Co-channel interference in cellular mobile radio systems with coded ...

Co-channel interference in cellular mobile radio systems with coded PSK and diversity  Ezio Biglieri

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Giuseppe Caire

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Giorgio Taricco

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Javier Ventura-Traveset

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SPECIAL ISSUE ON INTERFERENCE IN MOBILE WIRELESS SYSTEMS Abstract In this paper, the performance of coding and diversity in narrowband wireless cellular systems aected by fading, shadowing and co-channel interference is analyzed. We consider low-order diversity, that can be practically realized for both coherent and dierential phase shift keying. We assume that the shadowing random processes aecting all transmitted signals do not vary appreciably over the transmission duration. Fading, on the contrary, is assumed to vary more rapidly. Our main focus here is on outage probability. After choosing a performance indicator, its expectation is taken with respect to fading and co-channel interference, conditionally on shadowing. Hence, the resulting average performance indicator is a random variable. The probability that this random variable exceeds a speci ed threshold de nes the outage probability. We consider as performance indicators i) the channel cut-o rate R0 and ii) the bit error rate Pb of an actual coded scheme. As we are interested in interference-limited, rather than power-limited systems, we evaluate both R0 and Pb for very high signal-to-noise ratios. Results are parameterized by the frequency reuse factor and the diversity order.

Keywords: Fading channels, Coded modulation, Shadowing, Co-channel interference, Outage probability,

Cellular systems.

1 Introduction and motivation of the work Multiple access, a vital part of mobile wireless systems, may lead to co-channel interference (CCI) from other users sharing the same spectrum. This paper is devoted to the joint eects of coding and diversity on the performance of systems aected by fading, shadowing and CCI. We assume narrowband interference-limited systems, where CCI comes from far cells which use the same radio-channel (frequency reuse) and where the operating signal-to-noise ratio (C=N ) is suciently high so that the performance limitations are basically due to interference. The work of the rst three authors was supported by the Italian Space Agency (ASI). Dipartimento di Elettronica  Politecnico  Corso Duca degli Abruzzi 24  I-10129 Torino (Italy) z European Space Agency/ ESTEC  P.O. Box 299,  2200 AG Noordwijk (The Netherlands)

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1

Diversity is a well-known technique to cope with fading. Recent research by the authors [24] showed that the output statistics of a fading channel with diversity and perfect channel-state information (CSI) approach { asymptotically, as the diversity order M grows to in nity { those of an additive white Gaussian noise (AWGN) channel with the same average signal-to-noise ratio. Coded modulation may also be regarded as a way of introducing time diversity [13]: in fact, the eect of increasing the Hamming distance between pairs of possible symbol sequences transmitted over the at fading channel is the same as induced by increasing the number of branches in space diversity. The synergy between coded modulation and space diversity was investigated in [24, 25, 26]. Rather than the limiting behavior for high diversity order, we are interested in assessing the performance of systems which make use of low (practically realizable) diversity receivers together with coded coherent and dierentially-coherent phase-shift keying (CPSK/DPSK) modulation. We consider a system where both wanted and CCI signals are attenuated by the superposition of three independent factors: Rayleighdistributed fading, log-normally distributed shadowing, and deterministic distance-dependent path loss. This model is suited to macrocellular environments [1]. Our main focus here is on outage probability. The de nition of outage probability is based on a speci c system performance indicator  (e.g., bit error rate (BER), frame error rate). Assume that  depends on a set of rapidly-varying random processes U = fu1; : : :; un g and a set of slowly-varying random processes V = fv1; : : :; vmg. Distinction of process rates is based on the product of bandwidth B by transmission time TD , which are both nite constants: a process is deemed to be slow (resp., fast) if BTD  1 (resp., BTD  1) [14]. Approximation of slow processes as piecewise constant processes over TD allows one to consider the average  with respect to U and conditionally on V . This quantity will be denoted by  = E[jV ]

(1)

By choosing a performance threshold , we say that the communication system is in outage condition if  <  (resp.,  > ). Then, the outage probability is de ned as (resp:; > )

PO = P ( < )

(2)

A similar approach is followed in [14], where the system performance indicator  was the mutual information functional and fading was a slow process. In the system considered here, noise, fading and CCI are fast processes; shadowing and receiver mobility are slow processes. The time autocorrelation function of the fading process is modeled as J0 (2fD  ), where J0 (
) denotes Bessel function and fD the Doppler bandwidth [1]. Our analysis is based on the assumption TD fD  1. System performance versus outage probability is obtained in two steps: i) Choose the performance indicator  and evaluate  as a function of shadowing; ii) Calculate PO according to (2) as a function of the threshold . We consider two performance indicators leading to dierent conditional averages . i)  = R0, the 2

channel cut-o rate conditioned on shadowing [16]. ii)  = Pb , the BER corresponding to a given coded modulation scheme conditioned on shadowing. As we are interested in CCI-limited { rather than powerlimited { systems, R0 and Pb are evaluated for C=N ! 1. Both parameters turn out to be monotonically increasing functions of the signal-to-interference ratio C=I . Thus, PO can be obtained from the cumulative distribution function (CDF) of C=I FC=I ( ) = P (C=I < ) (3) which depends on the shadowing processes, the geometry of cellular coverage and frequency reuse. This paper is organized as follows. Section 2 describes the basic model of signal propagation and cellular coverage that lead to the calculation of FC=I ( ). In Section 3 we analyze coded CPSK/DPSK and derive error probability and channel cut-o rate conditionally on the shadowing. Numerical examples are shown in Section 4. They show the trade-os among coding complexity, diversity, and frequency reuse. Finally, conclusions are presented in Section 5.

2 Cellular system model In this section we de ne the propagation model and the cellular coverage of the system under analysis.

