1. Saeid Rahati 1 Digital Signal Processing Week 1: Introduction 1.Course overview 2.Digital Signal Processing 3.Basic operations & block diagrams 4.Classes of sequences Based on: http://www.ee.columbia.edu/~dpwe/e4810/ By: Dan Ellishttp://www.ee.columbia.edu/~dpwe/e4810/

2. Saeid Rahati 2 1. Course overview Grading structure  Homeworks (and small project): 10%  Midterm: 20%  One session  Presentation (Max. 20 minutes): 10%  One paper related to course topics  Final exam: 30%  One session  Project: 30%

3. Saeid Rahati 3 1. Course overview Course project Goal: hands-on experience with DSP Practical implementation Work in pairs or alone Brief report, optional presentation Recommend MATLAB Ideas on DSP_Pack CD or past class or internet

4. Saeid Rahati 4 MATLAB Interactive system for numerical computation Extensive signal processing library Focus on algorithm, not implementation

5. Saeid Rahati 5 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 Grayscale image i(x,y) : 2-D Video: 3 x 3-D: {r(x,y,t) g(x,y,t) b(x,y,t)}

6. Saeid Rahati 6 Example signals Noise - all domains Spread-spectrum phone - radio ECG - biological Music Image/video - compression ….

7. Saeid Rahati 7 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…

8. Saeid Rahati 8 Digital Signal Processing DSP = signal processing on a computer Two effects: discrete-time, discrete level

9. Saeid Rahati 9 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]

10. Saeid Rahati 10 Digital vs. analog Pros Noise performance - quantized signal Use a general computer - flexibility, upgrade Stability/duplicability Novelty Cons Limitations of A/D & D/A Baseline complexity / power consumption

11. Saeid Rahati 11 DSP example Speech time-scale modification: extend duration without altering pitch M

12. Saeid Rahati 12 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 = sampling period; 1/T = sampling frequency

13. Saeid Rahati 13 Sequences Can write a sequence by listing values: Arrow indicates where n =0 Thus,

14. Saeid Rahati 14 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

15. Saeid Rahati 15 3. Basic 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]

16. Saeid Rahati 16 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]

17. Saeid Rahati 17 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]

18. Saeid Rahati 18 Combination of basic operations Example

19. Saeid Rahati 19 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

20. Saeid Rahati 20 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

21. Saeid Rahati 21 Down-sampling An example of down-sampling 3

22. Saeid Rahati 22 Up-sampling Up-sampling is the converse of down- sampling: L-1 zero values are inserted between each pair of original values. L

23. Saeid Rahati 23 Up-sampling An example of up-sampling 3 not inverse of downsampling!

24. Saeid Rahati 24 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  

25. Saeid Rahati 25 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 

26. Saeid Rahati 26 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

27. Saeid Rahati 27 4. Classes of sequences Useful to define broad categories… Finite/infinite (extent in n ) Real/complex: x[n] = x re [n] + j·x im [n]

28. Saeid Rahati 28 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]

29. Saeid Rahati 29 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

30. Saeid Rahati 30 Basic sequences Unit sample sequence: Shift in time:  [n - k] Can express any sequence with           [n] +    [n-1] +    [n-2]..

31. Saeid Rahati 31 More basic sequences Unit step sequence: Relate to unit sample:

32. Saeid Rahati 32 Exponential sequences Exponential sequences= eigenfunctions General form: x[n] = A·  n If A and  are real:  > 1  < 1

33. Saeid Rahati 33 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 magnitud e varying phase

34. Saeid Rahati 34 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]

35. Saeid Rahati 35 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

36. Saeid Rahati 36 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

37. Saeid Rahati 37 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 ”

38. Saeid Rahati 38 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]

39. Saeid Rahati 39 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