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Fourier Series for Continuous –Time Periodic Signals 3 Ex... Fourier Series for Continuous –Time Periodic Signals 6... Fourier Series for Continuous –Time Periodic Signals 7 Ex... Fourie

Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

Fourier Analysis

I Introduction

II Introduction to Electrophysiology

III Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time

Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

Fourier Analysis

Fourier Analysis

1 Sinusoidal Signals and their Properties

a) Continuous – Time Sinusoids

b) Discrete – Time Sinusoids

c) Frequency Variables and Units

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

Continuous – Time Sinusoids (1)

Continuous – Time Sinusoids (2)

Continuous – Time Sinusoids (3)

The fundamental/first harmonic

The second harmonic

http://tex.stackexchange.com/questions/1273

75/replicate-the-fourier-transform-time-frequency-domains-correspondence-illustrati

Continuous – Time Sinusoids (4)

1 ( ) 0 cos( 2 0 ) 1 cos( 2 3 0 ) 2 cos( 2 5 0 )

Fourier Analysis

1 Sinusoidal Signals and their Properties

a) Continuous – Time Sinusoids

b) Discrete – Time Sinusoids

c) Frequency Variables and Units

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

Discrete – Time Sinusoids (1)

T

T

Discrete – Time Sinusoids (2)

Discrete – Time Sinusoids (3)

Discrete – Time Sinusoids (4)

Discrete – Time Sinusoids (5)

• The sequence x[n] = Acos(2πf0 + θ) is periodic in ω0 with

fundamental period 2π and periodic in f0 with fundamental

period one (periodic in frequency), therefore:

–π < ω ≤ π or 0 ≤ ω < 2π (the fundamental frequency range)

3. A cos[ω0(n + n0) + θ] = Acos[ω0n + (ω0n0 + θ)]: a time shift is

equivalent to a phase change

Discrete – Time Sinusoids (6)

Fourier Analysis

1 Sinusoidal Signals and their Properties

a) Continuous – Time Sinusoids

b) Discrete – Time Sinusoids

c) Frequency Variables and Units

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

Frequency Variables and Units

cycles ,

samples

f

radians ,

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals a) Fourier Series for Continuous – Time Periodic

Signals b) Fourier Transform for Continuous – Time

Aperiodic Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

Fourier Series for Continuous –

Time Periodic Signals (1)

c e Ω

0

3 3

c e Ω

0

4 4

Fourier Series for Continuous –

Time Periodic Signals (2)

0

jk t k

The Fourier series representation of a continuous – time periodic signal

k k k

c = c ∠θ

Fourier Series for Continuous –

Time Periodic Signals (3)

Ex 1

0 τ τ

Fourier Series for Continuous –

Time Periodic Signals (4)

2 2

Fourier Series for Continuous –

Time Periodic Signals (5)

k

π π

2 2

2

sin( F )

A c

2 2

2

sin( F )

A c

Fourier Series for Continuous –

Time Periodic Signals (6)

Fourier Series for Continuous –

Time Periodic Signals (7)

Ex 1

0 τ τ

Fourier Series for Continuous –

Time Periodic Signals (8)

Ex 2

0 τ τ

Fourier Series for Continuous –

Time Periodic Signals (9)

Ex 3

Find the Fourier series?

0 0

0 0

Fourier Series for Continuous –

Time Periodic Signals (10)

Fourier Series for Continuous –

Time Periodic Signals (11)

Convergence conditions (Dirichlet conditions):

1 The periodic signal x(t) is absolutely integrable over any period, that is,

x (t) has a finite area per period:

2 The periodic signal x(t) has a finite number of maxima, minima, and finite

discontinuities per period.

