Approximate bit error probability analysis of 2PSK and 4PSK with partially coherent, symbol-by-symbol detection in carrier phase noise

PYKarri and YK.Some

Abstract: The effect of carrier phase noise on the BEP of 2PSK and 4PSK with partially coherent, symbol-by-symbol detection is analysed using an approximate approach. Partially coherent detection is achieved using a decision-aided, maximum likelihood carrier phase estimator (Kam, 1986). The carrier phase noise is modelled as a Brownian motion. Assuming a small phase noise variance, it is shown that the Brownian phase noise, together with the additive, white, Gaussian channel noise, leads to a Gaussian phase error in the recovered reference and a random signal component for data detection. These results enable the BEP to be numerically evaluated, and the numerical results agree well with the results of computer simulations. Both the effects of carrier phase fluctuations on the matched filter output and the decorrelation of the carrier phase from one symbol interval to the next are taken into account.

1 Introduction

Coherent detection of M-ary phase shift-keying (MPSK) can be acheved with a number of digitally-implemented receiver structures, such as those in [l, 21. The bit error probabihties (BEPs) of these receivers have been analysed in [l, 21 for the case where the carrier phase can be assumed constant over a duration of many symbol intervals. Their performance in the presence of carrier phase noise has pre- viously been assessed only via computer simulations. This paper provides an approximate analysis of the BEPs of such receivers in the presence of a slowly fluctuating, Brownian motion carrier phase with small phase noise var- iance. Attention is focused on the symbol-by-symbol receiver of [l], though a similar analysis applies to the sequence estimator of [2]. The results are applicable in het- erodyne optical communications, where a Brownian motion phase noise model is common [MI. It is shown that phase noise, together with additive, white, Gaussian noise (AWGN) from the channel, leads to a Gaussian phase reference error and a random signal component for data detection for the receiver of [I]. The result allows the BEP to be evaluated using the approaches in [7, 81 for 2PSK and 4PSK, respectively. The numerical results obtained agree well with simulations, thus validating the analysis.

2 Problem formulation

The receiver is shown in Fig. 1. The complex envelope r”(t) of the MPSK signal received in the presence of AWGN

OEE, 1Wl IEE Proceedings online no. 19990109 DOL lO.lO49/ipcom: 19990109 Paper fmt mived 3rd October 1997 and in revised form 14th September 1998 The authors are with the Department of Electrical Engjneering, National Uni- versity of Singapore, 10 Kent Ridge Crescent, Singapo~ 119260, Republic of Singapore

and an unknown, randomly time-varying carrier phase is

Here, E, is the energy per symbol and T is the symbol duration. The data modulated phase a(t) assumes the con- stant value &) for t in the kth symbol interval [kT, (k + 1)T). The phase d k ) assumes, with equal probability, the values in the set {2ni/M)i$-1. The carrier phase $(t) is a Brownian motion and, thus, $(t) is a Gaussian random process with $(O) = 0 (with probability one), E[@(t)] = 0 for all t, and E[$(+$o)] = 002 min(t, z), where q2 is the vari- ance parameter. Assume q 2 T << 1 so that $(t) fluctuates slowly over a symbol interval. The initial phase & is uni- formly distributed over [-x, +n). Finally, Z(t) is the complex envelope of AWGN with E[li(t)] = 0 and E[Z(t)n”’(t - z)] = Nos(z) where the - denotes a complex quantity, and * its conjugate. The data-modulated phase a(t), carrier phase

t=(k+l)T matched filter

I I optimum

symbol-by-symbol detector defined by

eqn. 4

foLm coherent reference x(k) given by eqn. 3

Fig. 1 Matched fdter equations: i ( i ) = IMT, 0 L r < T g(t) = 0, elsewhere

Smccture of optimum sy&l-by;Syn?bl receiver

120 IEE Proc.-Commun., Vol. 146, No. 2, April 1999

Ht), initial phase & and AWGN E(t) are all mutually inde- pendent. Since the transmitted signal pulse is rectangular, ?(t) is matched-filtered with an impulse response g( t ) = UdT, for 0 s t < T and zero elsewhere. The filter output for the kth interval is sampled at t = (k + 1)T, giving the sam- ple Zo(k), where

( k f 1 ) T exp[Mt>l dt

ZO(k) = G e x P [j(cp(k) + 4011 1 kT

+ CO(k) ( 2 )

