ETSF15 Analog/Digital. Stefan Höst

Transcription

1 ETSF15 Analog/Digital Stefan Höst

2 Physical layer Analog vs digital Sampling, quantisation, reconstruction Modulation Represent digital data in a continuous world Disturbances Noise and distortion Synchronization Synchronization to signal or between nodes Digital data processing Information

3 Data vs Signal vs Information Data: Static representation of information Signal: Dynamic representation of information Information: Information content in data or signal 3

4 Analog vs Digital Analog Continuous time and amplitude signal Electrical/optical domain Digital Discrete time and amplitude Often binary representation s(t) t s[n] n

5 Digitalization of analog signals Performed in three steps: 1. Sampling: Discretization in time. Quantization: Discretization in amplitude 3. Encoding: Binary representation of amplitude levels In practice: ADC: Analog to Digital Converter DAC: Digital to Analog Converter 5

6 Sampling The process of discretizing time of a continuous signal. s n = s(nt & ) Sample time: T & Sample frequnecy: F & = 1/T & Loose information about time s(t) s(t) s[n] t t n 6

7 Shannon-Nyquist Sampling Theorem If s(t) is a band limited signal with highest frequency component F,-., then s(t) is uniquely determined by the samples s n = s(nt & ) if and only if F & = 1 T & F,-. The signal can be (perfectly) reconstructed with 5 s t = 1 s n sinc t nt & 6785 Fs/ is the Nyquist frequency and F,-. the Nyquist rate T & 7

8 Reconstruction Example y t = sin π >? t, Fs = 1 Hz

9 Aliasing y(t) = cos (π7t) F s = 10 Hz T Reconstruction to lowest possiblefrequency Reconstruct freq: f IJK = f + kf & s.t. F & / f IJK F & / y t = cos π3t

10 Sampling theorem proof Two important transforms + sinc(t) p(f) F t -1/ 1/ f x(t) = P n (t n) F x(f) = P n (f n) t f + 10

11 Sampling theorem proof Mathematical description of sampling + s(t) 1 T x( t T )=P n (t nt ) s s (t) =s(t) 1 T x( t T ) = = P n s[n] (t nt ) t t t -T T T 3T -T T T 3T F S(f) x(tf)= 1 T Pn (f n T ) = S s (f) =S(f) x(tf) = 1 T Pn S(f n ) T f f f -1/T 1/T /T -1/T 1/T /T + 11

12 Sampling theorem proof Reconstruction + S s (f) = 1 T Pn S(f n T ) T p(ft) T = S(f) -1/T 1/T /T -1/T 1/T f f f F 1 s s (t) = P n s[n] (t nt ) s(t) =s s (t) sinc( t ) sinc( t ) T T = P t nt n s[n]sinc( ) T = t t t + -T T T 3T -T T T 3T 1

13 Sampling theorem Aliasing Let F s <F max + S(f) S s (f) Ŝ(f) Sampling Reconstruction f -F F F f -F/ F/ f 13

14 Example y(t) = cos(π 7t) Y ( f ) = 1 ( δ ( f + 7) + δ ( f 7) ) Sampling with F s =10 Hz -7 7 f f -5 5 Reconstruct in [-F s /, F s /]: Y ˆ 1 ( f ) = ( δ ( f + 3) + δ ( f 3) ) ŷ(t) = cos(π 3t) -3 3 f 14

15 Quantization Uniform quantization for k bits M 1 D x Q M= k equidistant levels M D D D M D x Represent sample with k bits M 1 D 15

16 Encoding Representation of quantized samples in bits x(t) x[n] Quant level 0 1 x=

17 Quantisation distortion Distortion (noise): d(x, x Q ) = (x x Q ) Average distortion for uniform input: E ( X X ) Q = Δ/ 1 x Δ dx = Δ/ Δ 1 M D M D Signal to quantisation noise ratio x x Q d(x, x Q ) M D M D x x SQNR = E X E ( X X ) Q = (M Δ) /1 Δ /1 = M = k SQNR db = 10log k = k 6dB 17

18 Delta modulation Represent change in amplitude with 1 bit 1: +1 0: 1 Must use higher sampling rate s(n) n

19 Examples Telephony F max = 4 khz F s = 8 khz (samples per sec) 8 bit/sample => 64 kb/s CD F max = 0 khz F s = 44.1 khz (samples per sec) 16 bit/sample => kb/s channels (stereo) => 1.4 Mb/s 19

20 Principles of circuits Sample and hold (sampling) ADC (quantisation) DAC (reconstruction) 0

21 Sample and hold 1

22 ADC Analog to Digital Converter Sample and hold circuit freeze the analog value during conversion ADC methods Direct conversion (flash ADC) Integrating Wilkinson Sigma-delta Etc (see e.g.

23 ADC Direct conversion example (3 bit) 3

24 DAC Digital to Analog Converter Pulse width modulator R-R ladder Interpolating Binary weighting See more on 4

25 DAC Example Weighted resistor R-R ladder 5