Fourier Series for Continuous –

Time Periodic Signals (12)

x t( )

A

0 τ τ

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

a) Fourier Series for Continuous – Time Periodic

Signals

b) Fourier Transform for Continuous – Time

Aperiodic Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

Fourier Transforms for Continuous –

Time Aperiodic Signals (1)

x t( )

A

0 τ τ

Fourier Transforms for Continuous –

Time Aperiodic Signals (2)

The Fourier series representation of a continuous – time periodic signal

Fourier Transforms for Continuous –

Time Aperiodic Signals (3)

0 0.2 0.4 0.6 0.8 1

Fourier Transforms for Continuous –

Time Aperiodic Signals (4)

o )

Fourier Transforms for Continuous –

Time Aperiodic Signals (5)

Ex 3

0

, ( )

2 2

Fourier Transforms for Continuous –

Time Aperiodic Signals (6)

aperiodic ( ) :

x t

periodic ( ) :

Fourier Transforms for Continuous –

Time Aperiodic Signals (7)

X j π F

F

0 0

0

s in ( ) ( )

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

a) Fourier Series for Discrete – Time Periodic Signals b) Fourier Transform for Discrete – Time Aperiodic

Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

Fourier Series for Discrete – Time Periodic Signals (1)

Fourier Series for Discrete – Time Periodic Signals (2)

Fourier Series for Discrete – Time Periodic Signals (3)

Fourier Series for Discrete – Time Periodic Signals (4)

n e N

Fourier Series for Discrete – Time Periodic Signals (5)

Ex 3

Find the Fourier series of the rectangular pulse sequence.

2 1

N N

n

n

a a

1 L j k m L( )

N k

Fourier Series for Discrete – Time Periodic Signals (6)

Ex 3

Find the Fourier series of the rectangular pulse sequence.

2 1

N

k N

π π

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

a) Fourier Series for Discrete – Time Periodic Signals

b) Fourier Transform for Discrete – Time Aperiodic

Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

Fourier Transform for Discrete –

Time Aperiodic Signals (1)

0 1 2 3 4 5 6 7 8 8

Fourier Transform for Discrete –

Time Aperiodic Signals (2)

2 1

0

[ ]

N j kn

N k

Fourier Transform for Discrete –

Time Aperiodic Signals (3)

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time

Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier

Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

Continuous – time signals Discrete – time signals Time – domain Frequency – domain Time – domain Frequency – domain

=−∞

= 

0 0

0 0

− t -0.4-5 -4 - 3 -2 -1 0 1 2 3 4 5

-0.2 0 0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform a) Symmetry Properties

b) Operational Properties

6 Computational Fourier Analysis

Symmetry Properties (2)

2 0

1 2

Symmetry Properties (3)

0

0

1 2 1 2

( ) { [ ] cos( ) [ ]sin( ) [ ] ( ) cos( ) ( ) sin( )

[ ] ( )sin( ) ( ) cos( ) ( ) { [ ]sin( ) [ ]cos( )

n

j I

n

ω ω

ω ω

( ) { [ ] cos( ) [ ]sin( ) [ ] ( ) cos( ) ( ) sin( )

[ ] ( )sin( ) ( ) cos( ) ( ) { [ ]sin( ) [ ]cos( )

Symmetry Properties (5)

If x[n] is real x n R[ ] = x n[ ]; x n I[ ] = 0

0

1 2 [ ] R( j ) cos( ) I( j ) sin( )

( )sin( ) ( ) sin( ) sin( ) sin( )

( ) { [ ] cos( ) [ ]sin( ) [ ] ( ) cos( ) ( ) sin( )

[ ] ( )sin( ) ( ) cos( ) ( ) { [ ]sin( ) [ ]cos( )

Symmetry Properties (6)

0

0

1 2 1 2

( ) { [ ] cos( ) [ ]sin( ) [ ] ( ) cos( ) ( ) sin( )

[ ] ( )sin( ) ( ) cos( ) ( ) { [ ]sin( ) [ ]cos( )

x n π X e ω ωn X e ω ωn dω

Symmetry Properties (7)

0

0

1 2 1 2

( ) { [ ] cos( ) [ ]sin( ) [ ] ( ) cos( ) ( ) sin( )

[ ] ( )sin( ) ( ) cos( ) ( ) { [ ]sin( ) [ ]cos( )

0 ( j )

cos sin ( cos ) ( sin )

