ELEN E4810: Digital Signal Processing Week 1: Introduction

1. ELEN E4810: Digital Signal ProcessingWeek 1: Introduction • Course overview • Digital Signal Processing • Basic operations & block diagrams • Classes of sequences Dan Ellis

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

3. Grading structure • Homeworks: 20% • Mainly from Mitra • Collaboration • Midterm: 20% • One session • Final exam: 30% • Project: 30% Dan 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 Dan 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 Dan 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) Dan Ellis

7. Course at a glance Dan 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-DimensionGreyscale image i(x,y) : 2-DVideo: 3 x 3-D: {r(x,y,t) g(x,y,t) b(x,y,t)} Dan Ellis

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

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

12. DSP vs. analog SP • Conventional signal processing: • Digital SP system: Processor p(t) q(t) p[n] q[n] A/D Processor D/A p(t) q(t) Dan 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 Dan Ellis

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

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

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

17. Left- and right-sided • x[n] may be defined only for certain n: • N1 ≤ n ≤ N2: Finite length (length = …) • N1 ≤ n: Right-sided (Causal if N1 ≥ 0) • n ≤ N2: Left-sided (Anticausal) • Can always extend with zero-padding Left-sided Right-sided Dan Ellis

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

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

20. y[n] x[n] y[n] x[n] 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 Dan Ellis

21. Combination of basic operations • Example Dan 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 Dan Ellis

23. M 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: Dan Ellis

24. 3 Down-sampling • An example of down-sampling Dan Ellis

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

26. 3 Up-sampling • An example of up-sampling not inverse of downsampling! Dan 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 = xre + j·ximwhere magnitude|x|= √(xre2 +xim2)and phaseq = tan-1(xim/xre) • Polar form: x = |x|ejq=|x|cosq + j·|x|sinq ( ) Dan 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: rejq·sejf = rsej(q+f) • Phases modulo 2 Dan Ellis

29. Complex conjugate • Flips imaginary part / negates phase:conjugatex* = xre– j·xim = |x| ej(–q) • Useful in resolving to real quantities:x + x* = xre + j·xim + xre – j·xim = 2xrex·x* = |x| ej(q) |x| ej(–q) = |x|2 Dan Ellis

30. Classes of sequences • Useful to define broad categories… • Finite/infinite (extent in n) • Real/complex:x[n] = xre[n] + j·xim[n] Dan Ellis

31. Classification by symmetry • Conjugate symmetric sequence:xcs[n] = xcs*[-n] = xre[-n] – j·xim[-n] • Conjugate antisymmetric:xca[n] = –xca*[-n] = –xre[-n] + j·xim[-n] Dan Ellis

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

33. Basic sequences • Unit sample sequence: • Shift in time:d[n - k] • Can express any sequence with d:{a0,a1,a2..}= a0d[n] + a1d[n-1] + a2d[n-2].. Dan Ellis

34. More basic sequences • Unit step sequence: • Relate to unit sample: Dan Ellis

35. Exponential sequences • Exponential sequences= eigenfunctions • General form: x[n] = A·an • If A and a are real: |a| > 1 |a| < 1 Dan Ellis

36. scale varying phase varying magnitude Complex exponentials x[n] = A·an • Constants A, a can be complex :A = |A|ejf ; a = e(s + jw)x[n] =|A| esnej(wn + f) Dan Ellis

37. Complex exponentials • Complex exponential sequence can ‘project down’ onto real & imaginary axes to give sinusoidal sequences xre[n] = en/12cos(pn/6)xim[n] = en/12sin(pn/6)M xre[n] xim[n] Dan Ellis

38. Periodic sequences • A sequence satisfying is called a periodic sequence with a periodN where N is a positive integer and k is any integer. Smallest value of N satisfyingis called the fundamental period Dan Ellis

39. Periodic exponentials • Sinusoidal sequence and complex exponential sequence are periodic sequences of period Nonly 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 samplesr > 1r cycles per N samples M Dan Ellis

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

41. Sampling sinusoids • Sampling a sinusoid is ambiguous: x1 [n] = sin(w0n)x2 [n] = sin((w0+2p)n) = sin(w0n) = x1 [n] Dan Ellis

42. Aliasing • E.g. for cos(wn), w = 2pr ±w0all r appear the same after sampling • We say that a larger wappears aliased to a lower frequency • Principal value for discrete-time frequency: 0 ≤w0 ≤ pi.e. less than one-half cycle per sample Dan Ellis