2.1 Propagation model We follow [1] in modeling the signal attenuation in a mobile radio environment without line-of-sight propagation by the superposition of three components: i) A rapidly varying Rayleigh-distributed fading component ; ii) a slowly varying log-normally-distributed shadowing component  , and iii) a deterministic path-loss component dependent on the distance d between mobile and base station. The path loss is proportional to the r-th power of the propagation distance, with r to be selected according to theoretical considerations and experimental results. This model applies to both forward (i.e., base-to-mobile) and reverse (mobile-to-base) links, when the mobile unit is located at the farthest position of the cell border and is assumed to be out of sight from the base station. Rayleigh fading is due to a multiplicative zero-mean complex Gaussian random process which models the diuse component of multi-path propagation. The average received signal power for a user at distance d from the base station is given by

PRx = d?r 2PTx

(4)

where  is Rayleigh distributed with unit second moment, 10 log10  is Gaussian-distributed with mean zero and standard deviation sh, and PTx denotes the transmitted power. Experimental evidence [1, 3] shows that, approximately, r = 4 and sh = 8 dB. We consider a set of interferers indexed in I , denote by di the distance of the i-th interferer from the receiver, and assume that all the signals have the same transmitted 3

power PTx . Then, we de ne the signal-to-interference ratio C=I as the ratio between the received power of the useful signal and the total CCI received power averaged with respect to fading conditionally on shadowing. The assumption that dierent CCI signals are aected by independent fading processes yields ?r C=I = P d ?r i2I di i

(5)

As for the correlation between shadowing components, we assume their logarithms to be jointly Gaussian, in accordance with [6, p. 185-186]. Therefore, we can express them as the product of two components: one in the near eld of the receiver that is common to the wanted and interfering signals, and the other speci c to the signal:  = 0 0 and i = 0 i0 : We still assume, following [6], that near eld and speci c components are independent and identically distributed.

2.2 Cellular coverage We consider the most ecient regular cellular coverage based on an in nite hexagonal tessellation of the plane [20, 21]. In this case, the frequency reuse factor is given by Kf = I 2 + IJ + J 2 (where I and J are nonnegative integers called \shift parameters" of the frequency reuse pattern [21]). We can express the centers of the hexagonal tessellation of the plane as elements of the set

H = fme?j=6 + nej=6 : m; n 2 Zg:

p

(Zdenotes the set of integers). Here, without loss of generality, we normalize the cell radius to 1= 3 and assume that the cell with center at z = 0 is the reference cell containing the mobile unit under analysis. Then, by considering the forward link, the C=I in (5), for a mobile unit located at z = x + jy , can be calculated as ?r 0 C=I = X jzj  ?r 0 : (6) zi 2H0

jzi ? zj i

where H0 denotes the set of the centers of cells re-using the same radio channel as cell 0 [21]:

H0 = f(Ie?j=6 + Jej=6 )(m + nej=3) : m; n 2 Zgnf0g C=I de ned in (6) is a function of shadowing and of the position z, which is assumed to be slowly-varying during the transmission time of duration TD . We xed z to be a vertex of the reference cell. As shown in Appendix A, this choice of z minimizes C=I when  0 = 10 =


= N0 = 1. Then, it is a worst-case assumption on the mobile unit position in the absence of shadowing.

We conclude the section by expressing the CDF of C=I (3) in terms of the geometrical parameters I; J of the frequency reuse pattern (under the assumption that all base stations irradiate the same power). We have 1 0 r X j z j 0 A (7) FC=I ( ) = P @ 0 ? r i < 0 zi 2H0 jzi ? z j

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In spite of the simple appearance of this expression, its numerical computation may turn out to be a complex task. A standard approach [22, 23] consists of approximating the sum of independent log-normal random variables by a log-normal variable. This requires complex iterative computations of the rst few moments of the sum of independent log-normal variables which may turn out into limited accuracy of the results. We preferred to resort to numerical integration. Figures 1-3 show FC=I ( ) for Kf = 3; 4; 7; 9 and for sh = 6; 8, and 12 dB. These values are typical of urban environments [1]. In the following analysis, we assume sh = 8 dB. Integration was performed by the Monte Carlo method [8, p. 288] with 105 samples, considering all interfering signals such that jz jr =jzi ? z jr  10?3.

3 Performance analysis In this section we evaluate the cut-o rate and the BER of coded CPSK/DPSK with diversity, fading, and CCI, conditionally on shadowing, which is assumed to equally aect all diversity branches. For the sake of simplicity, we limit the analysis to the case of Rayleigh fading and symbol-synchronous CCI (see [26] for a more general setting). In many situations the latter assumption corresponds to the worst case. For example, [15] shows that, with raised-cosine pulses with roll-o between 0:3 and 1:0, symbolsynchronous CCI maximizes the BER. We consider N  1 independent CCI signals, with the constraint that the total interfering power be xed by the value of C=I .

3.1 Channel model In the transmission system under analysis (see Fig. 4) a binary information sequence enters a coded modulator which generates the encoded sequence x. The components xk of x take on values from a q -ary PSK signal set: xk 2 Xq = fej2i=q : i = 0; 1; : : :; q ? 1g The sequence x is rst passed through an ideal interleaver and { with DPSK { dierentially encoded. The transmitted signal is sent through M independent Nyquist channels (each corresponding to one diversity branch), aected by at fading, CCI and additive white Gaussian noise (AWGN). With perfect timing recovery, after sampling the useful and the interfering signal samples are ISI-free. Let yk;i and k;i denote respectively the output and the CSI of the i-th diversity branch obtained after demodulation, matched ltering, sampling, and deinterleaving. Then the diversity channel can be modeled as a discretetime vector channel with a single input xk and 2M outputs, viz., the components of the two M -vectors yk = (yk;1; yk;2; : : :; yk;M )T and k = (k;1; k;2; : : :; k;M )T . We have

y k = g k xk +

N X i=1

hik ki + nk

where ki 2 Xq denotes the k-th symbol of the i-th CCI signal, and 5

(8)

 nk = (nk;1; nk;2; : : :; nk;M )T is a zero-mean complex Gaussian random vector representing the AWGN samples. Noise samples are normalized so that var(nk;j ) = 12 E[jnk;j j2 ] = 1=2.  gk = (gk;1; gk;2; : : :; gk;M )T and hik = (hik;1; hik;2; : : :; hik;M )T random vectors modeling the fading affecting the wanted and the i-th interfering signals, respectively. As we consider Rayleigh fading, independent diversity and we assume the fading aecting the wanted and the CCI signals to be mutually independent, gk and hik are zero-mean complex Gaussian random vectors with covariances

8 > cov(gk ) > > < cov(hik ) > cov(gk hik ) > > : cov(hik hjk )

= 21 IM = 21 i IM = 0M = 0M i 6= j where IM and 0M denote the M  M identity and zero matrix, respectively. We de ne

I =

N X i=1

i

(9)

(10)

