2. 1 Digital Signal Processing (DSP) By: Prof. M.R.Asharif Department of Information Engineering University of the Ryukyus, Okinawa, Japan

3. 2 Introduction DSP is different from other areas in computer science by the type of data it uses: signals. DSP is the mathematics and algorithms that used to manipulate these signals. But first it should be converted into digital form.

4. 3 Introduction Digital Signal Processing (DSP) Deals with the transformation of signals that are discrete in both amplitude and time  Is area of science and engineering developed rapidly over the last 30 years.  DSP is a result of significant advances in:  Digital computer technology  Integrated circuit fabrication  DSP involves time and amplitude quantization of signals and relies on the theory of discrete time signals and systems.

5. 4 Digital Signal THE SIGNALS ARE ANALOG (CONTINUOUS) SUCH AS: HUMAN VOICE ELECTRICAL SIGNAL(VOLTAGE OR CURRENT) RADIO WAVE OPTICAL AUDIO AND SO ON WHICH CONTAINS A STREAM OF INFORMATION OR DATA.

6. 5 Figure 1 : Digitized process of signal

7. 6 Figure 2 : Complete Process of Digital Signal

8. 7 Signals that encountered in Engineering are usually Continuous, such as: changes of light intensity with distance; voltage that varies over time etc. Analog-to-Digital Conversion (ADC) and Digital-to-analog Conversion (DAC) are the processes that allow digital computers to interact with these everyday signals.

9. 8 Digital Signal is referred as signal that is sampled and quantized. By sampling frequency and number of bits for quantization, one can decide how much information contains in digital signal.

10. 9 Two processes are made in digitization, ADC: (1) Sample and Hold (S/H) and (2) Quantization S/H changes time variable from continuous to discrete. Quantization is the conversion of discrete-time continuous-valued to discrete-time discrete- valued (digital) signal. The difference of this is called Quantization Error. Sampling without quantization is used in switched capacitor filters.

11. 10 Sampling: Time Domain DSP: Converting a continuously changing waveform (analog) into a series of discrete levels (digital)

12. 11 Many signals originate as continuous-time signals, e.g. conventional music or voice By sampling a continuous-time signal at isolated, equally-spaced points in time, we obtain a sequence of numbers.

13. 12 DSP

14. 13 Sampling: Time Domain Sampling replicates spectrum of continuous-time signal at integer multiples of sampling frequency Fourier series of impulse train where  s = 2  f s  G()G() ss  s  s  s  F()F() 2  f max -2  f max

15. 14 Representation of Sampling Convert impulse train to discrete- time sequence x c (t)x[n]=x c (nT) x s(t) -3T-2T2T3T4T-TT0 s(t) x c (t) t x[n] -3-223410 n

16. 15 Nyquist Sampling Theorem  N is generally known as the Nyquist Frequency The minimum sampling rate that must be exceeded is known as the Nyquist Rate Let xc(t) be a bandlimited signal with Then xc(t) is uniquely determined by its samples x[n]= xc(nT) if

17. 16 Aliasing is one of the most important properties of sampling Continuous function is discretized by sampling We loose some information

18. 17 Aliasing Artifacts arising from sampling and consequent loss of information Spatial: Jaggies, Moire Temporal: Strobe lights, “Wrong” wheel rotations Spatio-Temporal: Small objects appearing and disappearing

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31. 30 Sampling Demo Original Audio Decimated by 2

32. 31 Sampling Demo Original Audio Decimated by 4

33. 32 Sampling Demo Original Audio Decimated by 5

34. 33 Sampling Demo Original Audio Decimation by 4 Decimation by 5 Decimation by 6 Decimation by 7

35. 34 Quantization Mapping Quantization Dequantization Continuous valuesBinary codes Continuous values

36. 35 Quantization Mapping (cont.) Symmetric quantizers –Equal number of levels (codes) for positive and negative numbers Midrise and midread quantizers

37. 36 2-Bit Uniform Midrise Quantizer 01 10 11 3/4 1/4 -1/4 -3/4 1.0 1.0 0.0 00

38. 37 2-Bit Uniform Midtread Quantizer 01 11 2/3 0.0 -2/3 1.0 1.0 0.0 00/ 10

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46. 45 Convolution The equation (the * denotes convolution) For input f(t), impulse response g(t) the out put of the system is equal to the input signal convolved with the impulse response that i.e. h(t)= f(t)  g(t) Convolution in the time domain is equal to multiplication in the frequency domain, and vice versa.