Here, {to@) = fi(t)/dT dt} is a stationary sequence of independent, complex Gaussian random variables, each with mean zero and variance E[lto(k)12] = No. The matched fdter here is not the optimum receive filter, because the transmitted pulse is modulated by the random multiplica- tive distortion exp[i@(t)]. The optimum receive filter for ths unknown, randomly modulated pulse has not been derived. The matched fdter is used here because it approximates the optimum receive filter in ths case of slow phase fluctua- tions. The optimum symbol-by-symbol receiver of [l, 91 performs coherent detection of data d k ) using a reference Z(k), where

is formed from the immediate past L received signals with decision feedback. Here, @(l) is the decision made earlier by the receiver on the data phase dl), I < k. The initial L sym- bols dl), 0 5 Z 5 L - 1, form a known preamble. Ideal deci- sion feedback, i.e. @(l) = dl) for all I, is assumed throughout. The effect of prior decision errors is only to reduce the signal-to-noise ratio (SNR) of the reference Z(k) [ 11. The receiver computes the decision statistics

q i (k ) = Re [Zo(IC)Z*(k) exp [-j27ri/M]] 2 = 0,1, . . . , M - 1

and declares that @(k) = 2 m / M if qm(k) = maxi ql{k). To analyse the BEP of the receiver in eqn. 4, one needs a sta- tistical characterisation of the signal sample So@) and the reference Z(k).

3

We begin by expressing each term Zl(l) = Zo(l) exp[-j@(l)] in Z(k) as

(4)

Characterisation of the reference f (k)

z1 (1 ) = G e x p [ j (e(z) + 4011 (1+1)T

x 1 ; exp[j($(t) - O(l))ldt + C l ( 0 1T

(5) where e(l) = gt = I T ) , tl(l) = to(l) exp[j@(l)], and the data phase cp(l) in Z0(l) has been cancelled by the decision @(I) . The sequence { 0 1(1)} is statistically identical to the sequence {to(l)). The term dEJ exp[i(e(l) + &)I is the matched fdter output for the symbol interval [ZT, ( I + 1)T) if AWGN is absent, and if the carrier phase stays constant at the value ql). We fKst consider the multiplicative distortion due to the integral term in eqn. 5, and define

h(l) = hl ( l ) + j h Q ( l )

1 2 [ 1 ;exP[.Mt) -W)) ld t ( 6 )

(1+1)T

IEE Proc.-Commun., Vol. 146, No. 2, April 1999

The statistics of g( l ) are not easy to determine [lo]. How- ever, for small phase noise o2T << l so that the square and higher powers of a2T can be dropped, the Appendix (Section 7.1) shows first that hxl) = Re[h(l)] has mean E[hAl)] = (1 - 02T/4) and variance zero. Thus, with proba- bility one, we have A i l ) = (1 - a$T/4). Second, hQ(l) = Im[h(Z)] is Gaussian with mean zero and variance oiT/3. Phase fluctuations thus reduce the matched fdter output dEs exp[i(0(l) + &)I by the factor hxl) = (1 - oiT/4), and introduce Gaussian noise hQ(l) in an orthogonal direction.

From the expression for Zl(k) = Zo(k) exp[-j@(k)] (see eqn. 5 with I = k), the desired direction for the reference Z(k) is that of the phasor exp[i(e(k) + &)I, since the func- tion of Z(k) is to enable the receiver in eqn. 4 to cancel expb(f3(k) + &)I in So(k), before deciding on the data exp[idk)] in the latter. Thus, for each Zl(l) in Z(k), we resolve the phasor exp[i(e(l) + &)I into the sum of a com- ponent in the desired direction exp[i(e(k) + A)] and a com- ponent in an orthogonal direction. Noting that { q k ) = $(t = kT)} forms a random walk given by

O(k + 1) = O(k) + w ( k ) (7) where {w(k) = $((k + 1)T) - HkT)} is a set of independent, identically distributed, Gaussian increments, each with mean zero and variance o , T , we can write, for all 1 < k,

exp [ j ( O ( l ) + 4011 k-1

= exp [ j (Q(k) + $o)l exp - j , 4 2 1 [ %=I 1

[ i=l 1 (8)