Then, and I denote the C=N ratios of the useful and interfering signals per branch, conditioned on shadowing. Note that both deterministic path-loss and shadowing are contained in , 1; : : :; N . C=I , as de ned in (6), is given by C=I = = I, where we truncated the in nite summation over H0 to N interferers. The total useful C=N , conditioned on shadowing, at the receiver input is given by M 1. The CSI vector depends on the type of receiver. For coherent detection with perfect CSI, k is given by

k = gk

(11)

For dierential detection, the CSI for the `-th channel symbol is provided by (` ? 1)-th channel symbol. Let us assume that interleaving maps the k-th transmitted symbol to the `-th channel position with ` =  (k). Disregarding the xed time shift due to interleaving and deinterleaving delay, we note that symbols (xk0 ; xk ) are mapped to adjacent channel positions ` ? 1 and ` by the interleaver and back to original positions k0 and k by the deinterleaver. Clearly,

k0 =  ?1 (` ? 1) =  ?1 ((k) ? 1)

(12)

Let K denote the set of time positions k in the encoded sequence x. If  ?1 ( (K) ? 1) \ K = ;, i.e., the set of time positions of x and the set of time positions mapped by the interleaver to the channel positions immediately before the corresponding positions of x are disjoint, the values of xk0 for k0 2  ?1( (K) ? 1) are irrelevant with dierential detection and can be arbitrarily set to 1. This is actually the case with ideal interleaving, so that we have N X (13) k = gk0 + hik0 ki 0 + nk0

i=1 In order to evidence the individual impact of antenna diversity, we disregard, in our analysis, the C=N increase that actually occurs when multiple receive elements are used. This is obtained by assuming that the transmit power of both the useful and of the interfering signals is lowered by a factor equal to the diversity order. 1

6

In the following we will use the time-correlation coecients de ned as   = E[gk;j ( gk0;j ) ] = J0 (2fD T )

i =

E[hik;j (hik0 ;j )]

i

= J0 (2fDiT )

i = 1; : : :; N

(14)

where fD and fDi denote the Doppler bandwidths of the wanted and of the i-th interfering signals, respectively, and T denotes the symbol time interval. We stipulate that the receiver is a linear combiner followed by a metric computer and a Viterbi decoder. The linear combiner produces the combined channel output sample

rk = yk yk

(15)

where the dagger y denotes Hermitian transpose. The sequence of samples rk is fed to a metric computer, based on the Euclidean metric

m(rk ; v) = 2 Re frk v  g

v 2 Xq

(16)

The set of branch metrics fm(rk ; v ) : v 2 Xq g is then used in the Viterbi decoder.

3.2 Error probability calculation The BER of a coded modulation scheme can be well approximated by using a truncated version of the union bound [19, 24, 25], which can be written as X a(x; xb)P (x ! xb ) (17) P '1 b

b (x;bx)2E

where b denotes the number of information bits entering the coded modulator at each trellis step, E is a set of dominant error events, and a(x; xb ) and P (x ! xb ) are the weight coecient and the pairwise error probability (PEP) associated with the error event (x; xb ) (for the details on the calculation of the weight coecients, see [10, p. 102], [2, p. 184] and references therein). Each error event is represented by a pair of sequences (x; xb ) stemming from the same state and merging after S steps in the trellis (S being the span of the error event). The key quantity to be computed in the following, in order to evaluate Pb , is the PEP P (x ! xb ). Let us assume that the two sequences x and xb dier in positions KL = fj1; : : :; jLg (L  S is the Hamming distance dH (x; xb )). Let x be the sequence sent, and r the received sequence of combined channel output samples (15). Then, the path-metric dierence accumulated by x and xb over the span S is
= where

X

k2KL


k


k = 2 Re frk dk g 7

(18) (19)

and dk = xk ? xbk . The PEP can be calculated by rst evaluating the Laplace transform 
(s) = E[e?s
]; then using [4, App. 4B]

1 P (x ! xb ) = P (
 0) = 2j

Z c+j1 c?j 1 
(s)

(20) 
(s) ds s

(21)

Here c 2 , the intersection of the convergence region of with the real positive axis. Since the convergence region of 
(s) is a vertical strip containing the imaginary axis, we have  = [0; a+)

(22)

where a+ is the real part of the leftmost singularity of 
(s) in the Re (s)  0 half-plane. A simple but ecient numerical method for evaluating (21) within any degree of accuracy was developed by the authors in [27]. This is based on the application of Gauss-Chebyshev quadrature. One might also choose to upper-bound the PEP by using the Cherno bound

P (x ! xb )  C (x; xb ) = min  (); 2


(23)

but we hasten to say that this may be rather loose. BER results obtained by exact evaluation of PEP are substantially better than those obtained by Cherno bound on PEP, as we show in [27], for example. Ideal interleaving implies that the random variables
k are statistically independent from each other. Therefore, we can write 
(s) as Y 
(s) = 
k (s): (24) k2KL

In Appendix B we show that the terms in the above product can be written in the form 
k (s) = E



?M  2 1 + A k s ? Bk s

(25)

where expectation is taken, conditionally on shadowing, with respect to the CCI variables, and where the coecients Ak and Bk depend on the detection scheme. For coherent detection we get

8 > < Ak = jdkj2 > : Bk = jdkj2( I + 1)

(26)

8 > < Ak =  jdkj2 + 2 Re [kdk ] > : Bk = jdk j2 h( + I + 1)2 ? j xk + k j2i :

(27)

For dierential detection we get

where we de ne

k =

N X i=1

i i ki (ki 0 ) 8

(28)

A dierence between coherent and dierential detection schemes is the following. With the former, coef cients Ak and Bk depend only on the total interfering power I . Hence, the PEP is independent on the number N of interferers. On the other hand, in the case of dierential detection, Ak and Bk depend on both I and k . Then, in order to compute the PEP, we need to know the distribution of k , which takes on values in the set ) ( X N S =  = i ivi : vi 2 Xq i=1

Computation of (25) may be prohibitively complex. In fact, if we consider, for example, the case of 6 interferers and quaternary DPSK, we have jSj = q N = 46 = 4096, with a computational complexity 1024 times higher than in the single interferer case. Nevertheless, when there are many interferers with approximately the same power, it is possible to resort to the Gaussian approximation of CCI and avoid the expectation in the LHS of (25). Assume that N ! 1 with i = I =N and consider the random vector zk = (zk;1; : : :; zk;M )T with components

zk;j = Nlim !1

N X i=1

hik;j ki

From the central-limit theorem [11, p. 212], zk;j is a zero-mean complex Gaussian i.i.d. random vector with covariance 21 I IM . Then, the limiting expressions of the received signal and of the CSI vectors are

