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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

+

T/2-0.26sin(2πFt)

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

+

T/2-0.26sin(2πFt)

+

T/3+0.26sin(2πFt)

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

+

T/2-0.26sin(2πFt)

+

T/3+0.26sin(2πFt)

+

T/4

-0.13cos(2πFt)

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

+

T/2-0.26sin(2πFt)

+

T/3+0.26sin(2πFt)

+

T/4

-0.13cos(2πFt)


T. The nth harmonic is at frequency nF .

The Basic Idea


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Tu(t)

=

T +0.65sin(2πFt)

+

T/2-0.26sin(2πFt)

+

T/3+0.26sin(2πFt)

+

T/4

-0.13cos(2πFt)


T. The nth harmonic is at frequency nF .

Some waveforms need infinitely many harmonics (countable infinity).

Real versus Complex Fourier Series


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


All the algebra is much easier if we use eiωt instead of cosωt and sinωt

u(t) = a0

2 +∑∞




• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞


Substitute: cosωt = 12e

iωt + 12e

−iωt sinωt = −i2 eiωt + i

2e−iωt



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e

i2πnFt + 12 (an + ibn) e

−i2πnFt)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt

• U+n = 12 (an − ibn)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt

• U+n = 12 (an − ibn) and U−n = 1

2 (an + ibn) .



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

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.



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt

• U+n = 12 (an − ibn) and U−n = 1

2 (an + ibn) .


• U+n is half the equivalent phasor in Analysis of Circuits.



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt

• U+n = 12 (an − ibn) and U−n = 1

2 (an + ibn) .



Tu(t)

Plot the magnitude spectrum-4F -3F -2F -F 0 F 2F 3F 4F

0

0.5

1

|U|



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



u(t) = a0

2 +∑∞



iωt + 12e


2e−iωt

u(t) = a0

2 +∑∞

n=1

(

12 (an − ibn) e


−i2πnFt)

=∑∞

n=−∞ Unei2πnFt

• U+n = 12 (an − ibn) and U−n = 1

2 (an + ibn) .



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


• Periodic signals

Tu(t)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation


• Periodic signals → Fourier Series

Tu(t)

u(t) =∑∞

n=−∞ Unei2πnFt



• The Basic Idea

• Real v Complex









• Convolution

• Correlation


• 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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


-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

)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


-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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


Fourier Series: u(t) =∑∞

n=−∞ Unei2πnFt

Fourier Analysis = “how do you work out the Fourier coefficients, Un ?”

Fourier Analysis


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩

= 〈cosωt+ i sinωt〉 =

{

1 if ω = 0

0 otherwise

Fourier Analysis


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise

⇒ Orthogonality:⟨

ei2πnFt × e−i2πmFt⟩

=

{

1 for m = n

0 for m 6= n

Fourier Analysis


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise



=

{

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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise



=

{

1 for m = n

0 for m 6= n


u(t)e−i2πmFt⟩

=⟨ (

∑∞n=−∞ Une

i2πnFt)

e−i2πmFt⟩

=∑∞

n=−∞ Un

⟨

ei2πnFte−i2πmFt⟩

Fourier Analysis


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise



=

{

1 for m = n

0 for m 6= n


u(t)e−i2πmFt⟩

=⟨ (

∑∞n=−∞ Une

i2πnFt)

e−i2πmFt⟩

=∑∞

n=−∞ Un

⟨


= Um [since all other terms are zero]

Fourier Analysis


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise



=

{

1 for m = n

0 for m 6= n


u(t)e−i2πmFt⟩

=⟨ (

∑∞n=−∞ Une

i2πnFt)

e−i2πmFt⟩

=∑∞

n=−∞ Un

⟨



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


Key idea:⟨

eiωt⟩


{

1 if ω = 0

0 otherwise



=

{

1 for m = n

0 for m 6= n


u(t)e−i2πmFt⟩

=⟨ (

∑∞n=−∞ Une

i2πnFt)

e−i2πmFt⟩

=∑∞

n=−∞ Un

⟨



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Average power in u(t): Pu ,

⟨

|u(t)|2⟩

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T

0u2(t)dt [u(t) real]

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T


Average power in Fourier component n:⟨

∣

∣Unei2πnFt

∣

∣

2⟩

=⟨

|Un|2 ∣∣ei2πnFt

∣

∣

2⟩

= |Un|2

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2

Power conservation (Parseval’s Theorem):

