Concepts of Multimedia Processing and Transmission IT 481, Lecture 2 Dennis McCaughey, Ph.D. 29 January, 2007

01/22/2007 Dennis McCaughey, IT 481, Spring Course Web Site s07.htm s07.htm WebCt site will be set up this week

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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:

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring Linear System Example From Circuits Kirchhoff's Laws: 1.The algebraic sum of the voltages around a loop equal zero 2.The algebraic sum of the currents at a node equal zero

01/22/2007 Dennis McCaughey, IT 481, Spring Derivation of the Relevant Equations

01/22/2007 Dennis McCaughey, IT 481, Spring Adding a Second Voltage Source

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring Matrix Algebra

01/22/2007 Dennis McCaughey, IT 481, Spring Example (Multiplication)

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring 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 writeLaplace's formula Where the C i,j represent the matrix cofactors, i.e. C i,j is ( − 1) i + j times the minor M i,j, which is the determinant of the matrix that results from A by removing the i-th row and the j-th column.cofactorsminor

01/22/2007 Dennis McCaughey, IT 481, Spring 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).

01/22/2007 Dennis McCaughey, IT 481, Spring Example Useful for 2x2 matrices

01/22/2007 Dennis McCaughey, IT 481, Spring Matlab “Codelet” % column delimiter =; row delimiter = ; A=[2,1,1;0,-1,2;0,2,-1] d = det(A) adjA = d*inv(A)

01/22/2007 Dennis McCaughey, IT 481, Spring Return to Circuit Example

01/22/2007 Dennis McCaughey, IT 481, Spring Linear System Representation

01/22/2007 Dennis McCaughey, IT 481, Spring Linear System Definition

01/22/2007 Dennis McCaughey, IT 481, Spring Linear System Response to a Series of Sampled data Inputs

01/22/2007 Dennis McCaughey, IT 481, Spring Linear System Input/Output This is denoted as the convolution of f(t) and h(t)

01/22/2007 Dennis McCaughey, IT 481, Spring Convolution Sum Example n g = n f + n h -1 f(k) = h(k) =0 for k >2

01/22/2007 Dennis McCaughey, IT 481, Spring Integer Arithmetic Example Multiplication of 2 Integers is a form of discrete convolution

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring Fourier Transform Example

01/22/2007 Dennis McCaughey, IT 481, Spring Relationship Between the Fourier Transform and Convolution

01/22/2007 Dennis McCaughey, IT 481, Spring Very Important Properties

01/22/2007 Dennis McCaughey, IT 481, Spring Important Fourier Transform Properties

01/22/2007 Dennis McCaughey, IT 481, Spring Combined Shifting and Scaling

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring Signal Sampling and Recovery Sampler (Rate 1/T) Low Pass Filter s(t) s(n) s(t) 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

01/22/2007 Dennis McCaughey, IT 481, Spring Sampling Theorem Demonstration -BB S(f) f S s (f) 01/(2T)1/T1/(3T)-1/T-1/(2T)-1/(3T) f Original Spectrum Sampled Signal Spectrum Low Pass Filter

01/22/2007 Dennis McCaughey, IT 481, Spring 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,…….}

01/22/2007 Dennis McCaughey, IT 481, Spring Digital Filter Forms Finite Impulse Response (FIR) Infinite Impulse Response (IIR) All 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.

01/22/2007 Dennis McCaughey, IT 481, Spring Z-Transform

01/22/2007 Dennis McCaughey, IT 481, Spring Z-Transform and Discrete Convolution Z-Transform of the output is the product if the Z-Transforms of the input and the filter response

01/22/2007 Dennis McCaughey, IT 481, Spring Calculating the Filter Impulse Response from its Z-Transform

01/22/2007 Dennis McCaughey, IT 481, Spring IIR-Example

01/22/2007 Dennis McCaughey, IT 481, Spring Matlab “Codelet” n =[0:20] y= 6*(0.6).^n-5*(0.5).^n bar(n,y,.01)

01/22/2007 Dennis McCaughey, IT 481, Spring Impulse Response

01/22/2007 Dennis McCaughey, IT 481, Spring Determine k for Unity Gain

01/22/2007 Dennis McCaughey, IT 481, Spring Filter Response

01/22/2007 Dennis McCaughey, IT 481, Spring Flow Chart

01/22/2007 Dennis McCaughey, IT 481, Spring Matrix Representation The filter behavior can be determined from the characteristics of A

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring The Discrete Fourier Transform Let x p (t) be a periodic signal with property, x p (t) = x p (t+T 0 ) where T 0 is the signal period. –Note: for the purposes if this discussion, any signal observed over a finite window (nT 0 <t<(n+1)T 0 ) can be considered periodic outside it.

01/22/2007 Dennis McCaughey, IT 481, Spring Relationship Between the DFT and the Z-Transform

01/22/2007 Dennis McCaughey, IT 481, Spring Frequency Response

01/22/2007 Dennis McCaughey, IT 481, Spring The Discrete Cosine Transform

01/22/2007 Dennis McCaughey, IT 481, Spring 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

01/22/2007 Dennis McCaughey, IT 481, Spring Summary Shannon’s Sampling Theorem Fourier Transform Linear Systems Digital Filters Utility of Matrix Representations