Fourier Series and Transforms Revision LectureThe Basic Idea Fourier Series and Transforms Revision...
Transcript of Fourier Series and Transforms Revision LectureThe Basic Idea Fourier Series and Transforms Revision...
Fourier Series and TransformsRevision Lecture
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 1 / 13
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
+
T/4
-0.13cos(2πFt)
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
+
T/4
-0.13cos(2πFt)
Fundamental Period: the smallest T > 0 for which u(t+ T ) = u(t).
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
+
T/4
-0.13cos(2πFt)
Fundamental Period: the smallest T > 0 for which u(t+ T ) = u(t).Fundamental Frequency: F = 1
T.
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
+
T/4
-0.13cos(2πFt)
Fundamental Period: the smallest T > 0 for which u(t+ T ) = u(t).Fundamental Frequency: F = 1
T. The nth harmonic is at frequency nF .
The Basic Idea
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 2 / 13
Periodic signals can be written as a sum of sine and cosine waves:
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Tu(t)
=
T +0.65sin(2πFt)
+
T/2-0.26sin(2πFt)
+
T/3+0.26sin(2πFt)
+
T/4
-0.13cos(2πFt)
Fundamental Period: the smallest T > 0 for which u(t+ T ) = u(t).Fundamental Frequency: F = 1
T. The nth harmonic is at frequency nF .
Some waveforms need infinitely many harmonics (countable infinity).
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn)
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn) and U−n = 1
2 (an + ibn) .
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn) and U−n = 1
2 (an + ibn) .
• U+n and U−n are complex conjugates.
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn) and U−n = 1
2 (an + ibn) .
• U+n and U−n are complex conjugates.
• U+n is half the equivalent phasor in Analysis of Circuits.
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn) and U−n = 1
2 (an + ibn) .
• U+n and U−n are complex conjugates.
• U+n is half the equivalent phasor in Analysis of Circuits.
Tu(t)
Plot the magnitude spectrum-4F -3F -2F -F 0 F 2F 3F 4F
0
0.5
1
|U|
Real versus Complex Fourier Series
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 3 / 13
All the algebra is much easier if we use eiωt instead of cosωt and sinωt
u(t) = a0
2 +∑∞
n=1 (an cos 2πnFt+ bn sin 2πnFt)
Substitute: cosωt = 12e
iωt + 12e
−iωt sinωt = −i2 eiωt + i
2e−iωt
u(t) = a0
2 +∑∞
n=1
(
12 (an − ibn) e
i2πnFt + 12 (an + ibn) e
−i2πnFt)
=∑∞
n=−∞ Unei2πnFt
• U+n = 12 (an − ibn) and U−n = 1
2 (an + ibn) .
• U+n and U−n are complex conjugates.
• U+n is half the equivalent phasor in Analysis of Circuits.
Tu(t)
Plot the magnitude spectrumand phase spectrum:
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals
Tu(t)
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series→ Discrete spectrum
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series→ Discrete spectrum
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
• Aperiodic signals
-5 0 50
0.5
1
Time (s)
u(t)
a=2
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series→ Discrete spectrum
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
• Aperiodic signals → Fourier Transform
-5 0 50
0.5
1
Time (s)
u(t)
a=2
u(t) =∫∞f=−∞ U(f)ei2πftdf
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series→ Discrete spectrum
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
• Aperiodic signals → Fourier Transform→ Continuous Spectrum
-5 0 50
0.5
1
Time (s)
u(t)
a=2
u(t) =∫∞f=−∞ U(f)ei2πftdf
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
-5 0 5-pi/2
0
pi/2
Frequence (Hz)
∠U
(f)
(rad
)
Fourier Series versus Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 4 / 13
• Periodic signals → Fourier Series→ Discrete spectrum
Tu(t)
u(t) =∑∞
n=−∞ Unei2πnFt
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
-4F -3F -2F -F 0 F 2F 3F 4F
-pi
0
pi
∠U
• Aperiodic signals → Fourier Transform→ Continuous Spectrum
-5 0 50
0.5
1
Time (s)
u(t)
a=2
u(t) =∫∞f=−∞ U(f)ei2πftdf
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
-5 0 5-pi/2
0
pi/2
Frequence (Hz)
∠U
(f)
(rad
)• Both types of spectrum are conjugate symmetric.• If u(t) is periodic, its Fourier transform consists of Dirac δ functions
with amplitudes {Un}.