ω ω ω

j L

j

e A

e

ω ω

0 2

0 4

0 6

0 8 1

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

a) Symmetry Properties

b) Operational Properties

6 Computational Fourier Analysis

Operational Properties (1)

Operational Properties (3)

Operational Properties (5)

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time

Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

Computational Fourier Analysis

(1), Summary

Computational Fourier Analysis

Computational Fourier Analysis

Computational Fourier Analysis

Computational Fourier Analysis

0

N n

Computational Fourier Analysis

0

0 1 2 3

N n

Computational Fourier Analysis

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

a) Algebraic Formulation of DFT

Algebraic Formulation of DFT

(1)

2 1

0 3

W

3

N =

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

W

0 6

W

3 6

W

5 6

W

6

N =

1 3

W

0 3

W

3

N =

2 0

N

n N n

e N

e N

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

a) Algebraic Formulation of DFT

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

a) Effects of Time – Windowing on Sinusoidal Signals b) Effects of Time – Windowing on Signals with

Continuous Spectra c) “Good” Window and the Uncertainty Principle

Effects of Time – Windowing on

Sinusoidal Signals (1)

-1 0 1

Effects of Time – Windowing on

Effects of Time – Windowing on

Effects of Time – Windowing on

-1 0 1 2 3

Effects of Time – Windowing on

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

a) Effects of Time – Windowing on Sinusoidal Signals b) Effects of Time – Windowing on Signals with

Continuous Spectra c) “Good” Window and the Uncertainty Principle

Effects of Time – Windowing on Signals with Continuous Spectra

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

a) Effects of Time – Windowing on Sinusoidal Signals b) Effects of Time – Windowing on Signals with

Continuous Spectra c) “Good” Window and the Uncertainty Principle

“Good” Windows and the Uncertainty Principle (1)

“Good” Windows and the Uncertainty Principle (2)

1 ( ) CTFT

“Good” Windows and the Uncertainty Principle (3)

4 6 8 10 12

“Good” Windows and the Uncertainty Principle (4)

0 0

π π

“Good” Windows and the Uncertainty Principle (5)

0 0

π π

“Good” Windows and

the Uncertainty Principle (6)

“Good” Windows and the Uncertainty Principle (7)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

9 Fast Fourier Transform

a) Direct Computation of the DFT

b) The FFT Idea Using a Matrix Approach

Direct Computation of the Discrete – Fourier Transform

1 0

[ ] [ ] , , , ,

N

kn N n

The Fast Fourier Transform Idea Using a Matrix Approach (1)

1 1

2

1 3

4

1 5

6

1 7

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

x x x x x x x

The Fast Fourier Transform Idea Using a Matrix Approach (2)

1 1

1 1 2

1 3

1 1 1 1 4

1 5

1 1 6

1 7

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

x x x x x x x

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

W

1 8

W

2 8

W

3 8

W

4 8

W

5 8

W

6 8

W

7 8

The Fast Fourier Transform Idea Using a Matrix Approach (3)

1 1

1 1 2

1 3

1 1 1 1 1 1 1 1 4

1 5

1 1 6

1 7

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

x x x x x x x

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] ; [ ] [ ] [ ] [ ]

The Fast Fourier Transform Idea

Using a Matrix Approach (4)

Top Even

Odd Bottom

N/2 – point Sequence

The Fast Fourier Transform Idea Using a Matrix Approach (5)

1 1

1 1 2

1 3

1 1 1 1 1 1 1 1 4

1 5

1 1 6

1 7

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

x x x x x x x

[ ] [ ] [ ] [ ] [ ]

x x x x x

Fourier Analysis

1 Sinusoidal Signals and their Properties

2 Fourier Representation of Continuous – Time Signals

3 Fourier Representation of Discrete – Time Signals

4 Summary of Fourier Series and Fourier Transforms

5 Properties of the Discrete – Time Fourier Transform

6 Computational Fourier Analysis

7 The Discrete Fourier Transform

8 Fourier Analysis of Signals Using the DFT

9 Fast Fourier Transform