Hereafter, we condition on e(k) and & being known, and define

IC-1 n

d ( l , k ) = c i l ( ~ , ~ c ) + j c i Q ( ~ , k ) = exp - j c w ( i )

1 < k (9) Assuming low phase noise oo2T << 1 and ignoring the second and higher powers of oJT, - the Appendix (Section 7.2) shows first that dA1, k) = Re[d(l, k)] has mean 1 - f(k - l)oaT and variance zero, and hence dxl, k) = 1 - i(k - l)oaT with probability one. Second, dQ(l, k ) = Im[d(Z, k)] is Gaussian, with mean zero and variance (k - l)aaT. Thus, from eqn. 8, exp[i(e(l) + &)I comprises a determinis- tic component dxl, k ) in the direction exp[i(e(k) + &)I and a Gaussian noise component dQ(l, k ) orthogonal to exp[-

Returning to Zl(l) in eqn. 5, we see from eqns. 6, 8 and + &)I.

9 that it can now be expressed as

W ) = JE,h(+-@, k ) exp [ j ( e@) + 40)1+ ~ ( 1 ) (10)

It remains to characterise the product term

b ( l , k ) = bl (1 , k ) + jbQ(1 , k ) 4 k ( l )d ( l , k ) (11) Using results for g( l ) and &Z, k ) above and assuming small phase noise oaT << 1 so that squares and higher powers of oJT are drppped, the Appendix (Section 7.3) shows that bXI, k ) = Re[b(l, k)] has mean 1 - i(k - I - f)oZT and vari- ance zero, i.e. bxl, k ) = 1-- f(k - 1 - i ) o , T with probability one, while bQ(l, k) = Im[b(l, k)] is Gaussian with mean zero and variance (k - I - +)oZT. The effects of carrier phase fluctuations on the matched fdter output and the effects of carrier phase decorrelation from one symbol interval to the next have been taken into account in b(l, k). The term dEybxl, k) exp[i(B(k)+ h)] is the component of Zl(l) parallel

121

to exp[i(qk) + &)I, whde dEs (ibQ(Z, k)) exp[i(6(k) + &)I is orthogonal to exp[i(8(k) + &)I, see Fig. 2. The former is known given 6(k) and &,, while the latter is the unknown 'noisy' component of F1(!).

? , ( I ) , showing how it contributes to formmg the

b(1, k) = h(l)d(l k) = b#, k) + jb& k) h(i) amunts for effect of carrier phase fluctuations on matched-fdter output &I, k) amounts for effects of camer phase decorrelation from one symbol interval

Having characterised Z1(Z), we can now examine the ref- erence Z(k) given in eqn. 3. Using the expression in eqn. 10 for Zl(Z) = Z0(Z) exp[j@(Z)] in eqn. 3, we have

Z ( k ) = ( L G ) - l [ [ E mz4 1

l = k - L

x exp [ j (W) + $ O N + G ( 0 l=k -L

(12) Factoring out exp[i(6(k) + &)I, and noting that the sequence { i j2(Z) = v2xZ) +jv2Q(l)}, where i j2(Z) = i j l ( l ) exp[- j(s(k) + &)I, is statistically identical to {ijl(Z)}, we can express .i?(k) as

(13) Here,

Z(k) = [ z l ( k ) + jzQ(k)I exP [ j ( e ( k ) + $ O ) ]

(14a) and

1 (14b)

Now, assume further that L q 2 T << 1 so that the second and higher powers of L q 2 T can be dropped. The Appen- dix (Section 7.4) then shows that xik) is Gaussian with mean 1 - Lao2T/4 and variance Nd2LE,, and xdk) is Gaussian with mean zero and variance A2, where

L-1

X 2 = 2 g i T x (g) ('-;)+%+LE, No (15) 2=1

These results show that for small phase noise and high SNR, the combined effect of phase noise and AWGN is to reduce the component of W(k) in the desired direction exp[j(@k) +&)I, and to introduce Gaussian noise parallel to and orthogonal to this direction. Rewriting eqn. 13 as

where IZ(k)l = d[x;(k) + xQ2(k)] and Z ( k ) = Iw4 exp [ j ( E @ ) + O(k) + 40)l (16)

122

~ ( k ) = arctan ( z Q ( k ) / z l ( k ) ) (17) we see that R(k) leads to a phase reference error ~ ( k ) . To find the distribution of E@), assume LEJiVo is high so that the variance of xxk) is negllgible compared to its mean. Then, xxk) = 1 - LoaTI4, with probability one. Also, from eqn. 15, we will have A2 << 1, and hence IxQ(k)l << 1 for most k. Thus, Ixe(k)l << Ixkk)l for most k, and the phase error ~ ( k ) in eqn. 17 can be approximated, using arctan(x) - x for 1x1 << 1, as

From eqn. 18, the phase error ~ ( k ) is Gaussian, with mean zero and variance d2(1 - Lq,2T/4)-2.

