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).