1. 2003-09-02Dan Ellis 1 ELEN E4810: Digital Signal Processing Week 1: Introduction 1.Course overview 2.Digital Signal Processing 3.Basic operations & block diagrams 4.Classes of sequences

2. 2003-09-02Dan Ellis 2 1. Course overview Digital signal processing: Modifying signals with computers Web site: http://www.ee.columbia.edu/~dpwe/e4810/ http://www.ee.columbia.edu/~dpwe/e4810/ Book: Mitra “Digital Signal Processing” (2nd ed.) 2001 printing Instructor: dpwe@ee.columbia.edu

3. 2003-09-02Dan Ellis 3 Grading structure  Homeworks: 20%  Mainly from Mitra  Collaboration  Midterm: 20%  One session  Final exam: 30%  Project: 30%

4. 2003-09-02Dan Ellis 4 Course project Goal: hands-on experience with DSP Practical implementation Work in pairs or alone Brief report, optional presentation Recommend MATLAB Ideas on website

5. 2003-09-02Dan Ellis 5 Example past projects Speech enhancement Audio watermarking Voice alteration with the Phase Vocoder DTMF decoder Reverb algorithms Compression algorithms on web site W

6. 2003-09-02Dan Ellis 6 MATLAB Interactive system for numerical computation Extensive signal processing library Focus on algorithm, not implementation Access: Engineering Terrace 251 computer lab Student Version (need Sig. Proc. toolbox) ILAB (form)

7. 2003-09-02Dan Ellis 7 Course at a glance

8. 2003-09-02Dan Ellis 8 2. Digital Signal Processing Signals: Information-bearing function E.g. sound: air pressure variation at a point as a function of time p(t) Dimensionality: Sound: 1-Dimension Greyscale image i(x,y) : 2-D Video: 3 x 3-D: {r(x,y,t) g(x,y,t) b(x,y,t)}

9. 2003-09-02Dan Ellis 9 Example signals Noise - all domains Spread-spectrum phone - radio ECG - biological Music Image/video - compression ….

10. 2003-09-02Dan Ellis 10 Signal processing Modify a signal to extract/enhance/ rearrange the information Origin in analog electronics e.g. radar Examples… Noise reduction Data compression Representation for recognition/classification…

11. 2003-09-02Dan Ellis 11 Digital Signal Processing DSP = signal processing on a computer Two effects: discrete-time, discrete level x(t)x(t) x[n]x[n]

12. 2003-09-02Dan Ellis 12 DSP vs. analog SP Conventional signal processing: Digital SP system: Processor p(t)p(t)q(t)q(t) p(t)p(t)q(t)q(t) A/DD/A p[n]p[n]q[n]q[n]

13. 2003-09-02Dan Ellis 13 Digital vs. analog Pros Noise performance - quantized signal Use a general computer - flexibility, upgrde Stability/duplicability Novelty Cons Limitations of A/D & D/A Baseline complexity / power consumption

14. 2003-09-02Dan Ellis 14 DSP example Speech time-scale modification: extend duration without altering pitch M

15. 2003-09-02Dan Ellis 15 3. Operations on signals Discrete time signal often obtained by sampling a continuous-time signal Sequence {x[n]} = x a (nT), n=…-1,0,1,2… T = samp. period; 1/T = samp. frequency

16. 2003-09-02Dan Ellis 16 Sequences Can write a sequence by listing values: Arrow indicates where n =0 Thus,

17. 2003-09-02Dan Ellis 17 Left- and right-sided x[n] may be defined only for certain n : N 1 ≤ n ≤ N 2 : Finite length (length = …) N 1 ≤ n : Right-sided (Causal if N 1 ≥ 0) n ≤ N 2 : Left-sided (Anticausal) Can always extend with zero-padding Right-sidedLeft-sided

18. 2003-09-02Dan Ellis 18 Operations on sequences Addition operation: Adder Multiplication operation Multiplier A x[n]x[n] y[n]y[n] x[n]x[n] y[n]y[n] w[n]w[n]

19. 2003-09-02Dan Ellis 19 More operations Product (modulation) operation: Modulator E.g. Windowing: multiplying an infinite- length sequence by a finite-length window sequence to extract a region x[n]x[n] y[n]y[n] w[n]w[n]

20. 2003-09-02Dan Ellis 20 Time shifting Time-shifting operation: where N is an integer If N > 0, it is delaying operation Unit delay If N < 0, it is an advance operation Unit advance y[n]y[n] x[n]x[n] y[n]y[n] x[n]x[n]

21. 2003-09-02Dan Ellis 21 Combination of basic operations Example

22. 2003-09-02Dan Ellis 22 Up- and down-sampling Certain operations change the effective sampling rate of sequences by adding or removing samples Up-sampling = adding more samples = interpolation Down-sampling = discarding samples = decimation

23. 2003-09-02Dan Ellis 23 Down-sampling In down-sampling by an integer factor M > 1, every M -th samples of the input sequence are kept and M - 1 in-between samples are removed: M

24. 2003-09-02Dan Ellis 24 Down-sampling An example of down-sampling 3

25. 2003-09-02Dan Ellis 25 Up-sampling Up-sampling is the converse of down- sampling: L-1 zero values are inserted between each pair of original values. L

26. 2003-09-02Dan Ellis 26 Up-sampling An example of up-sampling 3 not inverse of downsampling!