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49. 48 Z-Transform For time sequence x(n) the discrete Z-transform is defined as: Z is a complex number, but in Fourier transform ejw is represented in unit circle. And it is similar to laplace transform which is analogue. Z is infinite but it is possible to make it convergent.

50. 49 Z-Transform Example 1 For sequence x(n)=an where n  0 transformed by Z-transform ifthenwhere The condition for convergence

51. 50 Z-Transform Example 2 For sequence x(n)=a|n| where -  <n<n transformed by Z- transform The left part The right part is convergent when: Making a summation into parts

52. 51 Properties of Z-Transform Linearity Let x(n) and y(n) be any two functions and let X(z) and Y(z) be their respective transforms. Then for any consonants a and b If w(n)=x(n)*y(n) then Z Shifting Z convolution Z

53. 52 Digital Filter Design Filter can be defined as a system that modifies certain frequencies relative to others. Digital filter is a linear shift invariance system (LSI). The designing filter involves the following stages: 1)Desired characteristics (Specification) of the system. 2)Approximation of the specification using a casual discrete-time system. 3)The realization of the system (building the filter by finite arithmetic computation.

54. 53 What are FIR and IIR systems? A discrete system is said to be an FIR system if its impulse response has zero- valued samples for n > M > 0 Integer number M is called the length of the impulse response IIR system is a discrete system with an infinite impulse response FIR = Finite Impulse Response IIR = Infinite Impulse Response

55. 54 Follow graph(FIR digital filter,IIR digital filter): x(n) DDDD x x a0 X0 a1 X(n-1) x a2 X(n-2) x a3 X(n-3) x an X(n-N) FIR Digital Filter: It is also known as difference equation. IIR Digital Filter: + DD D xx x b1b(m-1) bm    N i in inxay 0 )(

56. 55 Basic FIR structures Direct form, Transposed form, Cascade form, Linear-phase, Symmetric

57. 56 Direct form 2nd order

58. 57 Transposed direct form 2nd order

59. 58 Cascade direct form

60. 59 Direct form (Tapped delay line)

61. 60 Transposed direct form

62. 61 Basic IIR structures Direct form, Transposed form

63. 62 Direct form I 2nd order

64. 63 Direct form II 1st order

65. 64 Cascade direct form II

66. 65 Digital Filter & Spectrum by Adjusting the Coefficient

67. 66 Spectrum π 2π2π 0

68. 67 Digital Filter & Spectrum by Adjusting the Coefficient

69. 68 Spectrum π 2π2π 0

70. 69 Sampling Listening Example n=[0:0.0001:1]'; Sampling frequency Fs=10000 y=sin(2*pi*1000*n); sound(y,Fs); plot(y(1:20))

71. 70 Sampling Listening Example n=[0:0.0002:1]'; Sampling frequency Fs=5000 y=sin(2*pi*1000*n); sound(y,Fs); plot(y(1:20))

72. 71 Sampling Listening Example n=[0:0.0003:1]'; Sampling frequency Fs=3333 y=sin(2*pi*1000*n); sound(y,Fs); plot(y(1:20))

73. 72 Sampling Listening Example n=[0:0.0004:1]'; Sampling frequency Fs=2500 y=sin(2*pi*1000*n); sound(y,Fs); plot(y(1:20))

74. 73 Applications Most DSP applications deal with analogue signals. The analogue signal has to be converted to digital form Information is lost in converting from analogue to digital When the signal is converted to digital form, the precision is limited by the number of bits available.

75. 74 Early applications of DSP radar & sonar, where national security was at risk; oil exploration, where large amounts of money could be made; space exploration, where the The Scientist and Engineer's Guide to Digital Signal Processing

76. 75 New Applications After PC has been developed enough (1980-1990), new commercial applications were expanded for DSP, such products as: Mobile telephones, compact disc players, and electronic voice mail.

77. 76 Thank you

78. 77 x(t) x(nT) y(nT) y(t)

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83. 82 Alias Demo http://www.dsptutor.freeuk.com/ http://www.dsptutor.freeuk.com/aliasing/A D102.html

84. 83 FIR Filter Demo http://www.ami.ac.uk/courses/topics/0021_ firdemo/index.html