8 < y k = g k xk + z k + n k (29) : k = gk0 + zk0 + nk0 : Assuming that ki is uniformly distributed over Xq , we have E[zk;j zk0 ;j 0 ] = 0 for all k0 6= k. Thus, by repeating the calculation of (25) with yk and k given by (29) instead of (8) and (13), we obtain coecients Ak ; Bk given by (see Appendix B) 8 > < Ak =  jdkj2 (30) > : Bk = jdkj2 ( + I + 1)2 ? 2 2 Numerical evidence (see Section 4) led us to conjecture that the performance of a DPSK system with a nite number of interferers satis es the following bounds:

PG (x ! xb )  P (x ! xb )  P1 (x ! xb )

(31)

where PG refers to the Gaussian approximation of CCI, and P1 to a single interferer. The lower bound to PEP is obtained by using coecients (30), while the upper bound follows by inserting N = 1 and 1 = I into (27). The following arguments are given to support (31). In the case of Gaussian approximation, yk and k in (29) can be seen as the received vector and the CSI of a diversity channel with dierential detection, no CCI and total AWGN zk + nk . The C=N per branch of this equivalent CCI-free channel is equal to

=( I +1). Since the decoder is based on the Euclidean metric (16), which is matched to a CCI-free channel, the PEP resulting from the Gaussian approximation is nearly a best case. On the other hand, the case of a single interferer with C=N per branch equal to I corresponds to the case when all the CCI signals add up coherently [18]. 9

3.3 Performance for high C=N A key factor that determines the performance of a system aected by fading and CCI is its irreducible BER (commonly referred to as \error oor") [15]. Here we evaluate the impact that coding and diversity have on the error oor by evaluating the Cherno bound to the PEP. In the case of CPSK we get

Y

?M 1 + 14 + 1 jdk j2 (32) I k2KL With DPSK, a closed-form expression cannot be obtained because the coecients in (27) depend on the CCI samples, thus preventing simple minimization with respect to . For this reason we computed the Cherno bound in the case of single interferer, with a further worst-case assumption (see Appendix C): C (x; xb) =

!?M 2 (  ? 

1 1 I) C (x; xb ) = 1 + 4 ( + + 1)2 ? ( ?  )2 jdk j2 I 1 I k2KL and in the case of Gaussian approximation: Y

C (x; xb ) =

Y k2KL

!?M 2 2  1 2 jd j 1+ 4 ( + I + 1)2 ? 2 2 k

(33)

(34)

The details of the derivation of (32),(33) and (34) are given in Appendix C. For ease of notation, in the following equations we denote the minimum squared Euclidean distance of Xq by jdminj2 = 4 sin2(=q ) and the C=I ratio by . Then, by letting ! 1 with = I = in (32), (33) and (34), we get the following upper bounds on the PEP oor ?LM Y  1 2?M  1 2 b lim P ( x ! x )  1 +  1 + (35) j d j j d j

!1 4 k 4 min k2KL for CPSK,

b 

lim !1 P (x ! x)



Y k2KL

!?M 2 (  ?  ) 1 1 2 jd j 1+ 4 ( + 1)2 ? ( ? 1 )2 k

!?LM 2 (  ?  ) 1 1 1 + 4 ( + 1)2 ? ( ?  )2 jdminj2 1

(36)

for DPSK (single interferer), and

!?M 2 2 1  2 b

lim !1 P (x ! x)  k2K 1 + 4 ( + 1)2 ? 2 2 jdk j L !?LM 2 2  1 2  1 + 4 ( + 1)2 ? 2 2 jdminj Y

(37)

for DPSK (Gaussian approximation). From the above equations we note 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 10

minimum-Hamming-distance terms, we have proved that the BER oor of a coded modulation scheme with minimum Hamming distance (or code diversity) Lmin and space-diversity order M depends exponentially on the \product diversity" LminM 2 . Moreover, when the fading time-correlation coecients are close to 1 (i.e.,  ' 1 ' 1), the bounds (36) and (37) become

8 !?LM 2 > 1 ( ? 1) 2 > (single interferer) > < 1 + 16 jdminj lim P (x ! xb )  > !?LM ?!1 2 1 2 > (Gaussian approximation) : 1 + 4 2 + 1 jdminj

Comparing these bounds with (35) for  1, we note that, in order to have the same error- oor bound, a system employing DPSK requires approximately a C=I increase of 6 dB (single interferer) or 3 dB (Gaussian approximation) with respect to a system employing CPSK with perfect CSI. Exact computations of the BER

oor of some coded modulation schemes, obtained by TUB (17) with exact PEP evaluation via the GaussChebyshev quadrature [27], support this claim (see Section 4).

3.4 Channel cut-o rate. We use the cut-o rate R0 [7] as the system performance indicator. Under the assumption of ideal interleaving, the channel is memoryless (conditionally on shadowing), so we can apply results of [2, p. 48]. With 
k (s) given in (25), and
k the path metric dierence corresponding to the span 1 error event (xk ; xbk ), we have X X R0 = 2 log2 q ? log2 min 
k () (38) 2 xk 2Xq bxk 2Xq

where  is the intersection of the real positive axis with the regions of convergence of 
k (s) for all the pairs (xk ; xbk ). For CPSK, minimization over  gives the following result:

3 2   ?M 7 6 X 75 1 + 41 + 1 j1 ? v j2 R0 = log2 q ? log2 641 + I v2Xq

(39)

v6=1

With DPSK we use again the single interferer (with a worst-case assumption) and the Gaussian approximation to nd upper and lower bounds on R0 as

8  P  ?M  2 > (  ?  1 1 I ) 2 single interferer < log2 q ? log2 1 + vv2X6=1q 1 + 4 ( + I+1)2?( ?1 I)2 j1 ? vj   P  R0 = >  2 2 2 ?M Gaussian approximation : log2 q ? log2 1 + vv2X6=1q 1 + 41 ( + I+1) 2 ?2 2 j1 ? v j

(40)

4 Results In this section we show some of the results that can be obtained by using the techniques previously outlined. We provide two types of charts. Charts of the rst type plot outage probability vs. R0. The implicit 2

A similar results was found in [17] for coded dierential 4PSK with a single interferer.