4 BEPresutts

& ( I c ) zQ(k)/zI(k) (18)

We return to the receiver in eqn. 4, and recall that ZO(k) from eqn. 2 can be expressed as

ZO@) = JE,f@) exp[jv(k)l .XP[j(W + $bo)] + Go(k) (19)

The complex term i ( k ) leads to a deviation of the signal component of io&) from the desired direction exp[i(6(k) + 444. This can be seen by rewriting it from eqn. 6 as

W) = Iw4 e x P [ m ) l

= , /hf(k) + h;(k) exp [ j arctan (hQ(k) /hI (k ) ) ]

(20)

Zo(k) = GIw4 exp[jv(WI

and then using eqn. 20 to rewrite eqn. 19 as

x exp[j(rl(k) + q k ) + $011 + C O ( k )

(21) Using results from the Appendix (Section 7.1) for i ( k ) and the assumption q 2 T << 1 , we have Ir(k)l << 1 for most k, and thus q(k) can be approximated as

Eqn. 22 shows that q(k) is approximately Gaussian, with mean zero and variance (1 - q2T/4)-2 (?jq2T, - ( 4 0 ~ ~ 7 ) . Now, using eqn. 21 for f0(k) and eqn. 16 for Z(k) in eqn. 4, we have

r l ( k ) h Q ( k ) / h l ( k ) (22)

Zo(k)Z*(k) = (JE,lL(k)l exp[jv(k)l

x exp[j(rl(k) - 4k))1+ W)) IZ(k)l (23)

where F3(k) = g,$k) exp[-j(E(k) + 6(k) + &)I, and ij3(k) is statistically identical to ijo(k). In eqn. 23, the common fac- tor lZ(k)l can be ignored. The result shows that the receiver in eqn. 4 detects the data cp(k) with a reduced signal com- ponent dEs li(k)l and a phase reference error

The reduced signa! component dEs lh(k)l can be approxi- mated by writing Ih(k)l as

t ( k ) = rlw - 4 k ) (24)

M h I ( k ) [1 + ( h 3 5 ) / 2 h ? ( k ) ) ] (25) arguing as in eqn. 22. Thus, lh(k)l has mean hxk) + (E[hQ2(k)v2hxk)) - 1 - q2T/12 and variance zeLo (only squares and higher powers of q 2 T involved), and Ih(k)l- 1 - q2T/12 with probability one. The BEP Pb(e) can now be

IEE Proc-Commun.. Vol. 146, No. 2, April 1999

obtained by averaging the BEP conditioned on a value a of the phase error Hk), i.e. Pb(e(&k) = a), with respect to Hk):

n

pb(e) = / pb (e l<(k) = P<(k) (a)da (26 ) -7r

From eqn. 24, Rk) is Gaussian with mean zero. In eqn. 24, ~ ( k ) and ~ ( k ) are independent, because they are defined in terms of the carrier phase process $(t) and the AWGN process E(t) over disjoint time intervals. Thus, the variance of 5(k) is the sum of the variances of q(k) and E@).

R2 - 11

R3 Fig.3 regwns

4PSK sigml set with Gray codmg of bits und optimum deckion

The conditional BEP Pb(el@) = a) is given, for 2PSK, by [71:

pb (e lE(k) = a ) =

(27) For 4PSK with Gray code bit mapping, we follow the approach in [8]. Defining the BEP as the expected number of bit errors made per bit detected [8, 111, and referring to Fig. 3, we find

pb(elE(k) = a ) = pb(elE(k) = a , 4 s ) = 0) = [(Ai + A2) + (A2 + As)] / 2

( 2 8 )

( 2 9 )

where Ai = Prob(Zo(k)%*(k) E Ril[(k) = u,cp(k) = 0)

and Ri is the optimum decision region for the symbol id2, i = 0, 1, 2, 3. The probabilities (A , + A2) and (A2 + A3) can be computed easily since RI U R2 and R2 U R3 are half- planes. We compute these probabilities for 0 = a s n; for -x s a < 0, the result is the same as that for -a, by symmetry. For 0 5 a 5 d4, we get

and

Similarly, for d4 < a s 3d4, we have

and

Finally, for 3d4 < a 5 x, we have

and

(32b) Using eqn. 27 in eqn. 26 for 2PSK, and using eqns.

