Summary II: Modulation and Demodulation

Transcription

1 Summary II: Modulation and Demodulation Instructor : Jun Chen Department of Electrical and Computer Engineering, McMaster University Room: ITB A1, ext junchen@mail.ece.mcmaster.ca Website: junchen/

2 Passband and Baseband Signals Definition 1. A signal is said to be baseband if S(f) = 0 for f > W for some W > 0. Definition. A signal is said to be passband if S(f) = 0 for f±f c > W for some f c > W > Time-domain representation of a passband signal. Any passband signal can be written as s p (t) = s c (t)cos(πf c t) s s (t)sin(πf c t), where s c (t) and s s (t) are real-valued signals. Usually, the waveforms s c (t) and s s (t) are referred to as the in-phase (or I) component and the quadrature (or Q) component of the passband signal s p (t), respectively.

3 . Time-domain representation of a baseband signal. by The complex envelope or complex baseband signal of s p (t) is represented s(t) = s c (t)+js s (t) 3. Time-domain relationship between passband and complex baseband. s p (t) = Re ( s(t)e jπf ct ) 4. Frequency-domain relationship between passband and complex

4 baseband. S(f) = S + p (f +f c ) S p (f) = S(f f c)+s ( f f c ) 5. Properties. Property 1. The I-channel is orthogonal to the Q-channel; i.e., < x c (t),x s (t) >= 0, where x c (t) = s c (t)cos(πf c t) and x s (t) = s s (t)sin(πf c t). Property. < u p (t),v p (t) >=< u c (t),v c (t) > + < u s (t),v s (t) >= Re ( u(t),v(t) ). Property 3. The energy of the passband signal is equal to that of its equivalent baseband signal; i.e., s p (t) = s(t).

5 M-ary PAM Constellation Definition 3. M-ary PAM constellation is defined as where M = b. A = {± (k 1)d } M/ k=1 Suppose that each constellation point is equally likely, then The mean of this constellation is m(a) = 1 M M/ k=1 ( ± (k 1)d ) = 0.

6 The average anergy E s is given by E s = 1 M = M = d M M/ k=1 M/ k=1 4 ( ( (k 1)d ( ) (k 1)d = d M M/ k=1 = d (M 1), 1 k 4 ) ) + ( (k 1)d) M/ k=1 k + M/ k=1 M/ k=1 1 ( 4k 4k +1 )

7 where we have used the fact that N k = k=1 N k = k=1 N(N +1)(N +1) 6 N(N +1) The minimum distance between any two distinct points is d min (A) = min = d a,a A,a a a a

8 PAM Modulation The baseband transmitted waveform for linear PAM is u(t) = n b[n]g TX (t nt), where each symbol b n is the PAM constellation point, the modulating pulse g TX (t) is a fixed baseband real waveform, T is termed the symbol interval and 1/T the symbol rate. Then, passband waveform is u p (t) = Re ( u(t)e jπf ct ) = u(t)cos(πf c t).

9 Square M-ary QAM Constellation Definition 4. Square M = b -ary QAM constellation is defined as A = {a+jb : a,b M arypamconstellation}. Suppose that each constellation point is equally likely, then The mean of this constellation is m(a) = 1 M = M M a,b M PAM a M PAM ( ) a+jb a+ j M M b M PAM b = 0

10 The average anergy is given by E s = 1 M = 1 M = M M a,b M PAM a,b M PAM a M PAM a+jb (a +b ) = 1 M a + b M PAM a,b M PAM b a + = d (M 1) 6 The minimum distance between any two distinct points is d min (A) = min = d a,a A,a a a a a,b M PAM b

11 QAM Modulation The baseband transmitted waveform for linear QAM is u(t) = n b[n]g TX (t nt), where each symbol b[n] is the M-ary QAM constellation point, the modulating pulse g TX (t) is a fixed real baseband waveform, T is termed the symbol interval and 1/T the symbol rate. Then, the passband waveform is u p (t) = Re ( u(t)e jπf ct ) = u c (t)cos(πf c t) u s (t)sin(πf c t), whereu c (t) = n Re(b n)g TX (t nt) andu s (t) = n Im(b n)g TX (t nt).