27. 2003-09-02Dan Ellis 27 Complex numbers.. a mathematical convenience that lead to simple expressions A second “imaginary” dimension ( j  √ -1 ) is added to all values. Rectangular form: x = x re + j·x im where magnitude |x| = √(x re 2 + x im 2 ) and phase  = tan -1 (x im /x re ) Polar form: x = |x| e j  = |x|cos  + j· |x|sin  

28. 2003-09-02Dan Ellis 28 Complex math When adding, real and imaginary parts add: (a+jb) + (c+jd) = (a+c) + j(b+d) When multiplying, magnitudes multiply and phases add: re j  ·se j  = rse j(  +  ) Phases modulo 2 

29. 2003-09-02Dan Ellis 29 Complex conjugate Flips imaginary part / negates phase: conjugate x* = x re – j·x im = |x| e j(–  ) Useful in resolving to real quantities: x + x* = x re + j·x im + x re – j·x im = 2x re x·x* = |x| e j(  ) |x| e j(–  ) = |x| 2

30. 2003-09-02Dan Ellis 30 Classes of sequences Useful to define broad categories… Finite/infinite (extent in n ) Real/complex: x[n] = x re [n] + j·x im [n]

31. 2003-09-02Dan Ellis 31 Classification by symmetry Conjugate symmetric sequence: x cs [n] = x cs *[-n] = x re [-n] – j·x im [-n] Conjugate antisymmetric: x ca [n] = –x ca *[-n] = –x re [-n] + j·x im [-n]

32. 2003-09-02Dan Ellis 32 Conjugate symmetric decomposition Any sequence can be expressed as conjugate symmetric (CS) / antisymmetric (CA) parts: x[n] = x cs [n] + x ca [n] where: x cs [n] = 1 / 2 (x[n] + x*[-n]) = x cs *[-n] x ca [n] = 1 / 2 (x[n] – x*[-n]) = -x ca *[-n] When signals are real, CS  Even ( x re [n] = x re [-n]), CA  Odd

33. 2003-09-02Dan Ellis 33 Basic sequences Unit sample sequence: Shift in time:  [n - k] Can express any sequence with           [n] +    [n-1] +    [n-2]..

34. 2003-09-02Dan Ellis 34 More basic sequences Unit step sequence: Relate to unit sample:

35. 2003-09-02Dan Ellis 35 Exponential sequences Exponential sequences= eigenfunctions General form: x[n] = A·  n If A and  are real:  > 1  < 1

36. 2003-09-02Dan Ellis 36 Complex exponentials x[n] = A·  n Constants A,  can be complex : A = |A|e j  ;  = e (  + j  )  x[n] = |A| e  n e j(  n +  ) scale varying magnitude varying phase

37. 2003-09-02Dan Ellis 37 Complex exponentials Complex exponential sequence can ‘project down’ onto real & imaginary axes to give sinusoidal sequences x re [n] = e n/12 cos(  n/6) x im [n] = e n/12 sin(  n/6) M x re [n]x im [n]

38. 2003-09-02Dan Ellis 38 Periodic sequences A sequence satisfying is called a periodic sequence with a period N where N is a positive integer and k is any integer. Smallest value of N satisfying is called the fundamental period

39. 2003-09-02Dan Ellis 39 Periodic exponentials Sinusoidal sequence and complex exponential sequence are periodic sequences of period N only if with N & r positive integers Smallest value of N satisfying is the fundamental period of the sequence r = 1  one sinusoid cycle per N samples r > 1  r cycles per N samples M

40. 2003-09-02Dan Ellis 40 Symmetry of periodic sequences An N -point finite-length sequence x f [n] defines a periodic sequence: x[n] = x f [ N ] Symmetry of x f [n] is not defined because x f [n] is undefined for n < 0 Define Periodic Conjugate Symmetric: x pcs [n] = 1 / 2 (x[n] + x*[ N ]) = 1 / 2 (x[n] + x*[N – n]) 0 ≤ n < N “ n modulo N ”

41. 2003-09-02Dan Ellis 41 Sampling sinusoids Sampling a sinusoid is ambiguous: x 1 [n] = sin(  0 n) x 2 [n] = sin((  0 +2  )n) = sin(  0 n) = x 1 [n]

42. 2003-09-02Dan Ellis 42 Aliasing E.g. for cos(  n),  = 2  r ±  0 all r appear the same after sampling We say that a larger  appears aliased to a lower frequency Principal value for discrete-time frequency: 0 ≤  0 ≤   i.e. less than one-half cycle per sample