3CL32 in eqn. 28, followed by eqn. 28 in eqn. 26 for 4PSK, we evaluate the BEP using numerical integration. The results are shown in Fig. 4 for 2PSK with o:T = 0.02 rad2 and L = 7, and in Fig. 5 for 4PSK with o2T = 0.002 rad2 and L = 11. In a heterodyne optical communication sys- tem, we have o2T = 24Av)T, where AV is the total line- width of the transmitter and receiver semiconductor laser oscillators. Thus, a value of oo2T = 0.02 rad2 would corre- spond to a system with a typical total laser hewidth of AV = 20MHz and a bit rate of T' = 2x Gbiffs. Computer sim- ulations with ideal decision feedback have been performed and the results agree reasonably well with numerical results from eqn. 26. The software simulates the operation of the

-7 l o ! ' 9 ' ' 10 ' ' 11 ' ' 12

SNR per bit (Eb/No), dB Fig.4 BEPofZPSK -+- analysis and numerical ink ation, q 2 T = 0.02 rad2, L = 7 4- simulation, c$T = 0.02 rad? L = 7 -A- coherent (theoretical) -X- partially coherent, no phase noise (theoretical), q 2 T = 0, L = 7

IEE Proc-Commun., Vol. 146, No. 2, April 1999 123

receiver eqn. 4, with Zo(k) and n(k) generated according to eqns. 2 and 3, respectively. The Brownian motion $(t) in eqn. 2 is generated in simulations by integrating white Gaussian noise. The discrepancy in Fig. 4 between analysis and simulations arises because the value Lq2T = 0.14 does not quite satisfy the assumption LoJT << 1. In Fig. 5, the value Lo?T = 0.022 satisfies the latter assumption much better, and analysis agrees very well with simulations at high SNR. Thus, within limits of simulation accuracy, the analysis here of the effects of phase noise on the BEP of PSK, for the symbol-by-symbol detector (eqn. 4), is vali- dated under the assumptions of high SNR and Lo2T << 1. The BEPs of the receiver (eqn. 4) with no phase noise (o2T I= 0), computed using results in [l], are also presented in Figs. 4 and 5 to show the degradation in the receiver’s performance due to phase noise.

IC

10

4 10

n m W

-5 10

-6 10

-7 10

8 9 10 11 12 SNR per bit (Eb/No), dB

Fig. 5 BEP of 4PSK -e- analysis and numerical integration via eqn. 26, q2T = 0.002 rad2, L = 1 I a- simulation, ah2T = 0.002 rad2, L = 11 -A- coherent (theoretical) -X- partially coherent, no phase noise (theoretical), %*T = 0, L = I 1

5 Conclusions

We have analysed the effects of phase noise on the BEP of the optimum symbol-by-symbol MPSK receiver of [l, 91. Phase noise is modelled as a Brownian motion with small variance (Lg’T << 1). We show that phase fluctuations w i t h a symbol interval lead to a reduction in the mean of the complex matched-fiter output, and introduce Gaussian noise orthogonal to the latter. This leads to a random sig- nal component for data detection. Next, the decorrelation of the carrier phase from one symbol interval to the next is shown to lead to two component contributions from each matched-fiter output sample to the recovered reference - a mean component in the direction of the reference whose magnitude decreases with time separation, and a Gaussian noise component orthogonal to the desired direction whose variance increases with time separation. The result is that carrier phase fluctuations, together with channel AWGN, lead to a phase reference for coherent detection with a Gaussian phase error. This enables one to evaluate the BEP of MPSK, and we have considered 2PSK and 4PSK.

124

The numerical BEP results agree well with simulations, thus justifying the analytical approximations made.

