Infocommunication. Sampling, Quantization. - Bálint TÓTH, BME TMIT -

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1 Infocommunication Sampling, Quantization - Bálint TÓTH, BME TMIT -

2 Overview PPT is for demonstration, not for learning! Analog signals problem: noise, distortion Digital signals what are the benefits? We re going to talk about A/D and D/A conversions Sampling The sampling theorem Band limited signals Quantization Quantization noise Signal-to-noise ratio (SNR) 2

3 Analog and digital signals Sampling makes us able to use modern technology: Sampling Audio: CD, MP3, cell phone Pictures: digital camera, printer Video: DVD Other: Computer, etc. Source: 3

4 Effects of sampling (example 1) Picture 4

5 Effects of sampling (example 2) Newspaper 5

6 Effects of sampling (example 3) Your eyes Source: eagleeye.co.ke 6

7 Effects of sampling (example 3) Your eyes 7

8 Effects of sampling (example 4) Music 8

9 Analog-to-digital conversion Relation between the analog and the sampled signal (Sampling theorem): X S f = f s X(f i f s ) i 9

10 Jean Baptiste Joseph Fourier Source: 10

11 Who s theorem? The sampling theorem is usually known as the Shannon Sampling Theorem due to Claude E. Shannon s paper A mathematical theory of communciation (1948). However, he himself said that is common knowledge in the communication art. The minimum required sampling rate f s (i.e. 2xB) is known as the Nyquist sampling rate or Nyquist frequency because of H. Nyquist s work on telegraph transmission in 1924 with K. Küpfmüller. The first formulation of the sampling theorem precisely and applied it to communication is probably a Russian scientist by the name of V. A. Kotelnikov in However, mathematician already knew about this in a different form and called this the interpolation formula. E. T. Whittaker published the paper On the functions which are represented by the expansions of the interpolation theory back in

12 Digital-to-analog conversion x t x k x s t The spectrum of a sampled signal: k time between samples: T X S f = x k e i2πfkt sampling frequency: f s = 1/T 12

13 Band limited signal, low-pass filter There is no spectral overlapping if X f i f S = 0, if f < B or f > B, i > 0 With ideal low-pass filter the reconstruction is granted if f s 2B (Nyquist frequency) 13

14 Sub- and oversampling Below the Nyquist-frequency the sampling points are farther from each other, the signal cannot be perfectly reconstructed (with lowpass filter) Subsampling Above the Nyquist-frequency the sampling points are nearer to each other, the signal can be better recognised from the samples. Oversampling 14

15 Aliasing, leakage This phenomena is caused by the poor choise of sampling frequency or by non-ideal filters. Aliasing: input filter (anti-aliasing filter). Leakage: output filter. 15

16 Aliasing example 16

17 Aliasing example with antialising 17

18 Aliasing example w/o antialising 18

19 Quantization Convert the signal to discrete values Source: 19

20 Quantization interval and steps m = 2m / N 20

21 Noise (linear quantization) Power of noise: Signal-to-noise ration for sinusoid signal: SNR = n [db] 21

22 Signal transmission and reconstruction with D/A and A/D Filter Sampler & holder Quantization Coder Decoder Filter 22

23 Signal transmission and reconstruction with D/A and A/D Filter Sampler & holder Quantization Coder Decoder Filter 23

24 Signal transmission and reconstruction with D/A and A/D Filter Sampler & holder Quantization Coder Decoder Filter 24

25 Signal transmission and reconstruction with D/A and A/D 3, 5, 7, 6, 4, 4, 5, 3, 2 Filter Sampler & holder Quantization Coder Decoder Filter 25

26 Signal transmission and reconstruction with D/A and A/D Filter Sampler & holder Quantization Coder Decoder Filter 3, 5, 7, 6, 4, 4, 5, 3, 2 26

27 Signal transmission and reconstruction with D/A and A/D Filter Sampler & holder Quantization Coder Decoder Filter 27

28 Signal transmission and reconstruction with D/A and A/D 28

Signals and Systems. Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI

Signals and Systems. Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI Signals and Systems Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Continuous time versus discrete time Continuous time