## Impact of diversity reception on fading channels with ... - CiteSeerX

Jun 20, 1996 - This work was supported in part by the Human Capital and Mobility .... Let x and bx di er in positions KL = fj1;:::;jLg (L S is their Hamming ...

Impact of diversity reception on fading channels with coded modulation. Part III: Co-channel interference  J. Ventura-Traveset1 1

G. Caire2

E. Biglieri2

G. Taricco2

European Space Agency/ESTEC, P.O. Box 299, 2200 AG Noordwijk (The Netherlands) 2

Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino (Italy)

June 20, 1996 Abstract In previous work we have studied the impact of diversity on coded digital communication systems operating over fading channels. In particular, we have shown that diversity may be thought of as a way of making the channel more similar to a Gaussian one. The present paper extends this analysis to fading channels a ected by co-channel interference (CCI). Several receiver models are examined, namely, with ideal coherent detection and perfect channel-state information (CSI), and with di erential and pilottone detection. We study the e ect of diversity on the irreducible error oor caused by CCI and fading, and the asymptotic behavior of the channel as the diversity order increases. Our results show that, when perfect CSI is available, diversity is able to turn asymptotically the channel into a CCI-free additive white Gaussian noise (AWGN) channel with the same signal-to-noise ratio (SNR). On the other hand, for di erential and pilot-tone detection, diversity achieves signi cant gains when the SNR is large enough. Calculation of the channel cut-o rate provides guidelines for the design of coded systems with CCI in fading environments. A wide range of examples, validated by computer simulation, illustrates our conclusions.  This work was supported in part by the Human Capital and Mobility Program of the Commission of the European Union.

1

1 Introduction In radio communication, the sensitivity to co-channel interference (CCI) has a direct impact on the design of the overall system. For example, the ability of a narrow-band mobile cellular communication system to cope with CCI a ects its capacity. In a satellite mobile system, the limits on CCI determine in which antenna beams a given frequency can be reused, and the necessary inter-beam rejection. A conspicuous amount of literature is available on the performance of uncoded modulation with CCI and fading (see, e.g., [9],[2] and references therein). In [4] an approximated method for calculating the performance of uncoded pilot symbol assisted modulation is proposed. E ect of diversity (with selection combining) is analyzed in [1] for uncoded PSK in fading and CCI. The cut-o rate as a means to compare coded systems with DPSK, fading and CCI has been proposed in [10]. An analysis of coded PSK with coherent and di erential detection, based on Cherno bound, is presented in [17]. Two companion papers [15, 16] study the e ect of space diversity (with maximum-ratio combining) on coded PSK over fading channels. In [15] we considered coherent detection with perfect CSI and showed that, as the diversity order increases, the channel is turned asymptotically into an AWGN channel with the same total signal-to-noise ratio (SNR). In [16] we considered di erential block detection, and showed that in this case diversity can improve performance for medium- to high-rate codes. Moreover, with fast fading, we showed that the error oor due to non-coherent detection decreases exponentially when the diversity order is increased. This paper is devoted to the e ect of diversity on a coded PSK system a ected by at fading and CCI. We consider coherent detection with perfect channel state information (CSI), di erential detection, and pilot-tone detection. In the latter scheme, CSI is provided by the noisy output of a pilot-tone extraction lter [5, 13]. In order to separate the intrinsic e ect of diversity from the sheer increase in SNR achieved by the combination of M signals, we assume that the average SNR per branch is 1=M of the total SNR. This can be realized in practice by decreasing the transmitted power of both wanted and interfering signals by a factor M . 2

We start by considering the general problem where both the wanted and interfering signals are a ected by correlated Rician at fading. We derive an expression for the pairwise error probability (PEP) of coded modulation which depends on some \fading kernels." Computation of the latter might not be easy, since it requires averaging over all the possible interfering sequences and, above all, since the fading autocorrelation and cross-correlation functions involved may not be known. Consequently, a special attention is paid to the special case of ideal interleaving and independent diversity, with the wanted signal a ected by Rician fading and the CCI by Rayleigh fading. This model, as suggested in [2], is typical of outdoor microcellular environments. We show how diversity, operating with maximum-ratio combining in conjunction with perfect CSI, can suppress both fading and CCI, thus turning (asymptotically as the diversity order grows to in nity) the original channel into an AWGN CCI-free channel with the same total SNR. For a nite diversity order, we prove that the error oor caused by CCI decreases exponentially with the \product diversity" Lmin M , where Lmin is the code diversity (i.e., the minimum Hamming distance of the coded modulation scheme) and M is the diversity order. For di erential and pilot-tone detection asymptotic convergence to an AWGN CCIfree channel cannot be achieved: nevertheless, the error oor is still exponentially reduced by diversity. The paper is organized as follows. In Section 2 we describe our system model. In Section 3 we derive expressions for calculating the PEP in the general case and in some simple, yet relevant, special cases. Section 4 deals with the error- oor analysis as SNR increases, and with the asymptotic behavior of the channel as the diversity order M increases. The cuto rate of the fading diversity channel with CCI is computed for coherent, di erential and pilot-tone detection. Section 5 provides some examples which illustrate the results of our analysis and support our conclusions, which are collected in Section 6.