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
So, to find a particular coefficient, Um, we work out⟨
u(t)e−i2πmFt⟩
=⟨ (
∑∞n=−∞ Une
i2πnFt)
e−i2πmFt⟩
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
So, to find a particular coefficient, Um, we work out⟨
u(t)e−i2πmFt⟩
=⟨ (
∑∞n=−∞ Une
i2πnFt)
e−i2πmFt⟩
=∑∞
n=−∞ Un
⟨
ei2πnFte−i2πmFt⟩
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
So, to find a particular coefficient, Um, we work out⟨
u(t)e−i2πmFt⟩
=⟨ (
∑∞n=−∞ Une
i2πnFt)
e−i2πmFt⟩
=∑∞
n=−∞ Un
⟨
ei2πnFte−i2πmFt⟩
= Um [since all other terms are zero]
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
So, to find a particular coefficient, Um, we work out⟨
u(t)e−i2πmFt⟩
=⟨ (
∑∞n=−∞ Une
i2πnFt)
e−i2πmFt⟩
=∑∞
n=−∞ Un
⟨
ei2πnFte−i2πmFt⟩
= Um [since all other terms are zero]
Calculate the average by integrating over any integer number of periods
Um =⟨
u(t)e−i2πmFt⟩
= 1T
∫ T
t=0u(t)e−i2πmFtdt
Fourier Analysis
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 5 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”
Key idea:⟨
eiωt⟩
= 〈cosωt+ i sinωt〉 =
{
1 if ω = 0
0 otherwise
⇒ Orthogonality:⟨
ei2πnFt × e−i2πmFt⟩
=
{
1 for m = n
0 for m 6= n
So, to find a particular coefficient, Um, we work out⟨
u(t)e−i2πmFt⟩
=⟨ (
∑∞n=−∞ Une
i2πnFt)
e−i2πmFt⟩
=∑∞
n=−∞ Un
⟨
ei2πnFte−i2πmFt⟩
= Um [since all other terms are zero]
Calculate the average by integrating over any integer number of periods
Um =⟨
u(t)e−i2πmFt⟩
= 1T
∫ T
t=0u(t)e−i2πmFtdt
Notice the negative sign in Fourier analysis: in order to extract the term inthe series containing e+i2πmFt we need to multiply by e−i2πmFt.
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
The average power in u(t) is equal to the sum of the averagepowers in all the Fourier components.
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
The average power in u(t) is equal to the sum of the averagepowers in all the Fourier components.
This is a consequence of orthogonality:⟨
|u(t)|2⟩
=⟨(∑∞
n=−∞ Unei2πnFt
) (∑∞
m=−∞ U∗me−i2πmFt
)⟩
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
The average power in u(t) is equal to the sum of the averagepowers in all the Fourier components.
This is a consequence of orthogonality:⟨
|u(t)|2⟩
=⟨(∑∞
n=−∞ Unei2πnFt
) (∑∞
m=−∞ U∗me−i2πmFt
)⟩
=⟨∑∞
n=−∞∑∞
m=−∞ UnU∗mei2πnFte−i2πmFt
⟩
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
The average power in u(t) is equal to the sum of the averagepowers in all the Fourier components.
This is a consequence of orthogonality:⟨
|u(t)|2⟩
=⟨(∑∞
n=−∞ Unei2πnFt
) (∑∞
m=−∞ U∗me−i2πmFt
)⟩
=⟨∑∞
n=−∞∑∞
m=−∞ UnU∗mei2πnFte−i2πmFt
⟩
=∑∞
n=−∞∑∞
m=−∞ UnU∗m
⟨
ei2πnFte−i2πmFt⟩
Power Conservation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 6 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Average power in u(t): Pu ,
⟨
|u(t)|2⟩
= 1T
∫ T
0u2(t)dt [u(t) real]
Average power in Fourier component n:⟨
∣
∣Unei2πnFt
∣
∣
2⟩
=⟨
|Un|2 ∣∣ei2πnFt
∣
∣
2⟩
= |Un|2
Power conservation (Parseval’s Theorem):
Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
The average power in u(t) is equal to the sum of the averagepowers in all the Fourier components.
This is a consequence of orthogonality:⟨
|u(t)|2⟩
=⟨(∑∞
n=−∞ Unei2πnFt
) (∑∞
m=−∞ U∗me−i2πmFt
)⟩
=⟨∑∞
n=−∞∑∞
m=−∞ UnU∗mei2πnFte−i2πmFt
⟩
=∑∞
n=−∞∑∞
m=−∞ UnU∗m
⟨
ei2πnFte−i2πmFt⟩
=∑∞
n=−∞ |Un|2
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 0 5 0
0 5 0 5 0
0 5 0 5 0
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 0 5 0
0 5 0 5 0
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 0 5 0
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2. 0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2.