Pu =⟨

|u(t)|2⟩

=∑∞

n=−∞ |Un|2

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2


Pu =⟨

|u(t)|2⟩

=∑∞

n=−∞ |Un|2


This is a consequence of orthogonality:⟨

|u(t)|2⟩

=⟨(∑∞

n=−∞ Unei2πnFt

) (∑∞

m=−∞ U∗me−i2πmFt

)⟩

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2


Pu =⟨

|u(t)|2⟩

=∑∞

n=−∞ |Un|2



|u(t)|2⟩

=⟨(∑∞

n=−∞ Unei2πnFt

) (∑∞


)⟩

=⟨∑∞

n=−∞∑∞

m=−∞ UnU∗mei2πnFte−i2πmFt

⟩

Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2


Pu =⟨

|u(t)|2⟩

=∑∞

n=−∞ |Un|2



|u(t)|2⟩

=⟨(∑∞

n=−∞ Unei2πnFt

) (∑∞


)⟩

=⟨∑∞

n=−∞∑∞


⟩

=∑∞

n=−∞∑∞

m=−∞ UnU∗m

⟨


Power Conservation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt


⟨

|u(t)|2⟩

= 1T

∫ T



∣

∣Unei2πnFt

∣

∣

2⟩

=⟨


∣

∣

2⟩

= |Un|2


Pu =⟨

|u(t)|2⟩

=∑∞

n=−∞ |Un|2



|u(t)|2⟩

=⟨(∑∞

n=−∞ Unei2πnFt

) (∑∞


)⟩

=⟨∑∞

n=−∞∑∞


⟩

=∑∞

n=−∞∑∞

m=−∞ UnU∗m

⟨


=∑∞

n=−∞ |Un|2

Gibbs Phenomenon


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−N Unei2πnFt



|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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−N Unei2πnFt



|n|>N |Un|2.


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−N Unei2πnFt



|n|>N |Un|2.


Gibbs phenomenon


• 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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−N Unei2πnFt



|n|>N |Un|2.


Gibbs phenomenon



• 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


Gibbs Phenomenon


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−N Unei2πnFt



|n|>N |Un|2.


Gibbs phenomenon



• 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


Coefficient Decay Rate


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Integration:v(t) =

∫ t

0u(τ )dτ ⇒ Vn = 1

i2πnF Un

provided U0 = V0 = 0.



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Integration:v(t) =

∫ t

0u(τ )dτ ⇒ Vn = 1

i2πnF Un


Differentiation:w(t) = du(t)

dt⇒ Wn = i2πnF × Un

provided w(t) satisfies the Dirichlet conditions.



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Integration:v(t) =

∫ t

0u(τ )dτ ⇒ Vn = 1

i2πnF Un





Coefficient Decay Rate:u(t) has a discontinuity ⇒ |Un| is O

(

1n

)

for large |n|



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Integration:v(t) =

∫ t

0u(τ )dτ ⇒ Vn = 1

i2πnF Un






(

1n

)

for large |n|

dku(t)dtk

is the lowest derivative with a discontinuity

⇒ |Un| is O(

1nk+1

)

for large |n|



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



n=−∞ Unei2πnFt

Integration:v(t) =

∫ t

0u(τ )dτ ⇒ Vn = 1

i2πnF Un






(

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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ 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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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

0

2

4 N=1

-2 0 2 4

0

2

4 N=3

-2 0 2 4

Periodic Extension


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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

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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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

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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∫ B

0u(t)e−i2πnFtdt.


-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

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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation



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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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


• Shifting: δ(x− a) is a pulse at x = a and is zero everywhere else



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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



• Multiplication: f(x)× δ(x− a) = f(a)× δ(x− a)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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




• Integration:∫∞−∞ f(x)× δ(x− a)dx = f(a)



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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





• Fourier Transform: u(t) = δ(t) ⇔ U(f) = 1



• The Basic Idea

• Real v Complex









• Convolution

• Correlation




-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





• 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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


Fourier Transform: u(t) =∫∞−∞ U(f)ei2πftdf

U(f) =∫∞−∞ u(t)e−i2πftdt

Fourier Transform


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




• 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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation






◦ 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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation







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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation







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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


Convolution: w(t) = u(t) ∗ v(t) ⇔ w(t) =∫∞−∞ u(τ )v(t− τ )dτ

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



[In the integral, the arguments of u( ) and v( ) add up to t]

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation




∗ acts algebraically like × : Commutative, Associative, Distributive over +.Identity element is δ(t): u(t) ∗ δ(t) = u(t)

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation





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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation







Example application:

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation








• Impulse Response: [, y(t) for x(t) = δ(t)]

h(t) = 1RC

e−t

RC for t ≥ 0

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation









h(t) = 1RC

e−t

RC for t ≥ 0

• Frequency Response: H(f) = 11+i2πfRC

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation









h(t) = 1RC

e−t

RC for t ≥ 0


• Convolution: y(t) = h(t) ∗ x(t)

Convolution


• The Basic Idea

• Real v Complex









• Convolution

• Correlation









h(t) = 1RC

e−t

RC for t ≥ 0


• Convolution: y(t) = h(t) ∗ x(t)

• Multiplication: Y (f) = H(f)X(f)

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation


Cross-correlation:w(t) = u(t)⊗ v(t) ⇔ w(t) =

∫∞−∞ u∗(τ − t)v(τ )dτ

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ

[In the integral, the arguments of u∗( ) and v( ) differ by t]

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ


⊗ is not commutative or associative (unlike ∗)

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ



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


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ






Normalized cross-correlation: w(t)√EuEv

has a maximum absolute value of 1

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ








• Cross-correlation is used to find the time shift, t0, at which twosignals match and also how well they match.

Correlation


• The Basic Idea

• Real v Complex









• Convolution

• Correlation



∫∞−∞ u∗(τ − t)v(τ )dτ








• 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).

fourier series and transforms revision lecturethe basic idea fourier series and transforms revision...

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