12 The Nyquist criterion for ISI-Free Let g RX (t) be the receive filter and x(t) = g TX (t) g RX (t), i.e., x(t) = g TX (τ)g RX (t τ)dτ Then, the received filtered waveform using g RX (t) in the noise-free case is given by z(t) = u(t) g RX (t) = u(τ)g RX (t τ)dτ = n b[n]x(t nt). Definition 5. The waveform x(t) satisfying x[kt] = δ[k] is said to be ideal Nyquist with symbol interval T.

13 Therefore, when x(t) is the ideal Nyquist pulse, z[kt] = b[k] and thus, there is no itersymbol interference (ISI). Theorem 1. The x(t) is the ideal Nyquist with symbol interval T if and only if k= X(f k/t) = T for all f Two typical ideal Nyquist pulses are as follows: Minimum bandwidth Nyquist pulse: In frequency-domain, X(f) = { 1, f 1 T, 0, otherwise.

14 In time-domain, x(t) = sinc ( t) T Raised cosine pulse: In frequency-domain, X(f) = 1, ( f 1 α T 1 sin ( ) ) ( f 1 T )πt 1 α α, T T, 0, f > 1+α T. f 1+α T, In time-domain, x(t) = sinc ( t T ) cos( πα T ) 1 ( αt T bandwidth, typically chosen in the range of 0 α < 1. ), where α is the fractional excess

15 Choice of the ISI-Free Modulation Pulse Choose the ideal Nyquist X(f) 0. Choose G TX (f) = X(f) and G RX (f) = G TX (f) In general, g TX (t) can be allowed to be complex. The following theorem is very important. Theorem. let g TX (t) be an L (R) function such that X(f) = G TX (f) satisfies the Nyquist criterion for the symbol interval T. Then, {g TX (t kt) : k Z} is a set of orthonormal functions. Conversely, if {g TX (t kt) : k Z} is a set of orthonormal functions, then, X(f) = G TX (f) satisfies the Nyquist criterion for the symbol interval T.

16 Detection of PAM Signals Consider a discrete-time baseband equivalent AWGN channel model: y = s+η, (1) where η is the Gaussian random variable with zero mean and variance σ, and s is independently and equally likely chosen from the M-ary PAM constellation. 1. Decision rule for the ML detector: ŝ = arg s A min y s () Notice that the M-ary PAM constellation can be represented by A = { ( ) k M 1 d} M 1 k=0. Let s = ( ) k M 1 d. Then, the optimization

17 problem () is equivalent to the following optimization problem: ˆk = arg 0 k M 1 min y ( k M 1 ) d (M 1)d = arg 0 k M 1 min y + kd = arg 0 k M 1 min 0, M 1, = y d + (M 1) y d < (M 1) y d > (M 1) k +0.5, 0.5, y d + (M 1), y d + (M 1) y d + (M 1) < y d + (M 1) +0.5, y d + (M 1) +1, y d + (M 1) +0.5 y d + (M 1) < y d + (M 1) where notation x denote the maximum integer not exceeding x. Therefore, the optimal estimate is given by ŝ = (ˆk ) M 1 d. +1,

18 . Decision regions: The correct decision regions for detecting s k = ( ) k M 1 d using the ML receiver are given as follows: Γ k = {y : ( k M 1 Γ 0 = {y : y (M 1)d + d } for the left edge point Γ M 1 = {y : y (M 1)d d ) d d < y < ( k M 1 } for the right edge point ) d + d } for the kth inner point 3. Average symbol error probability: Note that the conditional probability density function of the received signal y given s = s k is the Gaussian distribution with mean s k and variance σ, i.e., f(y s k ) = 1 k ) πσ e (y s σ (3) Hence, the conditional probability of making correction decision, P c sk is