PeN → 0 monotonically as N → ∞.
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2.
PeN → 0 monotonically as N → ∞.
Gibbs phenomenon
If u(t0) has a discontinuity of height h then:
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 .5 0 .5 1
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2.
PeN → 0 monotonically as N → ∞.
Gibbs phenomenon
If u(t0) has a discontinuity of height h then:
• uN (t0) → the midpoint of thediscontinuity as N → ∞.
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 -0.5 0 0.5 1
0
0.5
1 max(u41
)=1.089
[Enlarged View: u41(t)]
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2.
PeN → 0 monotonically as N → ∞.
Gibbs phenomenon
If u(t0) has a discontinuity of height h then:
• uN (t0) → the midpoint of thediscontinuity as N → ∞.
• uN (t) overshoots by ≈ ±9%× h att ≈ t0 ±
T2N+1 .
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 -0.5 0 0.5 1
0
0.5
1 max(u41
)=1.089
[Enlarged View: u41(t)]
Gibbs Phenomenon
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 7 / 13
Truncated Fourier Series: uN (t) =∑N
n=−N Unei2πnFt
Approximation error: eN (t) = uN (t)− u(t)
Average error power PeN =∑
|n|>N |Un|2.
PeN → 0 monotonically as N → ∞.
Gibbs phenomenon
If u(t0) has a discontinuity of height h then:
• uN (t0) → the midpoint of thediscontinuity as N → ∞.
• uN (t) overshoots by ≈ ±9%× h att ≈ t0 ±
T2N+1 .
• For large N , the overshoots movecloser to the discontinuity but do notdecrease in size.
0 5 10 15 20
0
0.5
1 max(u0)=0.500
N=0
0 5 10 15 20
0
0.5
1 max(u1)=1.137
N=1
0 5 10 15 20
0
0.5
1 max(u3)=1.100
N=3
0 5 10 15 20
0
0.5
1 max(u5)=1.094
N=5
0 5 10 15 20
0
0.5
1 max(u41
)=1.089
N=41
-1 -0.5 0 0.5 1
0
0.5
1 max(u41
)=1.089
[Enlarged View: u41(t)]
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Integration:v(t) =
∫ t
0u(τ )dτ ⇒ Vn = 1
i2πnF Un
provided U0 = V0 = 0.
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Integration:v(t) =
∫ t
0u(τ )dτ ⇒ Vn = 1
i2πnF Un
provided U0 = V0 = 0.
Differentiation:w(t) = du(t)
dt⇒ Wn = i2πnF × Un
provided w(t) satisfies the Dirichlet conditions.
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Integration:v(t) =
∫ t
0u(τ )dτ ⇒ Vn = 1
i2πnF Un
provided U0 = V0 = 0.
Differentiation:w(t) = du(t)
dt⇒ Wn = i2πnF × Un
provided w(t) satisfies the Dirichlet conditions.
Coefficient Decay Rate:u(t) has a discontinuity ⇒ |Un| is O
(
1n
)
for large |n|
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Integration:v(t) =
∫ t
0u(τ )dτ ⇒ Vn = 1
i2πnF Un
provided U0 = V0 = 0.
Differentiation:w(t) = du(t)
dt⇒ Wn = i2πnF × Un
provided w(t) satisfies the Dirichlet conditions.
Coefficient Decay Rate:u(t) has a discontinuity ⇒ |Un| is O
(
1n
)
for large |n|
dku(t)dtk
is the lowest derivative with a discontinuity
⇒ |Un| is O(
1nk+1
)
for large |n|
Coefficient Decay Rate
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 8 / 13
Fourier Series: u(t) =∑∞
n=−∞ Unei2πnFt
Integration:v(t) =
∫ t
0u(τ )dτ ⇒ Vn = 1
i2πnF Un
provided U0 = V0 = 0.
Differentiation:w(t) = du(t)
dt⇒ Wn = i2πnF × Un
provided w(t) satisfies the Dirichlet conditions.
Coefficient Decay Rate:u(t) has a discontinuity ⇒ |Un| is O
(
1n
)
for large |n|
dku(t)dtk
is the lowest derivative with a discontinuity
⇒ |Un| is O(
1nk+1
)
for large |n|
If the coefficients, Un, decrease rapidly then only a few terms areneeded for a good approximation.
