1. بسم الله الرحمن الرحيم Digital Signal Processing Lecture 2 Analog to Digital Conversion University of Khartoum Department of Electrical and Electronic Engineering Fourth Year 2015-2016

2. Content 1.Introduction 2.Analog to Digital conversion 3.Sampling of Analog Signal 4.Quantization 2015-2016Dr Iman AbuelMaaly2

3. 2015-2016 3 1. Introduction 1. Continuous time Signals (notations) A simple harmonic oscillation: Signal parameters: A : Amplitude Ω : Frequency in radians/sec θ : phase in radians Ω = 2π F F : frequency in cycles/sec or Hz. T=1/F is the period of the signal Dr Iman AbuelMaaly

4. 2015-20164 1. Introduction Dr Iman AbuelMaaly 1. Continuous time Signals (notations)

5. 2015-2016 5 1. Introduction Signal parameters: n : integer variable A : Amplitude ω : Frequency in radians/sample (ω = 2π f) θ : phase in radians f : frequency in cycles/sample.(f=1/N) N : is the period Dr Iman AbuelMaaly 2. Discrete –time Signals (notations)

6. 2015-2016 6 Example of a discrete-time sinusoidal signal: (ω =π/6 radians per sample (f =1/12 cycles per sample) and θ = π/3 ) n x(n) Dr Iman AbuelMaaly 1. Introduction

7. 2015-2016Dr Iman AbuelMaaly 7 Sampler Quantizer coder xa(t)xa(t) xq(n)xq(n) x(n)x(n) 110010..0 Analog signal Discrete signal Quantizedsignal Digital signal Discrete signal Quantizedsignal x(n)x(n) xq(n)xq(n) 2. Analog to Digital conversion

8. We view the A/D conversion as three steps process. Basic parts of an A/D conversion 2015-2016Dr Iman AbuelMaaly 8 SamplerQuantizercoder xa(t)xa(t)xq(n)xq(n)x(n)x(n) 110010..0 Analog signal Discrete signal Quantized signal Digital signal

9. 2. Analog to Digital conversion Sampling: Conversion of a continuous time signal into a discrete time signal. Quantization: Conversion of a discrete time continuous valued signal into a discrete time discrete valued signal. Coding: Each discrete valued signal is represented by b-bits (binary sequence) 2015-2016Dr Iman AbuelMaaly 9

10. Distortion and Noise Sampling does not result in a loss of information, nor does it introduce distortion in the signal if the signal is band limited (according to the sampling theory) Quantization is a non invertible process that results in signal distortion. This quantization distortion decreases when the the accuracy (number of bits) increases. 2015-2016Dr Iman AbuelMaaly 10

11. Analog to digital conversion cost is proportional to the sampling rate and accuracy 2015-2016Dr Iman AbuelMaaly 11

12. 3. Sampling of analog signals Uniform sampling Sampling of a continuous-time signal x a (t) can be done by obtaining its values at periodic times to get x a (nT) where T is the sampling period. This is described by the relation: 2015-2016Dr Iman AbuelMaaly 12

13. Uniform Sampling 2015-2016Dr Iman AbuelMaaly 13

14. Uniform Sampling Sampling 2015-2016Dr Iman AbuelMaaly 14

15. Uniform Sampling T= Sampling period (secs). = sampling rate ( samples/sec) or sampling frequency in Hz. 2015-2016Dr Iman AbuelMaaly 15

16. Uniform Sampling Notice For continuous time signals we use F and Ω For discrete time signals we use f and ω 2015-2016Dr Iman AbuelMaaly 16

17. Uniform Sampling Example For the analog sinusoid When sampled periodically at rate We get 2015-2016Dr Iman AbuelMaaly 17

18. Frequency variables : F and f From the above example we get the following: (1) Relative or normalized frequency (2) x(n) 2015-2016Dr Iman AbuelMaaly 18

19. For continuous-time sinusoids: (3) For discrete-time sinusoids: (4) 2015-2016Dr Iman AbuelMaaly 19 Frequency variables : F and f

20. By substituting (1) and (2) into (4), the frequency of a continuous time sinusoid when sampled at a rate must fall in the range: Or equivalently: 20 (1) (2) (4) 2015-2016Dr Iman AbuelMaaly

21. Periodic sampling of a continuous signal implies a mapping of the infinite frequency range for the variable F or Ω into a finite frequency range for the variable f or ω. The highest frequency in a discrete time signal is ω = π or f= 1/2. It follows that with a sampling rate F s, the corresponding highest values F and Ω are: and 2015-2016Dr Iman AbuelMaaly 21

22. Quantization and Continuous-amplitude Signals: Quantization is the process of converting a discrete–time continuous amplitude signal into a digital signal by expressing each sample value as a finite ( instead of infinite) number of digits. Q[x(n)] is the quantization of x(n) 2015-201622

23. The Quantization Error e q (n) e q (n) is the difference between the quantized value and the actual value of samples. Quantization process is done by either truncation or rounding. 2015-201623

24. Example: x (1) = 3.8 and x (2) =3.2 Truncationx q (1) = 3x q (2) = 3 Roundingx q (1) = 4x q (2) = 3 2015-201624 The Quantization Error e q (n)

25. Example 2: Consider a discrete time signal Obtained by sampling the analog signal with a sampling frequency 1 Hz 2015-201625

26. Example 2: 2015-201626 Sampling

27. Ilustration of Quantization 2015-201627

28. 2015-201628 Rounding =0.7

29. 2015-201629 Quantization Error

30. ∆ is the distance allowed between two successive quantization levels and is called quantization step size or resolutions. The rounding quantizer assigns each sample of x(n) to the nearest quantization level. The quantization error can not exceed half of the quantization step, i.e., 2015-201630

31. L is the number of quantization levels. If x max and x min are the maximum and minimum values of x(n) then, x max – x min is the dynamic range of the signal 2015-201631

32. In the previous example: x max = 1.0 x min = 0 L= 11 Then ∆ =0.1 Increasing the number of quantization level results in a decrease of the quantization step size and thus e q (n) decreases. 2015-201632

33. Why is quantization a non-invertible proces? Because it is a many to one mapping. i.e., all samples in a distance ∆/2 about a certain quantization level are assigned the same value. 8.02, 8.1, 8.1, 8.2, 8.2, 8.3, 8.3, 8.4 8 8.4 are all quantized to be 8 2015-201633

34. Quantization of Sinusoidal Signals The following figure shows the sampling and quantization of an analog sinusoidal signal 2015-201634

35. 2015-201635

36. By quantizing the analog signal instead of the discrete signal, we get the following quantization error: x a (t) denotes the time that x a (t) stays within the quantization levels. 2015-201636 Quantization of Sinusoidal Signals

37. 2015-201637 xa(t) Signal xa(t) is almost linear between quantization levels. eq(t) The corresponding quantization error eq(t) is eq(t)= xa(t)- xq(t)

38. The Mean Square Error Power p q The mean square error power p q is: Since, We have 2015-201638

39. The Mean Square Error Power b If the quantizer has b bits of accuracy and the quantizer covers the entire range 2A, the quantization step is Hence The average power of the signal x a (t) is 2015-201639

40. Signal To Quantization Noise Ratio SQNR The quality of the output of the A/D converter is usually measured by the signal to quantization noise ratio SQNR, Expressed in dB, SQNR is, SQNR increases 6dB for every bit added to the word length. 2015-201640

41. Next Lecture 2015-2016Dr Iman AbuelMaaly41