2 System model In the transmission system under analysis (see Fig. 1) a binary information sequence enters a coded modulator which generates the encoded sequence x. We consider q-ary PSK signaling, 3

so that the components of x are xk 2 Xq = fej2i=q : i = 0; 1; : : : ; q ? 1g. The sequence x is passed through an interleaver described by a mapping I (a permutation of integers). Interleaving modi es the time correlation of the fading and CCI samples. In our notations, unless otherwise stated the time index k will always refer to sequences before the interleaver (or after the deinterleaver). With di erential PSK (DPSK), the interleaved sequence is di erentially encoded. The coded sequence is shaped by a lter with a Nyquist (ISI-free) baseband impulse response p p(t), where p(t) has energy 1. The transmitted signal is sent through M channels, each corresponding to one diversity branch. Each channel is a ected by independent AWGN (with normalized two-sided power spectral density 1=2), at Rician fading, and CCI of the same type as the wanted signal. Let yki and ik denote, respectively, the output and the CSI of the i-th diversity branch obtained after demodulation, matched ltering, sampling (with ideal timing), and deinterleaving. Then, the diversity channel can be modeled as a discrete channel with a single input xk and 2M outputs. These are the two M -vectors yk = (yk1; yk2; : : :; ykM )T and k = ( 1k ; 2k ; : : :; Mk )T , where yk is given by

yk = gk xk + hk k + nk ;

(1)

(nk , gk , and hk to be de ned soon) and k depends on the detection method. For coherent detection with perfect CSI, di erential and pilot-tone detection, k is given as 8 > > > > >
> > > > :

0

0

0

0

k0 = I ?1(I (k) ? 1)

(2)

The meaning of symbols is as follows:

 nk = (n1k ; : : :; nMk )T is a zero-mean complex Gaussian random vector representing the AWGN samples. We assume that these samples are normalized by var(nik ) = 1 i 2 2 E [jnk j ] = 1=2.

 gk = (gk1; : : :; gkM )T and hk = (h1k ; : : :; hMk )T are complex Gaussian random vectors representing the fading a ecting the wanted and interfering signals. The components of these vectors have independent real and imaginary parts, with E [jgki j2] = and E [jhik j2] = I. 4

 and I are the average SNR of the wanted and interfering signals, respectively, per

diversity branch. Let ? = E=N0 denote the total SNR of the system. Then, with normalized diversity, we have ?

= 1 +1 r M where r  0 is the power-split ratio (PSR) [13] and r=(1 + r) is the fraction of transmitted power devoted to the pilot tone. For coherent detection with perfect CSI and for di erential detection, no tone is actually transmitted, and hence r = 0.

 = = I is the signal-to-interference ratio (SIR).   k = (k1; k2; : : :; kM )T is a zero-mean complex Gaussian random vector which repre-

sents the noise sample of the pilot-tone extraction lter. We assume an ideal bandpass lter of bandwidth Bp = 2Bd (normalized with respect to the symbol rate), where Bd is the normalized Doppler bandwidth [5]. The complex random variables ki have independent real and imaginary parts and variance 21 2, with

2 = E [jki j2] = Brp : We assume that the pilot-tone extraction lter and the receiver's matched lter have non-overlapping bandwidths, so that nk and  k are independent.

 k is the complex CCI sample, which belongs to the \scattered signal set" S = fs = ` w` a` : a` 2 Xq g. Here, w` are the samples of the total channel impulse response w(t) = p(t)  p(?t)ej sampled with an arbitrary delay with respect to the optimal P

sampling time and rotated by a phase  due to the phase di erence between the carriers.