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
-2 0 2 4 -2 0 2 4 -2 0 2 4
-2 0 2 4 -2 0 2 4 -2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4 -2 0 2 4
-2 0 2 4 -2 0 2 4 -2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
-2 0 2 4 -2 0 2 4 -2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
-2 0 2 4 -2 0 2 4 -2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Symmetric extension:• To avoid a discontinuity at t = T , we can instead make the period
2B and define u(−t) = u(+t).
-2 0 2 4
0
2
4 N=1
-2 0 2 4 -2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Symmetric extension:• To avoid a discontinuity at t = T , we can instead make the period
2B and define u(−t) = u(+t).
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Symmetric extension:• To avoid a discontinuity at t = T , we can instead make the period
2B and define u(−t) = u(+t).
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Symmetric extension:• To avoid a discontinuity at t = T , we can instead make the period
2B and define u(−t) = u(+t).
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
• Symmetry around t = 0 means coefficients are real-valued andsymmetric (U−n = U∗
n = Un).
Periodic Extension
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 9 / 13
If u(t) is only defined over a finite range, [0, B], we can make it periodicby defining u(t±B) = u(t).
• Coefficients are given by Un = 1B
∫ B
0u(t)e−i2πnFtdt.
Example: u(t) = t2 for 0 ≤ t < 2
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
Symmetric extension:• To avoid a discontinuity at t = T , we can instead make the period
2B and define u(−t) = u(+t).
-2 0 2 4
0
2
4 N=1
-2 0 2 4
0
2
4 N=3
-2 0 2 4
0
2
4 N=6
• Symmetry around t = 0 means coefficients are real-valued andsymmetric (U−n = U∗
n = Un).• Still have a first-derivative discontinuity at t = B but now we have
no Gibbs phenomenon and coefficients ∝ n−2 instead of ∝ n−1 soapproximation error power decreases more quickly.
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
high
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else
• Multiplication: f(x)× δ(x− a) = f(a)× δ(x− a)
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else
• Multiplication: f(x)× δ(x− a) = f(a)× δ(x− a)
• Integration:∫∞−∞ f(x)× δ(x− a)dx = f(a)
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else
• Multiplication: f(x)× δ(x− a) = f(a)× δ(x− a)
• Integration:∫∞−∞ f(x)× δ(x− a)dx = f(a)
• Fourier Transform: u(t) = δ(t) ⇔ U(f) = 1
Dirac Delta Function
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 10 / 13
δ(x) is the limiting case as w → 0 of a pulse w wide and 1w
highIt is an infinitely thin, infinitely high pulse at x = 0 with unit area.
-3 -2 -1 0 1 2 30
2
4δ
0.2(x)
x-3 -2 -1 0 1 2 3
0
0.5
1
δ(x)
x
• Area:∫∞−∞ δ(x)dx = 1
• Scaling: δ(cx) = 1|c|δ(x)
• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else
• Multiplication: f(x)× δ(x− a) = f(a)× δ(x− a)
• Integration:∫∞−∞ f(x)× δ(x− a)dx = f(a)
• Fourier Transform: u(t) = δ(t) ⇔ U(f) = 1
• We plot hδ(x) as a pulse of height |h| (instead of its true height of ∞)
Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13
Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf
U(f) =∫∞−∞ u(t)e−i2πftdt
Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13
Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf
U(f) =∫∞−∞ u(t)e−i2πftdt
• An “Energy Signal” has finite energy ⇔ Eu =∫∞−∞ |u(t)|2 dt < ∞
◦ Complex-valued spectrum, U(f), decays to zero as f → ±∞
-5 0 50
0.5
1
Time (s)
u(t)
a=2
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13
Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf
U(f) =∫∞−∞ u(t)e−i2πftdt
• An “Energy Signal” has finite energy ⇔ Eu =∫∞−∞ |u(t)|2 dt < ∞
◦ Complex-valued spectrum, U(f), decays to zero as f → ±∞
◦ Energy Conservation: Eu = EU where EU =∫∞−∞ |U(f)|
2df
-5 0 50
0.5
1
Time (s)
u(t)
a=2
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13
Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf
U(f) =∫∞−∞ u(t)e−i2πftdt
• An “Energy Signal” has finite energy ⇔ Eu =∫∞−∞ |u(t)|2 dt < ∞
◦ Complex-valued spectrum, U(f), decays to zero as f → ±∞
◦ Energy Conservation: Eu = EU where EU =∫∞−∞ |U(f)|
2df
-5 0 50
0.5
1
Time (s)
u(t)
a=2
