Principles of Communications

Transcription

1 Principles of Communications Weiyao Lin, PhD Shanghai Jiao Tong University Chapter 4: Analog-to-Digital Conversion Textbook: /2011 Meixia SJTU 1

2 Outline Analog signal Sampling Quantization Encoding Discrete-time time continuous-valued signal Discrete-time discrete-valued signal Bit mapping Pulse Code Modulation Digital transmission 2010/2011 Meixia SJTU 2

3 Sampling Sampling Theorem: Let the signal x() t have a bandwidth W, i.e., let X ( f) 0, for f W. Let xt () be sampled at time 1 interval T to yield the sequence { xnt ( s )} s n=. 2W Then it is possible to reconstruct t the original i signal x() t from the sampled values. 2010/2011 Meixia SJTU 3

4 Sampling Process xt () xs() t 0 (a) t x () t xs( () t st () 0 (c) t st () Ts The sampling process can be regarded a modulation process with carrier given by periodic impulses. It s also called pulse modulation (b) t 2010/2011 Meixia SJTU 4

5 The result of sampling can be written as xδ () t = x( nts) δ( t nts) = x() t δ( t nts) n= Taking Fourier transform n= 1 n 1 n Xδ ( f ) = X ( f ) δ f = X f Ts n= Ts Ts n= Ts 1 1 X ( f ) s > f s = < 2W 2 W T If or T s then the replicated spectrum of x(t) () overlaps and reconstruction is not possible, known as aliasing error The minimum sampling rate fs = 2 is known as Nyquist sampling rate s -1/Ts W 2010/2011 Meixia SJTU -W W f X δ ( f ) 1/Ts 5 f

6 Reconstruction To get the original signal back, it is sufficient to filter the sampled signal through a LPF with frequency response 1 f < W H( f) = 1 0 f Ts The reconstruction is W [ ] x() t = 2 W ' Tsx( nts)sinc2 W '( t nts) for n= 1 W W ' W T s 2010/2011 Meixia SJTU 6

7 Quantization Quantization is a rounding process, each sampled signal point is rounded to the nearest value from a finite set of possible quantization levels. Scalar quantization Each sample is quantized individually Vector quantization Blocks of samples are quantized at a time 2010/2011 Meixia SJTU 7

8 Scalar Quantization The set of real numbers R is partitioned into N disjoint subsets, denoted as Rk, each called quantization region For each region Rk, a representation point, called quantization level xk is chosen If the sampled signal belongs to region Rk, then it is represented by xk, i.e. Q ( x ) = x, for all x R, k = 1,..., N k k Quantization error exqx (, ( )) = x Qx ( ) 2010/2011 Meixia SJTU 8

9 Performance Measure of Quantization Signal-to-quantization noise ratio (SQNR) is defined by SQNR = EX 2 ( ) ([ ( )] 2 ) E X Q x For random variable X SQNR = 1 T /2 2 lim x ( t) dt T T T /2 1 T /2 2 lim ( x( t) Q( x( t)) ) dt T T T /2 For signal x(t) 2010/2011 Meixia SJTU 9

10 Example The source X(t) is stationary Gaussian source with mean zero and power spectral density S x ( f) 2 f < 100 Hz = 0 otherwise It is sampled at the Nyquist rate and each sample is quantized using the 8-level quantizer with a =, a = 60, a = 40, a = 20, a = 0, a = 20, a = 40, a = x = 70, x = 50, x = 30, x = 10, x = 10, x = 30, x = 50, x = What is the resulting distortion and rate? What is the SQNR? 2010/2011 Meixia SJTU 10

11 Uniform Quantizer Let the range of the input samples is [-a, a] and the number of quantization levels is N = 2^v. Then the length of each quantization region is given by 2a a Δ= = 1 2 v N Quantized values are the midpoints of quantization regions. Assuming that the quantization error is uniformly Δ Δ distributed on. Then (, ) Δ/ Δ a a Ee [ ] = xdx= = = /2 2 Δ 12 2 N 3 4 v Δ /2 v P 34 db X PX PX SQNR = = = 10log v One extra bit increases 2 Ee [ ] a a the SQNR by 6 db! 2010/2011 Meixia SJTU 11

12 Nonuniform Quantizer If we relax the condition that the quantization regions be of equal length, then we can minimize the distortion with less constraints; therefore, the resulting quantizer will perform better than a uniform quantizer The usual method of nonuniform quantization is to first pass the samples through a nonlinear filter and then perform a uniform quantization => Companding For speech coding, higher probability for smaller amplitude and lower probability for larger amplitude μ law compander A-law compander 2010/2011 Meixia SJTU 12

13 Compander μ law compander ( + μ x ) log 1 g ( x ) = sgn( x ), x 1 log(1 + μ) μ = /2011 Meixia SJTU 13

14 A law compander Compander 1+ log A x gx ( ) = sgn( x), x 1 y 1+ log A 4 A = x /2011 Meixia SJTU 14

15 Lloyd-Max Conditions Optimal Quantizer The boundaries of the quantization regions are the midpoints of the corresponding quantized values The quantized values are the centroids of the quantization regions. 2010/2011 Meixia SJTU 15

16 Vector Quantization The idea of vector quantization is to take blocks of source outputs of length n, and design the quantizer in the n-dim Euclidean space, rather than doing the quantization based on single samples in a one-dim space Optimal vector quantizer to minimize distortion Region Ri is the set of all points in the n-dim space that are closer to xi than any other xj, for all j\= I; i.e. { n x : x x x x, } R = R < j i i i j xi is the centroid of the region Ri, i.e. 1 xi = xf x( x) dx P ( x R i ) Ri R 2010/2011 Meixia SJTU 16

17 4.3 Encoding The encoding process is to assign v bits to N=2^v quantization levels. Since there are v bits for each sample and fs samples/second, we have a bit rate of R= vf s Natural binary coding bits/second Assign the values of 0 to N-1 1to different quantization levels l in order of increasing level value. Gray coding Adjacent levels differ only in one bit 2010/2011 Meixia SJTU 17

18 Examples Natural binary code (NBC), folded binary code (FBC), 2- complement code (2-C), 1-complement code (1-C), and Gray code Level no NBC FBC 2-C Gray code Amplitude level /2011 Meixia SJTU 18

19 Pulse Code Modulation (PCM) Systems Block diagram of a PCM system x() t { x n } { x ˆ n } { K 0110 K } Sampler Quantizer Encoder Bandwidth requirement: If a signal has a bandwidth of W and v bits are used for each sampled signal, then BW req R = = 2 vw Hz 2010/2011 Meixia SJTU 19

20 Differential PCM (DPCM) For a bandlimited random process, the sampled values are usually correlated random variables This correlation can be employed to improve the performance Differential PCM: quantize the difference between two adjacent samples. As the difference has small variation, to achieve a certain level of performance, fewer bits are required 2010/2011 Meixia SJTU 20

21 DPCM Quantizer encoder delay decoder delay 2010/2011 Meixia SJTU 21

22 Delta Modulation (DM) DM is a simplified version of DPCM, where the quantizer is a two-level quantizer with magnitude ±Δ In DM, only 1-bit per symbol is employed. So adjacent samples must have high correlation. m(t) m (t) m ( t ( ) i ) m t 1 m t ) i ( i The step size Δ is critical in designing a DM system. 2010/2011 Meixia SJTU 22