CHAPTER 4 SIGNAL SPACE. Xijun Wang

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1 CHAPTER 4 SIGNAL SPACE Xijun Wang

2 WEEKLY READING 1. Goldsmith, Wireless Communications, Chapters 5 2. Gallager, Principles of Digital Communication, Chapter 5 2

3 DIGITAL MODULATION AND DEMODULATION n Digital modulation Mapping the information bits into an analog signal for transmission over the channel. n Detection Determining the original bit sequence by decoding the received signal as the signal in the set of possible transmitted signals that is closest to the one received. n How to determine the distance between the transmitted and received signals? 3

4 SIGNAL AND SYSTEM MODEL n Every T seconds, the system sends K = log 2 M bits of information through the channel for a data rate of R = K/T bits per second (bps). n There are M = 2 K possible sequences of K bits n Each bit sequence of length K comprises a message m i 4

5 SIGNAL AND SYSTEM MODEL n Since the channel is analog, the message must be embedded into an analog signal for channel transmission. n Thus, each message m i M is mapped to a unique analog signal s i (t) S = {s 1 (t),..., s M (t)} n s i (t) is defined on the time interval [0, T) and has energy 5

6 SIGNAL AND SYSTEM MODEL n When messages are sent sequentially, the transmitted signal becomes a sequence of the corresponding analog signals 6

7 SIGNAL AND SYSTEM MODEL n The transmitted signal is sent through an AWGN channel and form the received signal r(t) = s(t) + n(t) n The receiver must determine the best estimate of which s i (t) S was transmitted during each transmission interval [kt, (k + 1)T) n This best estimate for si(t) is mapped to a best estimate n Probability of message error 7

8 ORTHOGONAL SPACE n Real orthonormal basis functions {φ 1 (t),..., φ N (t)} 8

9 SIGNAL VECTOR n Any set of M real energy signals S = (s 1 (t),..., s M (t)) defined on [0, T) can be represented as a linear combination of N M real orthonormal basis functions {φ 1 (t),..., φ N (t)} n Basis function representation a real coefficient representing the projection of s i (t) onto the basis function φ j (t) n The set of signal waveforms {s i (t)} can be viewed as a set of signal vectors {s i }={s i1,s i2,,s in }. 9

10 EXAMPLE ARBITRARY SIGNAL A set of basic function A set of basic function 10

11 EXAMPLE LINEAR PASSBAND MODULATION n Basis set n Orthogonal when f c T >> 1 n Complex baseband representation 11

12 SIGNAL SPACE n Signal constellation point of signal s i (t) Denote the coefficients {s ij } as a vector s i = (s i1,..., s in ) Signal constellation consists of all constellation points {s 1,..., s M } n Signal space representation Given the basis functions {φ 1 (t),..., φ N (t)} there is a one-to-one correspondence between the transmitted signal s i (t) and its constellation point s i The representation of s i (t) in terms of its constellation point s i 12

13 SIGNAL SPACE REPRESENTATION n If the signals {s i (t)} are linearly independent then N = M, otherwise N < M. n Signal space representations for common modulation techniques like MPSK and MQAM are two-dimensional 13

14 VECTOR CHARACTERIZATION IN THE VECTOR SPACE n The length of a vector in R N n The distance between two signal constellation points s i and s k n Inner product 14

15 WAVEFORM ENERGY n A special case of Parseval s theorem (K j =1) 15

16 NOISE IN SIGNAL SPACE n The representation of noise is similar to signal where n! (t) = N j=1 n j φ j (t) n! (t) = n(t) n! (t) n(t) = n! (t) + n! (t) is the noise within the signal space is the noise outside the signal space n The noise n(t) is then represented by n=(n 1,n 2,,n N ) 16

17 VECTORIAL VIEW OF DETECTION n The task of the receiver is to decide which of the prototypes within the signal space is closest in distance to the received vector r. 17

18 RECEIVER STRUCTURE 18

19 COHERENT DETECTOR remainder noise sufficient statistic 19

20 MATCHED FILTER n The sampled matched filter outputs (r 1,..., r n ) are the same as the (r 1,..., r n ) in coherent detector. h j (t) = φ j (T t) x j (t) = x j (T ) = t x(τ )h j (t τ )dτ = x(τ )φ j (T t + τ )dτ, j = 1,!N 0 T 0 x(τ )φ j (τ )dτ, j = 1,!N t 0 20

21 SUFFICIENT STATISTIC 21

22 MAP DETECTION n Posterior probability n MAP Minimize error probability n ML Assuming equally likely messages 22

23 ML DETECTION n Likelihood function n Log likelihood function the log likelihood function depends only on the distance between the received vector r and the constellation point s i 23

24 DECISION REGIONS n Computing the received vector r from r(t), n Finding which decision region Z i contains r, n Outputting the corresponding message m i. 24

25 DEGREE OF FREEDOM n Any continuous time signals of duration T which have most of their energy within the frequency band [ W/2, W/2] Complex signals has dimension approximately WT or has about WT degree of freedom. Real signals has dimension approximately 2WT or has about 2WT degree of freedom. n Any signal within this class can be approximated by specifying about WT (2WT) complex (real) numbers as coefficients in an orthogonal expansion. 25

26 DEGREE OF FREEDOM n Such theorems may seem counter-intuitive at first: How could a finite sequence of numbers, at discrete intervals, capture exhaustively the continuous and uncountable stream of numbers that represent all the values taken by a signal over some interval of time? n In general terms, the reason is that bandlimited continuous functions are not as free to vary as they might at first seem. Consequently, specifying their values at only certain points, suffices to determine their values at all other points. 26

Detection and Estimation of Signals in Noise. Dr. Robert Schober Department of Electrical and Computer Engineering University of British Columbia

Detection and Estimation of Signals in Noise Dr. Robert Schober Department of Electrical and Computer Engineering University of British Columbia Vancouver, August 24, 2010 2 Contents 1 Basic Elements