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
• Periodic Signals → Dirac δ functions at harmonics.Same complex-valued amplitudes as Un from Fourier Series
Tu(t)
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
Fourier Transform
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 11 / 13
Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf
U(f) =∫∞−∞ u(t)e−i2πftdt
• An “Energy Signal” has finite energy ⇔ Eu =∫∞−∞ |u(t)|2 dt < ∞
◦ Complex-valued spectrum, U(f), decays to zero as f → ±∞
◦ Energy Conservation: Eu = EU where EU =∫∞−∞ |U(f)|
2df
-5 0 50
0.5
1
Time (s)
u(t)
a=2
-5 0 50.10.20.30.40.5
Frequency (Hz)
|U(f
)|
• Periodic Signals → Dirac δ functions at harmonics.Same complex-valued amplitudes as Un from Fourier Series
Tu(t)
-4F -3F -2F -F 0 F 2F 3F 4F0
0.5
1
|U|
◦ Eu = ∞ but ave power is Pu =⟨
|u(t)|2⟩
=∑∞
n=−∞ |Un|2
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Example application:
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Example application:
• Impulse Response: [, y(t) for x(t) = δ(t)]
h(t) = 1RC
e−t
RC for t ≥ 0
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Example application:
• Impulse Response: [, y(t) for x(t) = δ(t)]
h(t) = 1RC
e−t
RC for t ≥ 0
• Frequency Response: H(f) = 11+i2πfRC
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Example application:
• Impulse Response: [, y(t) for x(t) = δ(t)]
h(t) = 1RC
e−t
RC for t ≥ 0
• Frequency Response: H(f) = 11+i2πfRC
• Convolution: y(t) = h(t) ∗ x(t)
Convolution
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 12 / 13
Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ
[In the integral, the arguments of u( ) and v( ) add up to t]
∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)
Multiplication in either the time or frequency domainis equivalent to convolution in the other domain:
w(t) = u(t) ∗ v(t) ⇔ W (f) = U(f)V (f)y(t) = u(t)v(t) ⇔ Y (f) = U(f) ∗ V (f)
Example application:
• Impulse Response: [, y(t) for x(t) = δ(t)]
h(t) = 1RC
e−t
RC for t ≥ 0
• Frequency Response: H(f) = 11+i2πfRC
• Convolution: y(t) = h(t) ∗ x(t)
• Multiplication: Y (f) = H(f)X(f)
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
⊗ is not commutative or associative (unlike ∗)
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
⊗ is not commutative or associative (unlike ∗)
Cauchy-Schwartz Inequality ⇒ Bound on |w(t)|
• For all values of t: |w(t)|2 ≤ EuEv
• u(t− t0) is an exact multiple of v(t) ⇔ |w(t0)|2 = EuEv
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
⊗ is not commutative or associative (unlike ∗)
Cauchy-Schwartz Inequality ⇒ Bound on |w(t)|
• For all values of t: |w(t)|2 ≤ EuEv
• u(t− t0) is an exact multiple of v(t) ⇔ |w(t0)|2 = EuEv
Normalized cross-correlation: w(t)√EuEv
has a maximum absolute value of 1
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
⊗ is not commutative or associative (unlike ∗)
Cauchy-Schwartz Inequality ⇒ Bound on |w(t)|
• For all values of t: |w(t)|2 ≤ EuEv
• u(t− t0) is an exact multiple of v(t) ⇔ |w(t0)|2 = EuEv
Normalized cross-correlation: w(t)√EuEv
has a maximum absolute value of 1
• Cross-correlation is used to find the time shift, t0, at which twosignals match and also how well they match.
Correlation
Fourier Series andTransformsRevision Lecture
• The Basic Idea
• Real v Complex
• Series v Transform
• Fourier Analysis
• Power Conservation
• Gibbs Phenomenon
• Coefficient Decay Rate
• Periodic Extension
• Dirac Delta Function
• Fourier Transform
• Convolution
• Correlation
E1.10 Fourier Series and Transforms (2015-6200) Revision Lecture: – 13 / 13
Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =
∫∞−∞ u∗(τ − t)v(τ )dτ
[In the integral, the arguments of u∗( ) and v( ) differ by t]
⊗ is not commutative or associative (unlike ∗)
Cauchy-Schwartz Inequality ⇒ Bound on |w(t)|
• For all values of t: |w(t)|2 ≤ EuEv
• u(t− t0) is an exact multiple of v(t) ⇔ |w(t0)|2 = EuEv
Normalized cross-correlation: w(t)√EuEv
has a maximum absolute value of 1
• Cross-correlation is used to find the time shift, t0, at which twosignals match and also how well they match.
• Auto-correlation is the cross-correlation of a signal with itself: used tofind the period of a signal (i.e. the time shift where it matches itself).