For di erential detection, we assume that both k and yk are rotated by the same phase of the last transmitted symbol (xk is the encoded symbol before di erential encoding). This phase rotation does not change the distribution of the noise samples, nor that of the CCI sequence. Hence, the metric distribution is left unchanged, as well as the PEP. The receiver is a based on linear combining and Viterbi decoding. Combining is performed before deinterleaving, and the combined channel output sample is given by

rk = yk yk 5

(3)

where the dagger y denotes Hermitian transpose. The sequence of samples rk is used to compute the Euclidean metric

m(rk ; xk ) = 2 Re frk xk g: b

(4)

b

The set of branch metrics fm(rk ; xk ) : xk 2 Xqg is then fed to the Viterbi decoder. b

b

3 Performance analysis The bit error rate (BER) of a coded modulation scheme can be well approximated by using a truncated version of the union bound [15, 16, 6]. Its computation is based on the knowledge of a set of dominant error events and weight coecients. An error event is a pair of sequences (x; x) stemming from the same state and merging after S steps in the trellis (S is the span of the error event). The pairwise error probability (PEP) P (x ! x) of each error event contributes to the union bound. b

b

Let x and x di er in positions KL = fj1; : : : ; jLg (L  S is their Hamming distance). If the sequence x was sent and r is the sequence of combined channel-output samples, then the path metrics di erence accumulated by x and x over the span S is b

b

=

X

k2KL

k

(5)

where k = 2 Re frk dk g and dk = xk ? xk . Note that, with correlated fading and nonideal interleaving, the PEP does actually depend on the positions KL, and not only on the Hamming distance L [8]. Since the positions k 2= KL do not contribute to , from now on we shall restrict time-index sets to their projection on KL. b

The PEP can be calculated from the inverse Laplace transform [11] 1 c+j1  (s) ds P (x ! x) = P (  0) = 2j (6)  s c?j 1 where  (s) = E [e?s ] : (7) Here c > 0, and the line s = c belongs to the region of convergence (ROC) of (s). Alternately, the PEP can be upper-bounded by using the Cherno bound Z

b

P (x ! x)  C (x; x) = min  () 2  b

b

6

(8)

where  is the intersection of the ROC of (s) with the real positive axis.

3.1 General expression of (s) Upon de ning the augmented vectors T = ( Tj1 ; : : :; TjL ), yT = (yjT1 ; : : :; yjTL ), and bT = ( T ; yT ), we can write  as the following Hermitian quadratic form:  = byFb; where F is de ned as

2

3

6 4

7 5

(9)

0 D I ; (10) M Dy 0 Im is the m  m identity matrix, 0 is the null matrix, D = diag (dj1 ; : : :; djL ), and denotes F=

Kronecker product. For a given interfering sequence  = (j1 ; : : : ; jL ), (9) turns out to be a Hermitian quadratic form of complex Gaussian random variables, so we can apply the result of [12, App. B]. Iterating expectations, we get exp ?sbyF(I2LM + 2sRF)?1b  (s) = E (11) det (I2LM + 2sRF) 2

3



6 4

7 5

where b and R are the conditional mean and covariance matrix of b given , respectively, and E denotes expectation with respect to . In general, b and R depend on the type of detection (i.e., on ) and on the joint statistical properties of the fading processes. To evaluate (11), let us de ne

 the fading and noise vectors g, h, n and  obtained by concatenating gk , hk , nk and  k , respectively, for k 2 KL;  the fading and noise vectors g0, h0 and n0, obtained by concatenating gk , hk and nk , respectively, for k0 2 KL0 = I ?1(I (KL) ? 1). KL0 is the set of time positions obtained after deinterleaving the positions I (KL) delayed by one time unit; 0

0

0

 the coded and interfering sequences x0 and 0, obtained by concatenating xk and k , respectively, for k0 2 KL0 . 0

0

 the diagonal matrices X = diag (x) IM ,  = diag () IM , X0 = diag (x0) IM , and 0 = diag (0) IM . 7

These de nitions allow us to write

y = Xg +  h + n and

8 > > > > >
> > > > :

s

s

IKI 1 1 g = E [g] = K K LM and h = E [h] = +1 KI + 1 LM Where 1m stands for the all-one m-vector. We further de ne the following fading (and noise) kernels g = E [(g ? g)(g ? g)y]; h = E [(h ? h)(h ? h)y] 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > :

0g gh 0gh 0n

= E [(g ? g)(g0 ? g)y]; 0h = E [(h ? h)(h0 ? h)y] = E [(g ? g)(h ? h)y];

(12)

= E [(g ? g)(h0 ? h)y]; 0hg = E [(h ? h)(g0 ? g)y]

= E [nn0y] which take into account the e ect of interleaving [15]. If KL \ KL0 = ;, then 0n = 0. We are now ready to calculate the vectors and matrices of equation (9). The conditional mean and covariance matrix of b are given by 2

3

6 4

7 5

b= y

and R = 12

2 6 4

 yy

y y

3 7 5

:

For all the detection schemes considered, we have 8 > > > > < > > > > :

y =

2s 4

s

3

K x + IKI  1 M K +1 KI + 1 5

y = Xg Xy + hy + 2 Re fXgh yg + ILM

(13)

The other terms are expressed in the following, according to the speci c detection scheme.