11

assumption with this choice is that a system with rate below R0 can achieve an arbitrarily low BER provided that a sucient amount of coding is employed (practical reliable communication). If R0 drops below the desired transmission rate, we consider the system in outage condition 3. Charts of the second type plot outage probability vs. the BER Pb of some actual coded schemes. Since we are interested in CCI-limited, rather than power-limited systems, we let C=N ! 1, so that both R0 and Pb depend only on C=I . Let R0 and P b denote the system performance requirement thresholds, denote again C=I by and let R0 ( ) and Pb ( ) be the limiting values of R0 and of Pb obtained by letting ! 1 with = I = . As anticipated in the Introduction, R0( ) and Pb ( ) are monotone functions of . In particular, R0 ( ) is non-decreasing and Pb ( ) is non-increasing. Then, we can compute outage probabilities by

8 < = R0?1(R0) : PO = P (R0( ) < R0) = FC=I ( )

and by

8 < = Pb?1(P b) : PO = P (Pb( ) > P b) = FC=I ( )

(practical reliable communication)

(41)

(actual coded schemes)

(42)

where FC=I ( ) is given in (7).

For evaluating Pb , we used the TUB approximation (17) with exact computation of the PEP via the Gauss-Chebyshev quadrature method of [27]. Three coded modulation schemes are considered. These are obtained by Gray-mapping onto 4PSK the best rate-1/2 binary convolutional codes with 4, 16, and 64 states with octal generators (7; 5), (23; 35) and (133; 171) [9, pp. 467]. For ease of reference, we denote these coded schemes by Q4, Q16, and Q64, respectively. We distinguish DPSK with a single interferer from DPSK with in nitely many interferers and vanishingly small equal power (Gaussian approximation).

Validation of the approximation of Pb. The BER vs C=N of Q64 at C=I = 10 dB with coherent and

dierential detection (the latter with single-interferer and Gaussian approximation) is plotted in Figs. 5 and 6, respectively. Simulation points show that the approximation obtained by the TUB (17) with exact evaluation of the PEP is very tight (much tighter than any Cherno union bound, especially for fading channels). Some simulation points for the case of N = 6 equal-energy interferers are also shown. It can be noticed that they lie between the single-interferer and the Gaussian approximation curves, thus supporting the claim that single interferer and Gaussian approximation are actually worst and best case assumptions (similar results were observed for all coded modulation schemes we examined). Curves of Fig. 5 and 6 con rm the existence of an irreducible BER (error oor) which is lowered by increasing the diversity order M.

Error oor. Figure 7 plots the error oor vs. C=I for Q64 with coherent and dierential detection (single

interferer and Gaussian approximation). The gains predicted by the error- oor bounds (35), (36), and (37), 3

The same approach was followed in [16] for DPSK with a single interferer and no diversity.

12

are con rmed, although the dierential single-interferer case performs from 1 to 2 dB better than expected from the worst-case analysis which leads to (36).

Outage probability and channel cut-o rate. From now on, we assume for the shadowing the stan-

dard deviation sh = 8 dB. This value is widely adopted in mobile radio communications (see, e.g., [6]) as an outcome of experimental results in urban environments [1, 3]. Figs. 8 to 13 show outage probability vs. system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and PSK cardinalities (2,4, and 8PSK). Each gure refers to a dierent frequency reuse factor (Kf = 3 or 9). Figures 8{9 refer to coherent detection. Figures 10{11 refer to dierential detection (single-interferer case). Figures 12{13 refer to dierential detection (Gaussian approximation). R0 is measured in bit/symbols, or equivalently, in bit/s/Hz for a system with zero excess bandwidth.

System tradeos. The above results are summarized in some Tables intended to illustrate the system

tradeos necessary to achieve reliable transmission at 1 bit/s/Hz. Table 1 shows the outage probabilities, the reuse factors, and the diversity order necessary to achieve R0 = 1 for the two types of detection considered here. Similarly, Table 2 does the same for code Q64 (whose Nyquist spectral eciency is 1 bit/s/Hz), showing what parameter values are necessary to achieve P b = 10?3 . The results from R0 and the speci c code Q64 are in almost all cases very close to each other, which con rms that R0 is a sensible criterion for system design when coding is used. The only exception is the case Kf = 9, where the code performance is slightly poorer than that predicted by the cut-o rate.

As an example of the tradeos mentioned above, observe that we have the same outage probability with a frequency reuse factor Kf = 4 and diversity order M = 4, and with Kf = 9 and M = 1 (i.e., no diversity). Table 3 shows the normalized radio capacity of a system using 4PSK, assuming practical reliable communication and an outage probability PO = 0:1. This is obtained as the maximum ratio of R0=Kf , where Kf takes on values in the set Kf = 3; 4; 7; 9 and R0 is given by the abscissas of curves like those shown in Figs. 8{13 for 4PSK modulation. It should be kept in mind that these values are based on the assumption that the mobile unit be at a vertex of the cell. This is generally reported in the literature, even though a spatially-averaged outage probability (under the assumption of some mobile-unit distribution) may be of some interest. Actual values in terms of users per cell can be obtained by multiplying the values in Table 3 by Bt =((1 + )Rb ), where Bt is the total system bandwidth, Rb is the user transmission bit-rate and  is the signal excess bandwidth.

Outage probability and BER. Figures 14 to 19 show the outage probability vs. system BER P b for codes Q4, Q16, and Q64. Each gure reports results for dierent diversity orders (M = 1; 2; 3; 4), with dierential and coherent detection, with reference to a speci c code (Q4, Q16, and Q64) and frequency reuse factors Kf = 3 and 9. 13

All results were obtained for an interference-limited system (C=N ! 1). The minimum C=N required to approach this condition increases with the diversity order M , as shown in Figures 5 and 6. A reasonable gure is around 20 dB.