19 determined by P c sk = Γ k f(y s k )dy (4) In order to evaluate this integral, we consider the following three cases: (a) For the left edge point, the conditional probability of making correct decision on s = s 0 = (M 1)d is given by P c s0 = Γ 0 f(y s 0 )dy = (M 1)d + d ( 1 πσ e ) y+ (M 1)d σ dy = 1 Q ( d ) σ Thus, the conditional probability of making wrong decision on s = s 0 = (M 1)d, P e s0 is P e s0 = 1 P c s0 = Q ( d σ). (b) Similar to Case (a), for the right edge point, the conditional probability

20 of making correct decision on s = s M 1 = (M 1)d is given by P c sm 1 = (M 1)d d ( 1 πσ e ) y (M 1)d σ dy = 1 Q ( d ) σ Hence, P e sm 1 = 1 p c sm 1 = Q ( d σ). (c) For the kth inner point, the conditional probability of making correct decision on s = s k = k (M 1)d is given by P c sk = (M 1)d k + d k (M 1)d d = 1 1 π d ( 1 πσ e y+ (M 1)d e t dt 1 π ) k σ dy = 1 d π e t d d e t dt dt = 1 Q ( d ). σ Thus, the conditional probability of making wrong decision on s =

21 s k = k (M 1)d, P e sk is P e sk = 1 P c sk = Q ( d σ). Therefore, over all average symbol error probability, P e is defined as P e = M 1 k=0 π(k)p e sk, (5) where π(k) denotes the prior probability for transmitting s = s k. For equally likely transmission, π(k) = 1 M and thus, P e = M 1 k=0 π(k)p e sk = (M 1) M Q( d ). (6) σ Since the average energy for the M-ary PAM constellation per symbol is E s = (M 1)d 1, if we define signal to noise ratio (SNR) per symbol to be

22 1E s M 1 SNR s = E s, then, d = and as a result, the average symbol error σ probability P e given by equation (6) can be represented in terms of the SNR s by SNR s b P e = (M 1) M Q( d ) (M 1) ( = σ M Q 6SNR ) s. M 1 Similarly, if we define signal to noise ratio (SNR) per bit to be SNR b = = SNR s log M = E b, where E σ b = E s b, then, the average symbol error probability P e given by equation (6) can be represented in terms of the SNR b by P e = (M 1) M Q 6 log M SNR b. M 1

23 Detection of QAM Signals Consider a discrete-time baseband equivalent AWGN channel model: y = s+η, (7) where η is the circularly symmetrical complex Gaussian random variable with zero mean and variance σ, and s is independently and equally likely chosen from the square M-ary QAM constellation S. 1. Decision rule for the ML detector: ŝ = arg s S min y s (8) Let y = y re + jy im and s = s re + js im. Since y s = (yre s re ) +(y im s im ) and s re and s im are independent, the

24 optimization problem () is equivalent to the following two optimization problems: ŝ re = arg sre Amin y re s re (9) ŝ im = arg sim Amin y im s im, (10) where A denotes the M-ary PAM constellation. If we let s re = ( m M 1 ) d and sim = ( n ) M 1 d, where m,n = 0,1,,, M 1, then, taking advantage of the result from equation (11), we have ˆm = 0, M 1, d + ( M 1) M 1) y re y re d + (, y re yre d < M 1) ( yre d > ( M 1) d + ( M 1) M 1) + 1, y re d + ( + 0.5, 0.5, y re d + ( M 1) M 1) y re d + ( < y re d + ( M 1) < y re d + ( + 0.5, M 1) + 1,

25 ˆn = EE 4TM4 and 0, M 1, y imd < ( M 1)d + 0.5, y imd > ( M 1)d M 1) y im d + ( M 1), y im d + ( y im d + ( M 1) + 1, y im d + ( 0.5, M 1) M 1) y im d + ( < y im d + ( + 0.5, y im d + ( M 1) < y im d + ( M 1) + 1. M 1) Therefore, the optimal estimate of s is given by ŝ = ŝ re +jŝ im, where ŝ re = (ˆm M 1 ) d, ŝ im = (ˆn M 1 ) d.. Decision regions: The correct decision region for detecting s m,n = ( m M 1 ) ( d+j n M 1 ) d using the ML receiver is given as follows:

26 (a) For four corner points: Γ 0,0 = Γ 0 Γ 0 Γ M 1,0 = Γ M 1 Γ 0 Γ 0, M 1 = Γ 0 Γ M 1 Γ M 1, M 1 = Γ M 1 Γ M 1 where notation Γ k denotes the kth decision region for the kth PAM constellation point, which defined in (3), and notation X Y denotes a set X Y = {(x,y) : x X,y Y}

27 (b) For 4( M ) edge points: for 1 k M. (c) For ( M ) inner points: Γ 0,k = Γ 0 Γ k Γ k,0 = Γ k Γ 0 Γ M 1,k = Γ M 1 Γ k Γ k, M 1 = Γ k Γ M 1 for 1 m,n M. Γ m,n = Γ m Γ n 3. Average symbol error probability: Note that the conditional

28 probability density function of the received signal y given s = s m,n is the joint Gaussian distribution, f(y s m,n ) = 1 (yre sm) +(y im sn) πσ e σ (11) wheres m ands n denotetherealandimaginarypartsofs m,n, respectively. Hence, the conditional probability of making correction decision on the QAM symbol s m,n, P c sm,n is determined by P c sm,n = Γ m,n f(y s m,n )dy re dy im (1) The following three points are the key to evaluate the integral (1): Making correction decision on the QAM symbol s m,n is equivalent to

29 making correction decisions on both the real part PAM symbol s re,m and the imaginary part PAM symbol s im,n. For a given s = s m,n real part and the imaginary part of the complex received signal are independent, i.e., the conditional probability density function (11) can be rewritten as f(y s m,n ) = 1 re sre,m) πσ e (y σ 1 im s im,n ) πσ e (y σ = f(y re s re,m ) f(y im s im,n ) (13) The decision region Γ m,n is separable, i.e., Γ m,n = Γ m Γ n. Therefore, we have P c sm,n = P c sre,m P c sim,n. Now, we can make use of the result for the PAM constellation by considering the following three cases:

30 (a) For four corner points, P c sm,n = P c sre,m P c sim,n = ( 1 Q ( d ) ) σ (b) For 4( M ) edge points, P c sm,n = P c sre,m P c sim,n = ( 1 Q ( d ) ) (1 Q ( d ) ) σ σ (c) For ( M ) inner points, P c sm,n = P c sre,m P c sim,n = ( 1 Q ( d ) ) σ Therefore, over all average symbol probability of a correct decision, P c is

31 determined by P c = M 1 m=0 ( 4 = 1 M ( 1 Q ( d M 1 n=0 ( = ) Q ( d M σ π(s m,n )P e sm,n ( 1 Q ( d ) ) +4( M ) (1 Q ( d ) ) σ σ ) ) +( M ) (1 Q ( d ) ) ) σ σ ) +4 (1 1 ) Q ( d ) M σ Hence, the average symbol error probability is given by ( P e = 1 P c = ) Q ( d ) 4 (1 1 ) Q ( d ) M σ M σ (14)

32 Since the average energyfor the M-ary QAM constellation per symbol is E s = (M 1)d 6E 6, then, d = s M 1 and as a result, the average symbol error probability P e given by equation (14) can be represented in terms of the SNR s by P e = 4 (1 1 ( )Q 3SNR ) s M M 1 4 (1 1 M ) Q ( 3SNR s M 1 ). Similarly, if we define signal to noise ratio (SNR) per bit to be SNR b = = SNR s log M = E b, where E σ b = E s b, then, the average symbol error SNR s b probability P e given by equation (14) can be represented in terms of the SNR b by P e = 4 (1 1 )Q M 3 log M SNR b M 1 4 (1 1 ) Q M 3 log M SNR b M 1.