8

Coherent detection

8 > > > > > > > > < > > > > > > > > :

s

=

K 1 K + 1 LM

 = g

(14)

y = Xg + ygh

Di erential detection 8 > > > > > > > > < > > > > > > > > :

s



s

I KI 0 1 = K K LM + +1 KI + 1 = g + 0h 0y + 2 Re fgh 0yg + ILM

(15)

y = X0g + 0h 0y + X0gh 0y + 0hg + 0n

Pilot-tone detection 8 > > > > > > > > < > > > > > > > > :



s

s

I KI 1 = K K 1 LM + +1 KI + 1 LM = g + h + 2 Re fgh g + 2ILM

(16)

y = Xg + h + Xgh + hg

By the above expressions, we can calculate the PEP by interchanging the expectation E with the integral (6), and evaluating (6) for all interfering sequences  and 0. Evaluation of (6) in the case of Rician fading and diversity cannot be performed easily by the residue method of [5, 6]. A simple but ecient numerical method for computing (6) with any desired accuracy, based on Gauss-Chebyshev quadrature, is described in [3, 16]. Whatever the computational method used, evaluation of the PEP amounts to averaging over an exponentially growing number of integrals, as many as the possible interfering sequences. Since our main goal here is to achieve insight in the e ect of diversity over CCI, we choose to zero in on a special case that can be more easily managed. This is described in next section.

3.2 Independent fading and ISI-free CCI From now on, we assume ideal interleaving of both wanted and interfering signals. We assume these signals not to be frame-synchronous, so that the CCI sequences (k ) and (0k ) 9

appear, at the decoder input, to be i.i.d. and uniformly distributed over the scattered signal set S , even though the original CCI signal is encoded. With ideal interleaving, given x and x the terms k in (5) are statistically independent, and  (s) can be factored as b

2

 (s) = E

4

3 Y

k2KL

k j (s) = 5

Y

k2KL

h

E k j (s)

i

(17)

where k j(s) denotes the Laplace transform of the pdf of k conditioned on the interfering symbols which appear in the expression of k (k in the case of coherent and pilot-tone detection, k and k in the case of di erential detection). Therefore, the complexity of calculating (s) grows with L only linearly, rather than exponentially as in the general case. 0

If, in addition to ideal interleaving, we assume independent diversity, the kernels (12) are the following diagonal and zero matrices: g = K +1 ILM , h = I ILM , 0g = K +1 ILM , 0h = I IILM , and gh = 0gh = 0hg = 0n = 0. Hereafter,  and I denote the fading correlation coecients for the wanted and interfering signals: i  i

i  i

 = E [(gk ) gk ] and I = E [(h k ) hk ] 0

0

I

In a typical urban environment [6] we have  = J0(2F ) and I = J0(2FI ) where F and FI denote the normalized Doppler bandwidths of the wanted and interfering signals, respectively. With these assumptions, we can write the conditional mean b and the covariance matrix R as b = (bj1 ; : : :; bjL )T and R = diag (Rj1 ; : : :; RjL ) where

2

3

2

3

6 4

7 5

6 4

7 5

 ;k yy ;k k 1

IM : bk =

1M and Rk = 2 y ;k y;k yk Hence, after straightforward algebraic manipulations we obtain 2 j (s) = 1 + A s1? B s2 exp ? 1 +CkAs ?s ?DkBs s2 k k k k "

10

M

!#

(18)

where

8 > > > > > > > > > > > >
> > > > > > > > > > > :

2 Re [ k yk dk ]

Bk = (y;k  ;k ? jy ;k j2)jdk j2 Ck =

(19)

Dk = [j k j2y;k + jyk j2 ;k ? 2 Re ( k y ;k yk )] jdk j2 For the sake of simplicity, we further assume that:

 the CCI signal is a ected by Rayleigh fading (i.e., KI = 0), as typical in an urban microcellular environment [2] where the wanted signal reaches through a direct path while the interfering signal has only a multi-path component;

 the CCI samples are ISI-free (i.e., S = Xq). In [1, 4] it is shown that, when p(t) is a raised-cosine pulse with roll-o  0:3, the worst-case performance is obtained for symbol-synchronous (i.e., ISI-free) CCI.