6 References

1 KAM, P.Y.: ‘Maximum likelihood carrier phase recovery for linear suppressedcmier digital data modulations’. IEEE Trans. Commun.,

KAM, P.Y., and SINHA, P.: ‘A Vitcrbi-type algorithm for efficient estimation of MPSK sequences over the Gaussian channel with unknown carrier phase’, IEEE Trans. Commun., 1995, COM-43, pp. 2429-2433

3 MACCHI, O., and SCHARF, L.L.: ‘A dynamic programming algo- rithm for phase estimation and data decoding on random phase chan- nels’, IEEE Trans. In$ Theory, 1981, IT-27, pp. 581-595 LIEBETREU, J.M.: ‘Joint camer phase estimation and data detection algorithms for multi-h CPM data transmission’, IEEE Trans. Cum- mun., 1986, COM-34, pp. 87G381

5 KIKUCHI, K., OKOSHI, T., NAGAMATSU, M., and HENMI, N.: ‘Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator’, Lightwave Technol., 1984, LT-2, pp. 10241033

6 SALZ, J.: ‘Coherent lightwave communications’, A T & T Tech. J. , 1985,64, (IO), pp. 215S2209

7 KAM, P.Y., TEO, S.K., SOME, Y.K., and TJHLJNG, T.T.: ‘Approximate results for the bit error probability of binary PSK with noisy phase reference’, IEEE Trans. Commun., 1993, COM-41, pp. 102&1022 SOME, Y.K., and KAM, P.Y.: ‘Bit error probability of QPSK with noisy phase reference’, IEE Proc., Commun., 1995, 142, (51, pp. 292- 296 KAM, P.Y., NG, S.S., and NG, T.S.: ‘Optimum symbol-by-symbol detection of uncoded digital data over the Gaussian channel with unknown carrier phase’, IEEE Trans. Commun., 1994, COM-42, pp. 25432552

IO FOSCHINI, G.J., and VANNUCCI, G.: ‘Characterizing filtered light waves corrupted by phase noise’, IEEE Trans. InJ Theory, 1988, ll- 34, pp. 1437-1448

11 VITERBI, A.J., and OMURA, J.K.: ‘Principles of digital communica- tion and coding’ (McGraw-Hill, New York, 1979), pp. 100, 243

1986, COM-34, pp. 522-527 2

4

8

9

7 Appendix

7.1 Derivations for h (I) in eqn. 6 Since oJT << 1, we have I ~ ( t ) - 6(1)1 << 1 rad for all t E [IT, (1 + 1)Q, with high probability. Expanding the expo- nential in eqn. 6 and retaining only up to the second power, we get

(33 ) When the means and variances of h,(I) and h,(l) are taken, the third and higher powers of $(t) - @I) lead to the second and higher powers of o ~ T , which can be neglected. Eqn. 33 gives

1T

and

Expanding the square in the integrand of eqn. 34, taking the expectation, and using

one obtains E[hXZj] = (1 - o?T/4). The variance of h,(I) is in the square of o?T, i.e. it is zero. The mean of hQ(I) is zero from eqn. 35. Its variance E[(he(l))’] is obtained by squaring the right side of eqn. 35, writing the product of two integrals as a double integral, and using eqn. 36. The

IEE Proc-Commun., Vol. 146, No. 2, April I999

result is oo2T/3. Since g(t) is a Gaussian process, hQ(l) is a Gaussian random variable.

7.2 Derivations for d (Il k) in eqn. 9 The term ZIT' w(i) in the exponent of eqn. 9 is Gaussian with mean zero and variance (k - l)oZT, which is much less than one, since L o ~ T << 1 and (k - I ) s L. Thus, we have

i=l

Thls gives dxl, k ) = 1 - -&Z%' w ( z ] ) ~ and dQ(l, k ) = -2s' w(i), and their means and variances follow easily. The hgher order terms in the expansion (eqn. 37) lead to the square and higher powers of oZT when computing the means and variances of dxl, k ) and dQ(l, k), and they are therefore dropped.

