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

2. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 2 Course Web Site http://teal.gmu.edu/~dgm/sp07/IT481- s07.htm http://teal.gmu.edu/~dgm/sp07/IT481- s07.htm WebCt site will be set up this week

3. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 3Overview 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

4. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 4 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

5. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 5 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

6. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 6 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

7. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 7 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

8. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 8 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:

9. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 9Examples 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

10. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 10 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

11. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 11 Derivation of the Relevant Equations

12. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 12 Adding a Second Voltage Source

13. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 13Superposition 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

14. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 14 Matrix Algebra

15. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 15 Example (Multiplication)

16. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 16 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

17. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 17 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

18. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 18 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).

19. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 19Example Useful for 2x2 matrices

20. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 20 Matlab “Codelet” % column delimiter =; row delimiter = ; A=[2,1,1;0,-1,2;0,2,-1] d = det(A) adjA = d*inv(A)

21. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 21 Return to Circuit Example

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

23. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 23 Linear System Definition

24. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 24 Linear System Response to a Series of Sampled data Inputs

25. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 25 Linear System Input/Output This is denoted as the convolution of f(t) and h(t)

26. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 26 Convolution Sum Example n g = n f + n h -1 f(k) = h(k) =0 for k >2

27. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 27 Integer Arithmetic Example Multiplication of 2 Integers is a form of discrete convolution

28. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 28 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

29. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 29 Fourier Transform Example

30. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 30 Relationship Between the Fourier Transform and Convolution

31. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 31 Very Important Properties

32. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 32 Important Fourier Transform Properties

33. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 33 Combined Shifting and Scaling

34. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 34 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

35. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 35 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

36. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 36 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

37. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 37 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,…….}

38. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 38 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.

39. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 39Z-Transform

40. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 40 Z-Transform and Discrete Convolution Z-Transform of the output is the product if the Z-Transforms of the input and the filter response

41. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 41 Calculating the Filter Impulse Response from its Z-Transform

42. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 42IIR-Example

43. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 43 Matlab “Codelet” n =[0:20] y= 6*(0.6).^n-5*(0.5).^n bar(n,y,.01)

44. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 44 Impulse Response

45. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 45 Determine k for Unity Gain

46. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 46 Filter Response

47. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 47 Flow Chart

48. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 48 Matrix Representation The filter behavior can be determined from the characteristics of A

49. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 49 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

50. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 50 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.

51. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 51 Relationship Between the DFT and the Z-Transform

52. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 52 Frequency Response

53. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 53 The Discrete Cosine Transform

54. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 54 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

55. 01/22/2007 Dennis McCaughey, IT 481, Spring 2007 55Summary Shannon’s Sampling Theorem Fourier Transform Linear Systems Digital Filters Utility of Matrix Representations