Finally, from (19) we obtain the coecients Ak ; Bk ; Ck , and Dk to be inserted in (18), namely,

Coherent detection

8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > :

Ak = Bk = Ck = Dk =

jdk j2 K +1

jdk j2 ( + 1) K +1 I

jdk j2K K +1

jdk j2K ( + 1) K +1 I

(20)

Di erential detection De ne the di erential CCI symbol vk = k k . Then  j d k j2 + 2 Re [v d ] A = 0

8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > :

I I k k K +1 2 2 Bk = jdk j2 K + 1 + I + 1 ? K + 1 xk + II vk 2 Ck = Kjdk+j K 1

jdk j2K [(1 ? ) + (1 ?  Re [v x ])(K + 1) + K + 1] Dk = 2(K I k k I + 1)2

k

"



11

#

(21)

Pilot-tone detection

j d k j2 + 2 Re [ d ] A = 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > :

I k k K +1 2 Bk = jdk j2 K + 1 + I + 1 K + 1 + I + 2 ? K + 1 xk + Ik 2 Ck = Kjdk+j K 1 2  2 Dk = Kjdk+j K 1 2 I (1 ? Re [k xk ]) + 1 + 

k

"





h

#

(22)

i

Optimization of the PSR for pilot-tone detection. When both wanted and CCI signals are Rayleigh faded (K = KI = 0), we can write the Cherno bound on the PEP as

C (x; x) = b

2 Y

k2KL

Ek

4

2 k 1 + 14 A Bk

?M 3

!

5

:

Then, inserting the de nitions of and I in terms of ? and the PSR r, we have

A2k = (?jdk j2 + 2?I Re [k dk ])2 Bk (? + ?I + M (1 + r))(? + ?I + M (1 + r)Bp=r) ? j?xk + ?Ik j2 : Straightforward optimization yields the PSR ropt achieving the minimum Cherno bound: M B ' B ropt = ??++??+I +MB p p I p in accordance with [13], but with slightly di erent assumptions. s

q

4 Asymptotic analysis It is well-known [1, 9] that the performance of a system a ected by fading and CCI is basically determined by its irreducible BER (error oor). This is caused by the fact that, no matter how large ? is, there is a non-zero probability that the wanted signal be faded while the CCI signal is not. On the other hand, we showed in [15] that diversity, at least with coherent detection and perfect CSI, converts asymptotically, as the diversity order M increases, a fading channel into an AWGN channel with the same overall SNR ?. The asymptotic e ectiveness of diversity as a means to combat error oor as well as CCI for a fading communication system is investigated in this section. We rst use the 12

Cherno bound and a worst-case approach to prove that, for all the detection techniques considered here, the error oor decreases exponentially with the product diversity LM . Next, we consider the combined channel output statistics as M ! 1, and we compute the channel cut-o rate.

4.1 Asymptotic performance for high SNR Let us the Cherno bound to assess the system performance at very high SNR (? ! 1). Consider rst coherent detection. In this case the coecients Ak ; Bk ; Ck and Dk do not depend on k . Therefore, minimization with respect to  (a straightforward task) yields 2

C (x; x) = b

Y

k2KL

6 6 4



exp ? 41 1+K 14 jdjkdkj2j2 1 + 41 0jdk j2 0

0

M

3 7 7 5

;

(23)

where 0 = ?=(M (K + 1)( I + 1)). From (8) and (23), as ? ! 1 we obtain 2



exp ? 41 1+K 14 jdjkdkj2j2 Y 6 6 b)  4 1 + 14 0jdk j2 k2KL 0

lim P (x ! x ?!1

0

3

M

7 7 5

2



j2 exp ? 41 1+K 14 jdjmin 6 d j2  64 1 + 1 0jd min minj2 4 0

0

LM

3 7 7 5

(24)

where 0 = =(K + 1) and jdminj2 is the minimum squared Euclidean distance of the signal set. Di erential and pilot-tone detection are more dicult to deal with. A closed-form expression for the Cherno bound cannot be obtained because the coecients in (21) and in (22) depend on the CCI samples k and k , so that minimization with respect to  must be performed after the expectation. Thus, we follow a worst-case approach. It can be easily shown that, for both types of detection and for  2 , k j () increases with Bk and Dk , and decreases with Ak and Ck . Then, the worst case (according to the Cherno bound criterion) occurs for a CCI sequence  minimizing Ak 's and Ck 's and maximizing Bk 's and Dk 's, for all k 2 KL. This is obtained for vk = k k = ?xk and k = ?xk , for di erential and pilot-tone detection, respectively. In case of Rayleigh fading we obtain the two bounds ?LM 2 1 (  ?  I) 2 1 + 4 ( + 1)2 ? ( ?  )2 jdminj di erential I (25) lim P (x ! x)  ?LM 2 ?!1 ( ? 1) 1 pilot-tone. 1 + 16 jdminj2 0

0

8 > > > > >
> > > > :