5 Conclusions We presented results on a CCI-limited cellular narrowband system based on coded CPSK/DPSK, in terms of performance vs outage probability parametrized by the frequency reuse factor and the diversity order. Our main system assumptions are that both the wanted and the CCI signals are aected by mutually independent Rayleigh fading, which is supposed to be signi cantly varying during a transmission of duration TD . On the contrary, the shadowing processes aecting signals are approximated as (random) constants during TD . Then, under the constraint of a strictly nite TD and decoding delay, the system performance is a random variable depending on the shadowing. This leads to a simple and general de nition of outage probability. We chose as system performance indicators the channel cut-o rate and the BER of some actual coded schemes. Outage probability was computed from the CDF of the C=I , obtained from the shadowing statistics in the case when the mobile unit is in a vertex of the useful cell. Our analysis allow the system designer to select the best trade-o among these parameters. For example, perusal of Table 2 shows that the same outage probability obtained with a frequency reuse factor Kf = 9 and no diversity can be obtained with Kf = 4 and diversity order M = 4. Therefore, a more than two-fold increase of system capacity can be achieved by inserting 4-branch diversity at the receiver. This result holds for coherent detection and, approximately, for dierential detection.

Acknowledgement The authors gratefully acknowledge the contribution of the Guest Editor Dr. Michael L. Honig and of the anonymous reviewers, whose comments helped improving the clarity of this manuscript.

APPENDIX A Minimum C=I ratio in cell 0 In this Appendix we show that, in the absence of shadowing, the minimum C=I is achieved at the reference cell vertexes. Let r (z) = P jzjzj ? zj?r (43) zi 2H0 i

denote the C=I de ned in (6) in the case of no shadowing, i.e., for  0 = i0 =


= N0 = 1. Given the p symmetry of the problem, it is sucient to prove that the minimum of (z ) is attained at z0 = 1= 3. 14

Let us set z = z0 (1 + "ej ). The rst order Taylor expansion of 1= (z ) about " = 0 is

(

1 3[2K (m2 + mn + n2 ) ? (I + J )m ? In] cos  1 X 1 = 1 + X f (z) (z0) m=?1 n=?1 3[Kf (m2 + mn + n2 ) ? (I + J )m ? In] + 1 ) p 3[( ? I + J ) m + ( I + 2 J ) n ] sin  + " 3[Kf (m2 + mn + n2 ) ? (I + J )m ? In] + 1 + O("2 )

with Kf = I 2 + IJ + J 2 . Hence, (z0 ) is a local minimum of cell 0 if the quantity in braces { hereafter denoted by 1 () { is negative for all  2 [2=3; 4=3]. Moreover, it is also a global minimum because of continuity and invariance by rotations of =3 radians. Then, we show that 1 () < 0 for all  2 [2=3; 4=3] as follows. 1. The inequality I 2 + IJ + J 2  I + J holds for all integer values of I and J . Proof: the discriminant of I 2 + IJ + J 2 ? I ? J is positive for I; J 2 (?1=3; 1). However, the inequality holds for the four pairs (0,0), (0,1), (1,0), and (1,1). Hence, it holds for all integer values of I and J . 2. The coecient of cos  in 1 () is positive. Proof: it follows from the inequality (I 2 + IJ + J 2 )(m2 + mn + n2 )  (I + J )(m + n)  (I + J )m + In; holding for all nonnegative integers I; J , which implies that both numerator and denominator of the coecient considered are positive. Hence, 1( ) < 0, and we are left to show that the inequality still holds at the extremes of the interval [2=3; 4=3]. 3. Since and

1(2=3) = ?

1 1 X X

3[Kf (m2 + mn + n2 ) ? (I + J )n ? Jm] 2 2 m=?1 n=?1 3[Kf (m + mn + n ) ? (I + J )m ? In] + 1 1 X 1 X

3[Kf (m2 + mn + n2 ) ? Im + Jn] 2 2 m=?1 n=?1 3[Kf (m + mn + n ) ? (I + J )m ? In] + 1 it is plain to show, as we did above, that 1 (2=3) < 0 and 1 (4=3) < 0, which concludes the proof of the result of this Appendix.

1(4=3) = ?

B Derivation of 
(s) Our derivation of 
(s) is based on a result from [5, App. B], reported hereafter with minor modi cations for easy reference.

15

Proposition. Given a (column) vector v of complex Gaussian random variables with mean  = E [v] and covariance matrix  = 21 (E [vvy] ?y), the Laplace transform Q(s) = E [exp(?sQ)] of the real quadratic form Q = vH Fv (with F = Fy Hermitian) is given by h i exp ?syF(I + 2sF)?1 Q(s) = (44) det(I + 2sF) Here, y denotes Hermitian transposition.

We compute 
k (s) in (24) by rst conditioning on the CCI variables k1 ; : : :; kN (and k10 ; : : :; kN0 , for dierential detection) and then averaging them out. The metric dierence
k in (19), conditioned on the CCI variables, can be expressed as the real quadratic form of complex Gaussian random variables
k = vky Fk vk ; where

2 3 vk = 4 k 5 ; yk

2 3  0 d k 5 I ; Fk = 4 M dk 0

(45)

dk = xk ? xbk ;

and where yk is de ned in (8), and k is de ned in (11) and (13) for coherent and dierential detection, respectively. Here, denotes the Kronecker product [12, p. 256]. As we consider Rayleigh fading, E [vk ] = 0. The covariance matrix k of vk , conditionally on the CCI variables, is 2 3 

x 1 k 5 IM k = 2 4

xk + I + 1 for coherent detection and 2 3  

+

1 I + 1  xk + k 5 IM k = 2 4  xk + k + I + 1 P for dierential detection, with  and i de ned in (14) and k = Ni=1 i i ki (ki 0 ), as in (28). Finally, 
k (s) can be computed as

h i h h ii 
k (s) = E e?s
k = E E e?s
k jk1 : : :kN ; k10 ; : : :; kN0

(46)

where, for what observed above, the inner conditional expectation can be computed by applying (44). Since E[vk ] = 0, the inner expectation reduces to 1= det(I + 2sk Fk ). By using the identity 2Re [xk dk ] = jdk j2 and the well-known properties of the Kronecker product, after straightforward determinant computation, we obtain the expression (25) of 
k (s) with coecients Ak ; Bk given by (26) and (27) for coherent and dierential detection, respectively. The case of dierential detection with Gaussian approximation of the interferers can be dealt with by using yk and k given in (29). By assuming interfering samples to be uniformly distributed over Xq , we get the covariance matrix 2 3 

+

+ 1  x I k 5 I k = 21 4 M  xk + I + 1 and by repeating the above steps we obtain the coecients reported in (30). 16

