Concepts of Multimedia Processing and Transmission

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

2. 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

3. 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

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 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 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 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 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: Dennis McCaughey, IT 481, Spring 2007

9. 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

10. 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

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

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

13. 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

14. Matrix Algebra Dennis McCaughey, IT 481, Spring 2007

15. Example (Multiplication) 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 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 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

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). Dennis McCaughey, IT 481, Spring 2007

19. Example Useful for 2x2 matrices 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) Dennis McCaughey, IT 481, Spring 2007

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

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

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

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

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

26. Convolution Sum Example ng = nf + nh -1 f(k) = h(k) =0 for k >2 Dennis McCaughey, IT 481, Spring 2007

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

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

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

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

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

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

35. 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

36. 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

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,…….} Dennis McCaughey, IT 481, Spring 2007

38. Finite Impulse Response (FIR) Infinite Impulse Response (IIR) Digital Filter Forms 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. Dennis McCaughey, IT 481, Spring 2007

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

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

42. IIR-Example 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) Dennis McCaughey, IT 481, Spring 2007

44. Impulse Response Dennis McCaughey, IT 481, Spring 2007

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

46. Filter Response Dennis McCaughey, IT 481, Spring 2007

47. Flow Chart Dennis McCaughey, IT 481, Spring 2007

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

50. 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