13

For slow fading (i.e., ; I ' 1) the two bounds coincide, so we obtain the same asymptotic performance for di erential and pilot-tone detection. However, as we will show in the following, di erential detection is more sensitive to fast fading than pilot-tone detection. It is seen from (24) and (25) that the limiting PEP decreases exponentially with the product of the Hamming distance of the error event L and the space diversity M . Since the union bound on the BER of a coded modulation scheme is asymptotically dominated by its minimum Hamming distance terms, we have proved the following

Proposition 1. Under the assumptions of Section 3.2, the BER oor of a coded modulation

scheme with fading and CCI depends exponentially on the product diversity LminM , where Lmin is the minimum Hamming distance (or code diversity) and M is the space diversity order. 2

4.2 Asymptotic performance for high diversity order Assume that the diversity order M grows without bound, and consider the statistics of the combined channel output samples rk given in (3). By multiplying rk by (1 + r)= M we obtain an equivalent combined channel output in the form p rk0 = ? ak xk + 1 bk + p1 ck + mk + zk (26) where ak ; bk ; ck ; mk and zk are random variables whose expressions can be easily obtained from (2) and (3) for the various detection techniques. As M ! 1, we are interested in the limiting distributions of these random variables. q

!

Coherent detection. ak = M1 jgk j2; bk = 0; ck = M1 gky hk k ; mk = p1 gky nk : and zk = 0 (27) M Here, gk = gk =p and hk = hk =p I have unit second-order moments. By the law of large numbers (LLN) [14] we have that, as M ! 1, ak ! 1 and ck ! 0 in probability. By the central limit theorem (CLT) [14] we have that mk is asymptotically distributed as a zero-mean complex Gaussian random variable with variance 1=2. Therefore, asymptotically, b b

b

b

b

b

14

the statistics of rk0 converge in distribution to the output statistics of a CCI-free AWGN channel with the same SNR. This implies that diversity removes both Rician fading and CCI, provided that the latter is a ected by Rayleigh (i.e., zero mean) independent fading. This result holds irrespective of the statistics of the CCI samples k : thus, we expect diversity to provide the same improvement also when CCI is a ected by ISI.

Di erential detection. ak = M1 gk0ygk ; b

b

bk = M1 h0yk hk vk ; b

b

ck = M1 gk0yhk k + h0yk gk k0xk ; h

b

"

b

b

zk = p1 n0yk nk ?

i

b

#

mk = p1 gk0ynk + n0yk gk xk + p1 h0yk nk k0 + n0yk hk k M p Here, gk0 = gk0 = and h0k = h0k =p I have unit second order moments. By the LLN, ak ! , bk ! I vk , and ck ! 0 in probability. By the CLT, mk is asymptotically distributed as a zero-mean complex Gaussian random variable with variance (1+1= ). However, as M ! 1 the variance of the term zk is unbounded. Yet, when ? is suciently high, i.e., when zk is negligible, the channel approaches an AWGN channel with CCI, whose SNR is decreased by a factor 2=(2(1+1= )) (which corresponds to the typical loss of about 3 dB with respect to the coherent case when the fading is slow and the SIR is large) and whose SIR is increased by a factor =2I . 

b

b

b

b



b

b

Pilot-tone detection. ak = M1 jgk j2; b

bk = M1 jhk j2k ; b

s

zk = 1p+ r  yk nk ?

ck = M1 gky hk k + hyk gk xk ; h

b b

b b

i

+ r gy n +  y g x + p1 hy n +  y h  mk = 1 M k k k k k k k k kk By the LLN, as M ! 1 we have ak ! 1, bk ! k , and ck ! 0 in probability. By the CLT, mk is asymptotically distributed as a zero-mean complex Gaussian random variable with variance 21 (1 + 1= )(1 + Brp )(1 + r). However, also in this case, as M ! 1 the variance of the term zk is unbounded. As a result, when ? is suciently high (i.e., when the term zk is negligible), the channel approaches an AWGN channel with CCI, whose SNR is decreased by the factor 1=((1 + r)(1 + 1 )(1 + Brp )) and whose SIR is increased by a factor . "



b

b

15

#

b

b



Channel cut-o rate. We use the cut-o rate R0 to examine the performance of the

diversity channel with CCI for di erent diversity orders and di erent coding rates. With the assumptions of Section 3.2, the channel a ected by CCI is memoryless and we can apply the results of [7] by properly de ning the channel Cherno factor. This can be written as

C(xk ; xk ) = E [k j()]

(28)

b

where k j (s) is de ned in (18) and k is the path metric di erence corresponding to the error event (xk ; xk ), whose span is S = 1. Then, R0 is given by b

R0 = 2 log2 q ? log2 min 2

X

X

xk 2Xq bxk 2Xq

C(xk ; xk )