7.3 Derivations for 6 (Il k) in eqn. 1 1 From eqns. 6,9 and 1 1 , we have

br(1, 5) = h r ( l ) d r ( l , k ) - h ~ ( l ) d ~ ( l , k ) and

(38)

bQ(1, k ) = hI(l)dQ(l, k ) + hQ(l)dI(l, k ) (39) Consider first bxl, k ) in eqn. 38. Using the results for hAI) and dil, k) in Sections 7.1 and 7.2, respectively, we have

hr(Z)dr(Z,k) = 1 - ( k - I + i) c$T/2,

with probability 1 (40)

ignoring higher powers in o2T. The second term, hQ(I)dQ(l, k), is ajandom variable. Its mean is obtained by using hQ(l) = Im[h(l)] from eqn. 6 and dQ(l, k) = Im[d(l, k)] from eqn. 37, bringmg the expression for dQ(l, k) inside the inte- gral for hQ(I), talung the expectation of the terms inside the integral using eqn. 36, and then evaluating the integrals to give

Its variance can be shown to involve the square of o2T, and will be taken to be zero. Thus

E [ h ~ ( l ) d ~ ( l , k ) ] = -$T/2 (41)

h ~ ( Z ) d ~ ( l , k ) = -a:T/2, with probability 1 (42)

and substituting eqns. 40 and 42 into eqn. 38 gives the desired result for bil, k).

For bQ(l, k ) in eqn. 39, note that dxl, k ) and hxI) are con- stants with probabhty 1, whde dQ(l, k) and hQ(I) are zero- mean, correlated Gaussian variables with correlation given in eqn. 41. Thus, bQ(l, k) is Gaussian, and its mean and variance can be easily obtained.

7.4 Characterising x,(k) and xo(k) Consider xxk) in eqn. 14u, where bxl, k ) = 1 - i(k - 1 - i)o$T with probability one, and (LdEs)-'vAZ) is Gaussian with mean zero and variance Nd2L2E,. Carrying out the summation in eqn. 14u, xxk) is then Gaussian with mean 1 - Lo( l74 and variance Nd2LEs.

For xe(k) in eqn. 14b, the term Eki-L ( L ~ E $ ) - ' v ~ ~ ( I ) is Gaussian with mean zero and variance Nd2LE,, while L-lZ$ji-L b&, k) is zero-mean Gaussian with variance computed as

L \ l=k--L ' 1

K-1 K-I

l1=K-L 12=K--L l 1 f l z

(43) Thus, xQ(k) is Gaussian with mean zero and variance A' equal to Nd2LEs, plus the variance of L-'E& bQ(l, k) given by eqn. 43. We now evaluate eqn. 43, and show that A2 is given by eqn. 15.

For the first term on the righthand side of eqn. 43, we have a(bQ(l, ~k))~] = (k - 1 - 3)o iT from Section 7.3, and performing the summation gives (Ll2 - 1/6)oiT/L. For the second term, we use eqn. 39 and evaluate the expectation of each product term. Thus, we have

bQ (11 7 k)bQ (12 2 k ) = (hI(ll)dQ(11, k ) + hQ(ll)dl(ll, k ) )

x (hI(l2)dQ(l2, k ) + hQ(l2)dl(h, k ) ) (44)

The expectation of the first product gives

E [hI(ll)dQ(ll, k)hI(lZ)dQ(l2, k ) ] = hI(ll)hI(l2)E [dQ(ll, k)dQ(l2, k ) ]

= hr(li)hr(lz)o,2T(k - 1 ) (45)

where 1 = max(ll, 12), and where we have used the fact that {w(k) = ~ ( k + l)n - gkn} is a set of independent, identi- cally distributed, Gaussian increments, each with mean zero and variance oZT. The expectation of the second product is

E[hl(~l)dQ(11,k)hQ(~2)d1(~2,~)]

= hI(I l )dI(b, k ) E [dQ(Il, k)hQ(12)] r

The last two steps of eqn. 46 follow from the fact that the increments of g(t) are independent, and the expectations in the last step are evaluated by frst bringing the expectation under the integral and using eqn. 36. The expectation of the third product can be similarly shown to be

125 IEE Proc.-Commun., Vol. 146, No. 2, April 1999

since hQ(l1) and hQ(l2) are integrals of ~ t ) over disjoint intervals and are thus independent. Substituting eqn. 4548 into eqn. 44, and eqn. 44 into the second term on the righthand side of eqn. 43, and then performing the indicated double summation gives us the result for the term

from which we finally get the expression for the variance A2 in eqn. 15.

126 IEE Proc.-Commun., Vol. 146, No. 2, April 1999