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### Concepts of Multimedia Processing and Transmission

IT 481, Lecture 2

Dennis McCaughey, Ph.D.

29 January, 2007

Course Web Site

- http://teal.gmu.edu/~dgm/sp07/IT481-s07.htm
- WebCt site will be set up this week

Dennis McCaughey, IT 481, Spring 2007

Overview

- Need for an understanding and ability to apply top level signal/image processing concepts and algorithms
- As a communication tool to aid in understanding the course material
- To allow the class to implement and observe the results of the key processing/compression required for the efficient storage and communication of multimedia data
- Not a course in DSP but a basic expertise is required
- Exercises will be confined to home work and not on the mid-term or final

Dennis McCaughey, IT 481, Spring 2007

Required Signal Processing Concepts

- Continuous-time Signal Processing
- Linear Filtering and Convolution
- Fourier Transform
- Relationship between the Fourier Transform and Convolution
- Extensions to Image Processing
- Discrete-Time Signal Processing
- Shannon’s Sampling Theorem
- Discrete Fourier Transform
- Linear Filtering and Convolution
- Relationship between the Fourier Transform and Convolution
- Extensions to Image Processing

Dennis McCaughey, IT 481, Spring 2007

Basic Toolsets

- Linear Algebra
- Vector Spaces
- Linear Operators
- Matrix and Vector Algebra
- Matlab
- Programming tool for signal/image processing
- Allows “hands-on” demonstration of signal/image processing algorithms
- Linear algebra intensive

Dennis McCaughey, IT 481, Spring 2007

Importance of Linear Systems

- A great deal of engineering situations are linear, at least within specified ranges
- Exact solutions of the behavior of linear systems can be usually found by standard techniques
- The techniques remain the same irrespective of whether the problem at hand is one on electrical circuits, mechanical vibration, heat conduction, motion of elastic beams or diffusion of liquids etc.
- Except for a very few special cases, there are no exact methods for analyzing nonlinear systems

Dennis McCaughey, IT 481, Spring 2007

Matrix Algebra and Linear Systems

- Every Linear operator on a finite dimensional vector space has a matrix representation
- Matrix representation provides a useful tool for examining the properties of a linear operator, even if the implementation does not explicitly employ a matrix
- In fact, a direct matrix implementation is often computationally inefficient
- What is a vector space?
- What is a finite dimensional vector space?
- We will define both and develop applicability through a simple electrical circuits example

Dennis McCaughey, IT 481, Spring 2007

Linear Vector Space

- Definition
- A vector space V is a set of elements called vectors with two operations, called addition (designated by +) and multiplication by scalars (designated by juxtaposition), such that the following axioms or conditions are satisfied:

Dennis McCaughey, IT 481, Spring 2007

Examples

- The sets of real and complex numbers
- The system of directed line segments in 3-space
- The set of a real polynomials in a variable t
- The set of all n-tuples of real numbers

Dennis McCaughey, IT 481, Spring 2007

Linear System Example From Circuits

- Kirchhoff\'s Laws:
- The algebraic sum of the voltages around a loop equal zero
- The algebraic sum of the currents at a node equal zero

Dennis McCaughey, IT 481, Spring 2007

Derivation of the Relevant Equations

Dennis McCaughey, IT 481, Spring 2007

Adding a Second Voltage Source

Dennis McCaughey, IT 481, Spring 2007

Superposition

The output is the sum of the response to the sum the separate inputs

The superposition theorem states that the response in any element of a linear network containing two or more sources is the sum of the responses obtained by each source acting separately and with all other sources set equal to zero

Dennis McCaughey, IT 481, Spring 2007

Matrix Algebra

Dennis McCaughey, IT 481, Spring 2007

Example (Multiplication)

Dennis McCaughey, IT 481, Spring 2007

Matrix Inversion

- For the inverse to exist the matrix determinant must be non zero
- The matrix must be square, i.e. the row and column dimensions must be equal
- Examples for some small matrices

Dennis McCaughey, IT 481, Spring 2007

Matrix Determinant

It is also possible to expand a determinant along a row or column using Laplace\'s formula, which is efficient for relatively small matrices. To do this along row i, say, we write

Where the Ci,j represent the matrix cofactors, i.e. Ci,j is ( − 1)i + j times the minorMi,j, which is the determinant of the matrix that results from A by removing the i-th row and the j-th column.

Dennis McCaughey, IT 481, Spring 2007

Matrix Classical Adjoint

It may (or may not) be helpful to attach names to the steps in the process. You can let M~ij be the (n-1) x (n-1) matrix minor, that is, the matrix that results from deleting row i and column j of A. Then Mij = det( M~ij). Let cof(A) be the cofactor matrix mentioned above. Then adj(A) = transpose of cof(A).