C Cherno bounds In this Appendix we compute the Cherno bounds for CPSK and DPSK with fading, diversity and CCI. In general, the Cherno bound C (x; xb ) on the PEP P (x ! xb ) can be obtained by applying (23). Here, minimization with respect to  2  may be dicult since each term 
k (s), appearing in the expression (24) of 
(s), is de ned by the expectation (25) with respect to the CCI variables. However, if Ak =(2Bk ) does not depend on k and on k , all the terms in (24) attain their minimum on  at opt = Ak =(2Bk ), so that we can write ?M # Y " 1 C (x; xb ) = (47) E 1 + 2 optAk k2KL

In the case of CPSK, we obtain opt = 2( I1+1) . By applying (47), we get (32). In the case of DPSK, this simple approach cannot be followed, since Ak =(2Bk ) and Ak depend on both k and k (see (27)). Then, we restrict to the cases of single interferer and Gaussian approximation. In the rst case, k = 1 I k1 (k10 ) is a q -PSK symbol, scaled by the (real) factor 1 I . In order to remove the dependency on k , we note that the terms (1 + Ak  ? Bk 2)?M with Ak ; Bk given in (27), attain their maximum over all the possible values of k (for any xed  2 ) at k = ?1 I xk , that is, when the CCI variable k is in phase opposition to the useful signal variable xk . This is the case where interference coherently subtracts to the useful signal, minimizing the useful C=N . By making this worst-case assumption and substituting k = ?1 I xk into Ak and Bk , we can apply (47) and we obtain (33). In the case of DPSK with Gaussian approximation, Ak and Bk are given in (30). We can apply directly (47) and we obtain (34).

References [1] W. C. Jakes, Jr., Microwave Mobile Communications. New York: J. Wiley & Sons, 1974. [2] S. H. Jamali and T. Le-Ngoc, Coded-Modulation Techniques for Fading Channels. New York: Kluwer Academic Publishers, 1994. [3] W. C. Y. Lee, Mobile Cellular Telecommunications Systems. New York: Mac-Graw Hill, 1989. [4] J. Proakis, Digital Communications. New York: McGraw-Hill, 1983. [5] M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques. New York: McGraw-Hill, 1966. [6] A. J. Viterbi, CDMA. Principles of Spread Spectrum Communications. Reading, MA: Addison-Wesley, 1995. [7] J. M. Wozencraft and I. M. Jacobs, Principles of Communications Engineering. New York: Wiley, 1967. [8] P. J. Davis and P. Rabinowitz, Methods of numerical integration. New York: Academic Press, 1975. [9] S. Benedetto, E. Biglieri and V. Castellani, Digital Transmission Theory. Englewood Clis, NJ: Prentice-Hall, 1987.

17

[10] E. Biglieri, D. Divsalar, P. McLane and M. K. Simon, Introduction to Trellis-Coded Modulation with Applications. New York: MacMillan, 1991. [11] J. B. Thomas, Introduction to probability, New York: Springer-Verlag, 1986. [12] P. Lancaster, Theory of matrices, New York: Academic Press, 1969. [13] C.-E. W. Sundberg and N. Seshadri, \Coded modulation for fading channels: An overview," European Transactions on Telecommunications, Vol. 4, No. 3, pp. 325{334, May-June 1993. [14] L. Ozarow, S. Shamai, and A. D. Wyner, \Information Theoretic Considerations for Cellular Mobile Radio," IEEE Trans. on Veh. Tech., Vol. 43, No. 2, May 1994. [15] F. Adachi and M. Sawahashi, "Error Analysis of MDPSK/CPSK with Diversity Reception under Very Slow Rayleigh Fading and Cochannel Interference," IEEE Trans. Veh. Tech., vol. 43, pp. 252-263, May 1994. [16] T. Matsumoto and F. Adachi, \Performance Limits of Coded Multilevel DPSK in Cellular Mobile Radio," IEEE Trans. Veh. Tech., vol. 41, pp. 329-336, Nov. 1992. [17] T. Matsumoto and F. Adachi, \Combined Convolutional Coding/Diversity Reception for QDPSK Land Mobile Radio," IEICE Transactions, Vol. E. 74, No. 6, June 1991. [18] N. Beaulieu and A. Abu-Dayya \Bandwidth ecient QPSK in Cochannel Interference and Fading," IEEE Trans. Commun., vol. 43, n. 9, Sept. 1995. [19] P. Ho and D. K. P. Fung, \Error performance of interleaved trellis coded PSK modulations in correlated Rayleigh fading channels," IEEE Trans. Commun., vol. 40, n. 12, Dec. 1992. [20] D. C. Cox, \Cochannel Interference Considerations in Frequency Reuse Small-Coverage-Area Radio Systems," IEEE Trans. Commun., vol. 30, pp. 135-142, Jan. 1982. [21] V. H. MacDonald, \The Cellular Concept," Bell Syst. Tech. J., vol. 58, pp. 15-41, Jan. 1979. [22] L. F. Fenton, \The sum of log-normal probability distributions in scatter transmission systems," IRE Trans. Commun. Syst., vol. 8, pp. 57-67, Mar. 1960. [23] S. C. Schwartz and Y. S. Yeh, \On the Distribution Function and Moments of Power Sums With Log-Normal Components," Bell Syst. Tech. J., vol. 61, n. 7, pp. 1441-1462, Sept. 1982. [24] J. Ventura-Traveset, G. Caire, E. Biglieri and G.Taricco, "Impact of diversity reception on fading channels with coded modulation. Part I: Coherent detection," submitted to IEEE Trans. Commun., Sept. 1995. [25] || "Impact of diversity reception on fading channels with coded modulation. Part II: Dierential block detection," submitted to IEEE Trans. Commun., Sept. 1995. [26] || "Impact of Diversity Reception on Fading Channels with Coded Modulation. Part III: Cochannel Interference," submitted to IEEE Trans. Commun., Nov. 1995. [27] E. Biglieri, G. Caire, G. Taricco, and J. Ventura, \Simple method for evaluating error probabilities," Electronic Letters, vol. 32, n. 3, Feb. 1996, pp. 191-192.

