CHAPTER 4 SIGNAL SPACE Xijun Wang

WEEKLY READING

1. Goldsmith, “Wireless Communications”, Chapters 52. Gallager, “Principles of Digital Communication”,

Chapter 5

2

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

SIGNAL AND SYSTEM MODEL

n Every T seconds, the system sends K = log2M bits of information through the channel for a data rate of R = K/T bits per second (bps).

n There are M = 2K possible sequences of K bits n Each bit sequence of length K comprises a message

mi

4

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 mi∈M is mapped to a unique analog signal si(t)∈S = {s1(t), . . . , sM(t)}

n si(t) is defined on the time interval [0, T) and has energy

5

SIGNAL AND SYSTEM MODEL

n When messages are sent sequentially, thetransmitted signal becomes a sequence of thecorresponding analog signals

6

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 si(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

ORTHOGONAL SPACE

n Real orthonormal basis functions {φ1(t), . . . , φN(t)}

8

SIGNAL VECTOR

n Any set of M real energy signals S = (s1(t), . . . , sM(t)) defined on [0, T) can be represented as a linear combination of N ≤ M real orthonormal basis functions {φ1(t), . . . , φN(t)}

n Basis function representation

n The set of signal waveforms {si(t)} can be viewed asa set of signal vectors {si}={si1,si2,…,siN}.

9

a real coefficient representing the projection of si(t) onto the basis function φj(t)

EXAMPLE – ARBITRARY SIGNAL

10

A set of basic function A set of basic function

EXAMPLE – LINEAR PASSBAND MODULATION

n Basis set

n Orthogonal when fcT >> 1

n Complex baseband representation

11

SIGNAL SPACE

n Signal constellation point of signal si(t) ¨ Denote the coefficients {sij} as a vector si = (si1, . . . , siN) ¨ Signal constellation consists of all constellation points {s1,

. . . , sM}

n Signal space representation ¨ Given the basis functions {φ1(t), . . . , φN(t)} there is a

one-to-one correspondence between the transmitted signal si(t) and its constellation point si

¨ The representation of si(t) in terms of its constellation point si

12

SIGNAL SPACE REPRESENTATION

n If the signals {si(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

VECTOR CHARACTERIZATION IN THE VECTOR SPACE n The length of a vector in RN

n The distance between two signal constellation points si and sk

n Inner product

14

WAVEFORM ENERGY

n A special case of Parseval’s theorem (Kj=1)

15

NOISE IN SIGNAL SPACE

n The representation of noise is similar to signal

whereis the noise within the signal space

is the noise outside the signal space

n The noise n(t) is then represented by n=(n1,n2,…,nN)

16

n(t) = n!(t)+ n!(t)

n!(t) = njφ j (t)j=1

N

∑n!(t) = n(t)− n!(t)

VECTORIAL VIEW OF DETECTION

n The task of thereceiver is to decidewhich of theprototypes withinthe signal space isclosest in distance tothe received vectorr.

17

RECEIVER STRUCTURE

18

COHERENT DETECTOR

19

remainder noise

sufficient statistic

MATCHED FILTER

20

n The sampled matched filter outputs (r1, . . . , rn) are the same as the (r1, . . . , rn) in coherent detector.

hj (t) = φ j (T − t)

x j (t) = x(τ )hj (t −τ )dτ0

t

∫ = x(τ )φ j (T − t +τ )dτ0

t

∫ , j = 1,!N

x j (T ) = x(τ )φ j (τ )dτ0

T

∫ , j = 1,!N

SUFFICIENT STATISTIC

21

MAP DETECTION

n Posterior probability

n MAP¨ Minimize error probability

n ML¨ Assuming equally likely messages

22

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 si

23

DECISION REGIONS

n Computing the received vector r from r(t),

n Finding which decision region Zi contains r,

n Outputting the corresponding message mi.

24

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 byspecifying about WT (2WT) complex (real) numbersas coefficients in an orthogonal expansion.

25

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, sufficesto determine their values at all other points.

26