Dennis McCaughey, IT 481, Spring 2007

Matlab “Codelet”

% column delimiter =; row delimiter = ;

A=[2,1,1;0,-1,2;0,2,-1]

d = det(A)

adjA = d*inv(A)

Dennis McCaughey, IT 481, Spring 2007

Return to Circuit Example

Dennis McCaughey, IT 481, Spring 2007

Linear System Representation

Dennis McCaughey, IT 481, Spring 2007

Linear System Definition

Dennis McCaughey, IT 481, Spring 2007

Linear System Response to a Series of Sampled data Inputs

Dennis McCaughey, IT 481, Spring 2007

Linear System Input/Output

This is denoted as the convolution of f(t) and h(t)

Dennis McCaughey, IT 481, Spring 2007

Integer Arithmetic Example

- Multiplication of 2 Integers is a form of discrete convolution

Dennis McCaughey, IT 481, Spring 2007

Fourier Transform - Non-periodic Signal

- Let x(t) be a non-periodic function of t
- The Fourier Transform of x(t) is
- The Inverse Fourier Transform is

Dennis McCaughey, IT 481, Spring 2007

Fourier Transform Example

Dennis McCaughey, IT 481, Spring 2007

Relationship Between the Fourier Transform and Convolution

Dennis McCaughey, IT 481, Spring 2007

Very Important Properties

Dennis McCaughey, IT 481, Spring 2007

Important Fourier Transform Properties

Dennis McCaughey, IT 481, Spring 2007

Combined Shifting and Scaling

Dennis McCaughey, IT 481, Spring 2007

Discrete Time Systems

- Computer applications deal with discrete time or sampled data systems
- Need a theory that connects sampled data and continuous time systems
- This is provided by Shannon’s Sampling Theorem

Dennis McCaughey, IT 481, Spring 2007

Signal Sampling and Recovery

Sampler

(Rate 1/T)

Low Pass Filter

s(t)

s(t)

s(n)

Shannon’s sampling theorem states that the original signal s(t) can be recovered from its sampled version if the sampling rate, 1/T is greater than 2B where B is the one sided bandwidth of the signal

Dennis McCaughey, IT 481, Spring 2007

Sampling Theorem Demonstration

S(f)

f

-B

B

Original Spectrum

Low Pass Filter

Ss(f)

Sampled Signal Spectrum

f

-1/(3T)

-1/(2T)

-1/T

0

1/T

1/(2T)

1/(3T)

Dennis McCaughey, IT 481, Spring 2007

Idealized Discrete-Time System Processing Flow

- Assume x(t) is band limited
- Implicit in the D/A converter is an ideal LPF
- What forms can the Digital Filter employ?

h(n) is the “impulse or characteristic” response of the filter.

It is given by the sequence h(n) ={y(0), y(1), y(2)…….} when the input sequence x(n) = {1, 0, 0,…….}

Dennis McCaughey, IT 481, Spring 2007

Finite Impulse Response (FIR)

Infinite Impulse Response (IIR)

Digital Filter FormsAll of the D\'s are zero for an FIR filter. The main advantage of IIR filters is that they can produce a steeper slope for a given number of coefficients. The main advantage of FIR filters is that the group delay is constant. This provides the capability of obtaining both a steep cutoff and perfect phase response. This is impossible to achieve with an analog filter.

Dennis McCaughey, IT 481, Spring 2007

Z-Transform

Dennis McCaughey, IT 481, Spring 2007

Z-Transform and Discrete Convolution

Z-Transform of the output is the product if the Z-Transforms of the input and the filter response

Dennis McCaughey, IT 481, Spring 2007

Calculating the Filter Impulse Response from its Z-Transform

Dennis McCaughey, IT 481, Spring 2007

IIR-Example

Dennis McCaughey, IT 481, Spring 2007

Impulse Response

Dennis McCaughey, IT 481, Spring 2007

Determine k for Unity Gain

Dennis McCaughey, IT 481, Spring 2007

Filter Response

Dennis McCaughey, IT 481, Spring 2007

Flow Chart

Dennis McCaughey, IT 481, Spring 2007

Matrix Representation

The filter behavior can be determined from the characteristics of A

Dennis McCaughey, IT 481, Spring 2007

Observations on the Z-Transform

- Useful tool for implementing convolutions
- We can develop a recursion relationship for y(n) given a filter impulse (characteristic) response h(n) and an input sequence x(n).
- Recursions often provide very advantageous implementations
- So far the development has been as an “algebraic” tool with no physical basis
- What are the frequency response characteristics of a digital filter described by H(z)?
- This will require the development of the Discrete Fourier Transform (DFT)

Recursion

Dennis McCaughey, IT 481, Spring 2007

The Discrete Fourier Transform

- Let xp(t) be a periodic signal with property, xp(t) = xp(t+T0) where T0 is the signal period.
- Note: for the purposes if this discussion, any signal observed over a finite window (nT0 <t<(n+1)T0) can be considered periodic outside it.

Dennis McCaughey, IT 481, Spring 2007

Relationship Between the DFT and the Z-Transform

Dennis McCaughey, IT 481, Spring 2007

Frequency Response

Dennis McCaughey, IT 481, Spring 2007

The Discrete Cosine Transform

Dennis McCaughey, IT 481, Spring 2007

DCT as It Applies to Images/Video

The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image\'s visual quality).

The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain

Dennis McCaughey, IT 481, Spring 2007

Summary

- Shannon’s Sampling Theorem
- Fourier Transform
- Linear Systems
- Digital Filters
- Utility of Matrix Representations

Dennis McCaughey, IT 481, Spring 2007

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