33 Union Bound Consider a discrete-time AWGN channel model: y = s+η, where η is the white Gaussian random vector with zero mean and variance per each real dimension being σ, and s is independently and equally likely chosen from a multi-dimensional constellation A = {a n } N n=1. Let Γ n is the correct decision region for a n with the ML detector. Then, the error probability P e is given by P e = N π(n)p e s=a n = N π(n)p ( y N Γ m s = a n ) n=1 n=1 m=1,m =n = N N π(n)p ( ) y Γ m s = a n. n=1 m=1,m =n

34 Since P ( y Γ m s = a n ) Q ( dm,n σ ), where dm,n = a m a n, we have P e N N n=1m=1,m =n π(n)q ( d m,n ). σ This upper bound is called the union bound for the ML detector. Example 1. Let A = { ( 1 0 ) ( 03 ) (,, 0 3 ) ( 1, 0 ) }. Then, the union bound of the ML receiver for this constellation is given by 5 ( 1 ) Q σ + 1 ( Q 3 ). σ

35 Nearest Neighbors Approximation ( dmin (A) σ ), the terms related to When the signal noise ratio is high, Q the minimum distance in the union bound are performance-dominant terms. For any a i A, Let us define the nearest neighbors set to be N(a i ) = {a : a Aand a a i = d min (A)} Let N dmin (i) denote the number of the nearest neighbors of a i. Sometimes, N dmin (i) is also called the kissing number of a i. The average nearest neighbors number, denoted by N dmin, is defined as N dmin = N π(i)n dmin (i). i=1

36 Therefore, the nearest neighbors approximation for the ML receiver is described by ( dmin (A) ) P e N dmin Q. σ Example. Consider the constellation defined in Example 1. Notice that d min (A) =. N(a 1 ) = {a,a 3,a 4 }, N(a ) = {a 1,a 3 } N(a 3 ) = {a 1,a,a 4 }, N(a 4 ) = {a 1,a 3 }. Hence, N dmin (1) = N dmin (3) = 3, N dmin () = N dmin (4) = and N dmin = =.5. For this constellation, the ( nearest ) neighbors approximation for the ML receiver is given by P e.5q. 1 σ

37 Typical Examples Example 3. Find the mean, average anergy and minimum distance of the following constellations if each constellation point is equally likely: 1. Orthogonal constellation A = { Ee i } M i=1, where each e i is an M 1 column vector with i-th component being 1 and the other components being all zeros.. Biorthogonal constellation A = { Ee i, Ee i } M/ i=1, where each e i is an (M/) 1 row vector with i-th component being 1 and the other components being all zeros. 3. SimplexconstellationA = { Es i } M i=1, whereeachs i isan M 1column vector with i-th component being M 1 M and the other components being all 1 M.

38 Solutions: 1. The mean of A is given by m(a) = 1 M M i=1 Eei = E M (1,1,,1)T The average anergy of A is given by E(A) = 1 M M Ee i = E i=1 Since the distance between any two distinct constellation points is Ee i Ee k = E for i k, d min (A) = E.

39 . The mean of A is given by m(a) = 1 M ( M/ i=1 Eei + M/ ( ) Ee i ) i=1 = 0 The average anergy of A is E(A) = 1 M ( M/ i=1 Ee i + M/ i=1 Ee i ) ) = E Similarly, the distance between any two distinct constellation points is either E or E and thus d min (A) = E.

40 3. The mean of A is given by m(a) = 1 M M Esi i=1 = E( M 1 M (M 1) 1 1,,M M M (M 1) 1 M )T = 0 The average anergy of A is E(A) = 1 M = M Es i = E M i=1 (M 1)E M M i=1 ( ( M 1 M ) +(M 1) ( 1 M )) Since the distance between any two distinct constellation points is

41 Es i Es k = d min (A) = E. E ( M 1 M ( 1 M) ) = E for i k,