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Fourier and Wavelet Signal Processing

Martin VetterliAmina Chebira, Ali Hormati

Spring 2011

Ecole Ploytechnique Federale de Lausanne (EPFL)Audio-Visual Communications Laboratory (LCAV)

Sequences and Discrete-time Systems 1

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Outline• Discrete-time sequences• Discrete-time systems• Discrete-time Fourier transform (DTFT)• Z-transform• Discrete Fourier transform (DFT)• Continuous-time Fourier transform (CTFT)• Continuous-time Fourier series (CTFS)• Sampling and Fourier Transform Relations (exercise session)

Readings:• Chapter 2: Sequences and Discrete-Time Systems• Chapter 3: Functions and Continuous-Time Systems

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Discrete-time Sequences

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• Infinite-length sequences :• Measures of some physical quantity• Geometric series

• Finite-length sequences :• Truely finite• Periodic

• A finite-length sequence can be seen as a period of a periodic sequence

Discrete-time Sequences

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5Sequences and Discrete-time Systems

• : absolutely summable sequences

• : square summable sequences

• : bounded sequences

• are all complete normed vector spaces Banach spaces

every Cauchy sequence is convergent

• is also a Hilbert space An inner product produces the norm

• Geometric View

Sequence Spaces

Example: one sided geometric series

Example: Kronecker Delta sequence

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Discrete Sequences

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7Sequences and Discrete-time Systems

Discrete Sequences

• Often, a finite length sequence is seen as a glimpse of an infinite sequence using windows

• Box window

• Raised cosine window (Hanning, Hamming)

• Trade off

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Deterministic Auto/Cross Correlation

• To find the similarity between sequences and their shifts

• Auto Correlation

• Cross Correlation

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9Sequences and Discrete-time Systems

Multidimensional Sequences

• Two dimensional sequences

• A part of a larger, invisible 2D sequence

• Example : images

• Generally assumed to be circularly extended at the borders

• Example :

Discrete-time Systems

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• Operators having discrete-time sequences as inputs and outputs

• The most important class is the linear shift-invariant (LSI) systems

• Practically relevant

• Easy to analyze

• Subscript in means that the operator may change over time

• Example :

• can operate on both and !11Sequences and Discrete-time Systems

Discrete-time Systems

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Types of Discrete-time Systems

• Linear Systems (Superposition)

• Shift-by-k operator

• Memoryless Systems

• Output at time only depends on input at time

• Modulation operator

• Shift-invariant Systems

• Shifted input by results in the shifted output by

• : System does not change over time

• Listening to the same track later should be the same

• Causal Systems

• The output at time only depends on the previous inputs

• Real systems are causal

• Stable Systems

• BIBO Stability:

• accumulator is not BIBO stable

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13Sequences and Discrete-time Systems

Types of Discrete-time Systems

• General linear form

• Shift-invariance difference equations

• Example: Accumulator

• A difference equation is LSI initial conditions are zero initially

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Linear Shift-Invariant Systems

• Impulse response• Output to the Kronecker Delta

• LSI system is completely characterized by and convolution

• Convolution

• BIBO stability

An LSI system is BIBO stable if and only if its impulse responseis absolutely summable.

Matrix View

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Discrete-time Fourier Transform(DTFT)

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• Eigensequence of convolution operator

• The discrete-time Fourier transform of a sequence is a function of given by

• The inverse DTFT of a -periodic function is given by

• To denote such a DTFT pair, we write

• The DTFT of is called the spectrum of .

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Discrete-time Fourier Transform (DTFT)

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DTFT Convergence Properties

• If then – is continuous

– Convergence is uniform

• For– Non- uniform convergence

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Gibbs Phenomenon

• Gibbs phenomenon

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Discrete-time Fourier Transform (DTFT)

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• Linearity

• Shift in time

• Shift in frequency

• Convolution in time

• Convolution in freq

• Deterministic cross-correlation

• Parseval’s equality

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Properties of DTFT

-transform

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• Motivation: What about sequences not in and like Heaviside ?

• are also eigensequences of the convolution

• If it is well defined and convergent

• Definition( -transform)

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-transform

• Convergence

The sufficient condition is

• Examples :

• Shift-by-k sequence

• Right-sided geometric series

• Left-sided geometric series

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-transform

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• Difference Equations

• Consider a causal solution with zero initial condition

Impulse response

A causal discrete-time linear system with a finite number of coefficients is BIBO stable, if and only if the poles of it’s

(reduced) -transform are inside the unit circle.

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Rational -transforms

• The same properties as DTFT, but for a larger class of signals

• Linearity

• Shift in time

• Scale in

• Time reversal

• Differentiation

• Convolution in time

• Deterministic cross correlation

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Properties of -transform

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• Autocorrelation

• A rational function is the deterministic autocorrelation of a stable real sequence , if and only if :

i. It’s complex poles and zeros appear in quadruples:

ii. It’s real poles and zeros appear in pairs:

iii. It’s zeros on the unit circle are double zeros:

With possibly double zeros at . There are no poles on the unit circle

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Spectral Factorization

Proof :

• Use the fact than is real

• Use the fact than is symmetric

• Then :

• This leads to quadruples, and pairs if pole/zero is already real

• On unit circle

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Spectral Factorization of real sequences

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Discrete Fourier Transform(DFT)

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• The circular convolution between a length-N sequence and a length-N impulse response of an LSI system is :

• Equivalence of linear and circular convolution

Linear and circular convolutions between a length-M sequence and a length-Limpulse response are equivalent, when the period of the circular convolution,N, satisfies :

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Circular Convolution

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• Circular Convolution of two finite-length sequence

• Borders are circularly extended

• DFT is a tool for fast computation of linear convolutions

• We have eigensequences for circular convolution

• Let’s check

• The DFT of length-N sequence

• The inverse DFT is given by

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Discrete Fourier Transform (DFT)

• DFT matrix diagonalizes circulant matrices

• DFT is an orthonormal basis

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DFT: Matrix View

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• For a given length-N sequence, DFT is the samples of DTFT

spectrum at

• Formally

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Relation between DFT & DTFT

DFT

DTFT

Sampling the spectrum

• Linearity

• Shift in time

• Shift in frequency

• Convolution in time

• Convolution in frequency

• Deterministic auto correlation

• Deterministic cross correlation

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Properties of DFT

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Continuous-time Fourier Transform(CTFT)

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• Continuous-time LTI systems are completely characterized by convolution with their impulse response.

• Complex exponentials are eigenfunctions of the convolution diagonalizing the convolution operator

• The Fourier transform (FT) of a function is a function of given by

• The inverse Fourier transform of is

• To denote such a Fourier-transform pair, we write :

• The Fourier transform of is called the spectrum of

• Properties of CTFT are very similar to DTFT & DFT34Sequences and Discrete-time Systems

Continuous-time Fourier Transform (CTFT)

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Continuous-time Fourier Series(CTFS)

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• What is the Fourier transform of a continuous-time periodic signal ?

• The Fourier series of a periodic function with period , , is a function of given by

• The inverse Fourier series of is

• To denote such a Fourier-series pair, we write

• The CTFT of a periodic signal is composed of Diracs at the harmonic position

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Continuous-time Fourier Series

CTFT

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• Discrete-time sequences & systems

• LSI systems

• Convolution operator

• Matrix View

• Eigensequences : Complex exponentials

• Transforms

• DTFT

• -transform

• DFT

• CTFT

• CTFS

• Spectral factorization

• Can be used for filter design37Sequences and Discrete-time Systems

Wrap Up