(29)

b

where  is the intersection of the real positive axis with the ROCs of C(xk ; xk ), for all the pairs (xk ; xk ). This expression can be written in closed form for coherent detection, and evaluated numerically for di erential and pilot-tone detection. From the cut-o rate we observed that the worst-case performance occurs when also the wanted signal is a ected by Rayleigh fading (i.e., K = 0), for all the detection techniques considered. This is the case when diversity provides the most remarkable bene ts, and will be considered next in our examples. b

b

5 Results In this section we present some examples of our analysis. Approximation of the BER is obtained by using a truncated union bound (TUB), while the PEP is computed exactly by using the expressions of (s) derived in Section 3.2 and the Gauss-Chebyshev quadrature rules of [3].

Cut-o rate. We rst analyze the achievable performance of di erent types of detection

in terms of the channel cut-o rate of an 8-PSK system with K = 0. Fig. 2 shows R0 vs. ? (the total SNR) with coherent detection and SIR=10 dB, for M = 1, 2, 4, 8, and 16. Note that, as M increases, R0 approaches the cut-o rate of the CCI-free AWGN channel (also shown for reference) for all values of Eb =N0. For nite M , the existence of the error oor 16

is indicated by the fact that, even for ? ! 1, R0 is bounded away from the maximum (3 bit/symbol). Similarly, Fig. 3 shows R0 vs. SNR with di erential detection and SIR=10 dB, for M = 1; 2; 4; 8; 16. There are two sets of curves, one for F = FI = 0:01 (slow fading) and the other for F = FI = 0:1 (fast fading). Note that, for both slow and fast fading, as the diversity order M increases, R0 gets higher at high rates and lower at low rates, where diversity deteriorates system performance. This is typical of non-coherent detection (see [11] about non-coherent combining loss). When fading is fast, the error oor is due to both CCI and fading. For the same values of M and ?, R0 is reduced by fast fading. However, asymptotically for large M and ?, both e ects are removed. Fig. 4 shows R0 vs. SNR for pilot-tone detection and SIR=10 dB, for M = 1; 2; 4; 8; 16. Also in this case we report two sets of curves, one for slow fading (F = FI = 0:01), the other for fast fading (F = FI = 0:1). Here, r = Bp and we assume Bp = 2F . Pilot-tone detection has the same behavior as di erential detection for low rates (non-coherent combining loss). Fast fading reduces the cut-o rate because the larger Doppler bandwidth increases the noise output from the pilot-tone extraction lter, but it does not in uence the error oor. q

E ect of diversity on BER performance. One of the central points in our analysis

is that diversity is able to reduce the error oor of a coded modulation system. This is con rmed by the performance of a 4-state Ungerboeck 8-PSK coded modulation scheme with SIR = 10 dB. Figs. 5{7 show the BER vs. Eb=N0 of the coded system with coherent detection (and perfect CSI), di erential detection, and pilot-tone detection, respectively. We note that for all the detection schemes considered the error oor decays exponentially as the diversity order M increases. Pilot-tone and di erential detection are compared in Figs. 6 and 7 with slow fading (F = FI = 0:01) and Fig. 8 with fast fading (F = FI = 0:1). They show that pilot-tone detection performance is slightly better than di erential detection in case of slow fading, and the di erence is enhanced when fading is fast.

E ect of the code. Fig. 9 shows the e ect of di erent coded modulations on the error

oor with coherent detection. The codes considered are denoted U4, U8 (4 and 8-states 8PSK Ungerboeck codes), and Q64 (obtained by mapping the \standard" 64-states rate 1/2 17

binary convolutional code with generators 171; 133 (octal notation) to Gray-encoded 4-PSK modulation [18]). The curves show the error oor vs. SIR of the three coded-modulation schemes. They con rm the exponential decay of the error oor with the product of the diversity order M by the minimum Hamming distance Lmin (1,2, and 6 for U4, U8, and Q64, respectively), as stated in Proposition 1.

Optimization of the PSR for pilot-tone detection. Minimization of the Cherno bound on the PEP for the case of Rayleigh fading (K = 0) yields the optimal PSR ropt ' q

Bp, the same value obtained in [13] without diversity and CCI. This is con rmed by Fig. 10, which shows the BER vs. PSR performance in the case of slow fading (F = FI = 0:01). In general, performances are not critical with respect to variations of the PSR around its optimal value, especially for moderate diversity order.

