk
ω0ω) dω
Evaluate integrals to obtain
f(t) =∞∑
k=−∞
Dk
sin ω0
2 (t − 2πkω0
)
t − 2πkω0
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
Now we use trig identity
f(t) =∞∑
k=−∞
Dk
∫ ω02
0cos(ωt −
2πk
ω0ω) dω
Evaluate integrals to obtain
f(t) =∞∑
k=−∞
Dk
sin ω0
2 (t − 2πkω0
)
t − 2πkω0
Recall that ω0 = 2πT then
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
But this is a result fromSTEP 1: show that for any signal f(t) with bandwidth bounded byωm
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kT
for some coefficients Dk
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
STEP 1: We demonstrated for any signal f(t)
f(t) =∞∑
k=−∞
Dksin π(t/T − k)
t − kTfor some coefficients Dk
STEP 2: how to find Dk?Take limit t → nT
f(nT ) = limt→nT
f(t) =+∞∑
k=−∞
Dk limt→nT
sin π(t/T − k)
t − kT
Then for k 6= n limt→nT sin π(t/T − k) = 0.For k = n use l’Hopital rule to obtain
f(nT ) = Dnπ
TDk = f(kT )
T
π
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
Proof of the Sampling Theorem ( after Kotelnikov)
f(t) =∞∑
k=−∞
f(kT )sin π(t/T − k)
π(t/T − k)
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 12/13
The Last Slide
Example: application of the Sampling Theorem
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 13/13
The Last Slide
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 13/13
The Last Slide
MATH 5710, Advanced Calculus II, March 23, 2020 – p. 13/13