6 Conclusions We have studied the impact of diversity on coded-modulation PSK systems a ected by CCI and at Rician fading. Coherent detection (with perfect CSI), di erential and pilottone detection are analyzed with arbitrary fading correlation. General expressions of their pairwise error probabilities are provided. Focusing on independent fading and on the Rice-Rayleigh case, the e ect of diversity was investigated by studying cut-o rate and BER. Asymptotic analysis for unlimited power or diversity shows the limiting behavior of the error oor and the convergence of the fading diversity channel to a Gaussian channel (at least for coherent detection). In particular, we have shown that with slow fading di erential and pilot-tone detection give substantially equivalent results, whereas the latter performs better under fast-fading conditions. The results of our asymptotic analysis hold also for the cases of CCI a ected by ISI and when more than one CCI signal is present. For this reason we conjecture that diversity provides similar performance improvements also in these more general cases. In conclusion, our analysis shows that the synergy of coding and diversity plays a fundamental role in increasing the performance of interference-limited systems. 18

19

0 xk k Pulse Source m-k Coded xInterleaver Shaping Generator Modulator x(t) T 8 > > g1(-t) ? ?mgM (t) > R > > m    > > > A C> > > h1 (t) ? > N H> > ?m > m >  + > S A> > > .. ... . M N< CCI ?m I N> m >  + > > > S E> > hM (t) 6 > > S L> n1 (-t) m ?    +?mnM (t) > > > > + I > > > y 1 (t) y M (t) > O > > : N ? ? Matched C Filtering O 0 1 M yk ... Viterbi rk Deinterleaver rk0 BI 0 M yk Decoder N  I N G

Figure 1: Model of a coded transmission system with fading, diversity, and linear combining, a ected by CCI and AWGN.

20

3 AWGN M=16 M=8 M=4 M=2 R0(bit/symbol)

2

M=1

1

0 0

10

20

30

Γ (dB)

Figure 2: Cut-o rate with coherent detection, 8-PSK, SIR=10 dB.

21

3 AWGN M=16 M=8 M=4

R0(bit/symbol)

2

M=2

1 M=1

0 0

10

20

30

Γ (dB)

Figure 3: Cut-o rate with di erential detection, 8-PSK, SIR=10 dB. The diagram reports two sets of curves for F = FI = 0:01 (solid) and F = FI = 0:1 (dashed).

22

3 AWGN M=16 M=8

2 R0(bit/symbol)

M=4

M=2 1 M=1

0 0

10

20

30

Γ (dB)

Figure 4: Cut-o rate with pilot-tone detection, 8-PSK, SIR=10 dB. The diagram reports two sets of curves for F = FI = 0:01 (solid) and F = FI = 0:1 (dashed).

23

100 M=1

10-1 10-2 M=2

10-3

BER

10-4 10

M=4 M=8

-5

M=16

10-6 10-7 10-8 10-9 10-10 0

10

20

30

Eb/N0 (dB)

Figure 5: Bit error rate of U4 vs. Eb=N0 with coherent detection, SIR=10 dB. Solid curves: TUB, dots: simulation results.

24

100 M=1 10-1 M=2 10-2 M=4

10-3 M=8

BER

10-4

M=16

10-5 10-6 10-7 10-8 10-9 10-10 0

10

20

30

Eb/N0 (dB)

Figure 6: Bit error rate of U4 vs. Eb=N0 with di erential detection, SIR=10 dB, F = FI = 0:01 (slow fading). Solid curves: TUB, dots: simulation results.

25

100 M=1 10-1 M=2 10

-2

10-3

M=4

BER

10-4 M=8 10-5 M=16 10-6 10-7 10-8 10-9 10-10 0

10

20

30

Eb/N0 (dB)

Figure 7: Bit error rate of U4 vs. Eb=N0 with pilot-tone detection, SIR=10 dB, F = FI = 0:01 (slow fading). Solid curves: TUB, dots: simulation results.

26

100 10

M=1

-1

M=2

10-2 M=4

10-3

BER

10-4 M=8

10-5 10-6 10-7 10-8

M=16

10-9 10-10 0

10

20

30

Eb/N0 (dB)

Figure 8: Bit error rate of U4 vs. Eb=N0 with di erential (dashed curves) and pilot-tone (solid curves) detection, SIR=10 dB, F = FI = 0:1 (fast fading).

27

100

BER

10-5

10-10

10-15 Q64,M=1,4 U8,M=1,4 U4,M=1,4

10-20 0

10

20

30

SIR (dB)

Figure 9: Comparison of the error oors of codes U4, U8, and Q64 vs. SIR with coherent detection and diversity orders M = 1 and 4.

28

0.1

M=4

BER

0.01

0.001

M=16

0.0001

0.00001 0.01

0.1

1

r

Figure 10: Bit error rate of U4 vs. PSR r with pilot-tone detection, Eb=N0 = 10 dB, SIR=10 dB, F = FI = 0:01, M = 4 and 16.

29