18

Coherent detection (perfect CSI)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.44 0.20 0.12 0.080 0.27 0.10 0.052 0.032 0.074 0.016 0.0065 0.0035 0.034 0.0060 0.0022 0.0011

Dierential detection (single-interferer assumption)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.71 0.49 0.40 0.35 0.54 0.32 0.24 0.20 0.23 0.093 0.059 0.046 0.13 0.044 0.026 0.020

Dierential detection (Gaussian approximation)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.65 0.41 0.30 0.23 0.48 0.24 0.16 0.12 0.18 0.061 0.033 0.021 0.099 0.027 0.013 0.0081

Table 1: Outage probability versus frequency reuse factor Kf and diversity order M . Results obtained for a channel cut-o rate R0 = 1 bit/s/Hz.

19

Coherent detection (perfect CSI)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.48 0.22 0.13 0.085 0.31 0.11 0.055 0.034 0.088 0.018 0.0073 0.0038 0.042 0.0068 0.0024 0.0012

Dierential detection (single-interferer assumption)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.74 0.51 0.41 0.36 0.57 0.33 0.25 0.20 0.25 0.099 0.063 0.048 0.15 0.047 0.028 0.021

Dierential detection (Gaussian approximation)

Kf 3 4 7 9

Outage probability PO M =1 M =2 M =3 M =4 0.68 0.42 0.31 0.24 0.51 0.26 0.17 0.12 0.21 0.066 0.035 0.022 0.11 0.030 0.014 0.0086

Table 2: Outage probability versus frequency reuse factor Kf and diversity order M . Results obtained for code Q64 (nominal spectral eciency: 1 bit/s/Hz) at P b = 10?3 .

M 1 2 3 4

CPSK 1.58
10?1 (Kf = 7) 2.50
10?1 (Kf = 4) 3.35
10?1 (Kf = 4) 3.92
10?1 (Kf = 4)

DPSK (single-interferer) 9.88
10?2 (Kf = 9) 1.55
10?1 (Kf = 9) 1.86
10?1 (Kf = 7) 2.15
10?1 (Kf = 7)

DPSK (Gaussian approx.) 1.11
10?1 (Kf = 9) 1.78
10?1 (Kf = 7) 2.25
10?1 (Kf = 7) 2.54
10?1 (Kf = 7)

Table 3: Normalized capacity for 4PSK schemes. These values correspond to the largest value of R0 =Kf at PO = 0:1 over the set of frequency reuse factors Kf = 3; 4; 7; 9. 20

1

Kf=3 Kf=4

Kf=7

Kf=9

FC/I(β)

0.1

0.01

0.001 0

10

20

30

β (dB)

Figure 1: FC=I ( ) for dierent values of frequency reuse factor Kf and sh = 6 dB.

21

1

Kf=3

FC/I(β)

Kf=4

Kf=7 Kf=9

0.1

0.01 0

10

20

30

β (dB)

Figure 2: FC=I ( ) for dierent values of frequency reuse factor Kf and sh = 8 dB.

1

Kf=3

0.5

FC/I(β)

Kf=4 Kf=7 Kf=9 0.2

0.1 0

10

20

30

β (dB)

Figure 3: FC=I ( ) for dierent values of frequency reuse factor Kf and sh = 12 dB. 22

x` Pulse k Source m-k Coded x- Shaping Interleaver Generator Modulator x(t) 8 1 g (t) ? gM (t) ? T > >  R > > A C> h1 (t) ? > > ? N H> + > S > .. ... < CCI . M A N ? I N> + > S E> hM (t) 6 S L> > n1 (t) ? ?nM (t) I > + + > O > > y1 (t) yM (t) N :

m m



m m








m

m m

?

?








Viterbi rk Deinterleaver r` Decoder

C O M B I N I N G

y`

m



Matched Filtering

1

M

y`

...

Figure 4: Model of a coded transmission system with fading, diversity, and linear combining, aected by CCI and AWGN.

23

100 10-1 10-2 10-3

BER

10-4 M=1

10-5 10-6 10-7 M=2

10-8 10-9

M=4

M=3

10-10 0

10

20

30

C/N

Figure 5: BER vs C=N of Q64 with diversity orders M = 1; 2; 3; 4 for C=I = 10 dB and coherent detection.

100 10-1 10-2 M=1 10-3

BER

10-4 10-5 10-6 M=2

10-7 10-8 M=4

10-9

M=3

10-10 0

10

20

30

C/N (dB)

Figure 6: BER vs C=N of Q64 with diversity orders M = 1; 2; 3; 4 for C=I = 10 dB and dierential detection (Continuous curves: single interferer. Dotted curve: Gaussian approximation). Simulations points are also shown (Hollow: single interferer and Gaussian approximation. Filled: N = 6 equal-energy interferers). 24

100

BER

10-5

10-10

10-15

10-20 0

10

20

30

C/I (dB)

Figure 7: Asymptotic BER (error oor) vs C=I of Q64 with diversity orders M = 1; 2; 3; 4. Solid curves: coherent detection. Dotted: dierential detection (single interferer). Dot-dashed: dierential detection (Gaussian approximation).

PO

1

0.1

M=1 M=2 M=3 M=4

0.01 0

1

2

3

R0 (bit/s/Hz)

Figure 8: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3 with coherent detection. 25

1

PO

0.1

0.01

M=1 M=2 M=3 M=4

0.001

0.0001 0

1

2

3

R0 (bit/s/Hz)

Figure 9: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9 with coherent detection.

1

PO

0.5

M=1 M=2 M=3 M=4

0.2

0.1 0

1

2

3

R0 (bit/s/Hz)

Figure 10: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3 with dierential detection (single interferer). 26

1

PO

0.1

M=1 M=2 M=3 M=4

0.01

0.001 0

1

2

3

R0 (bit/s/Hz)

Figure 11: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9 with dierential detection (single interferer).

PO

1

0.1

M=1 M=2 M=3 M=4

0.01 0

1

2

3

R0 (bit/s/Hz)

Figure 12: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3 with dierential detection (Gaussian approximation). 27

1

0.1

PO

M=1 M=2 M=3 M=4

0.01

0.001 0

1

2

3

R0 (bit/s/Hz)

Figure 13: Outage probability vs system cut-o rate R0 for dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9 with dierential detection (Gaussian approximation). 100 M=1

PO

M=4 10-1

10-2 10-5

10-4

10-3

10-2

10-1

Pb

Figure 14: Outage probability vs system BER (P b ) for code Q4, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation). 28

100

M=1

PO

10-1

10-2

M=4

10-3 10-5

10-4

10-3

10-2

10-1

Pb

Figure 15: Outage probability vs system BER (P b ) for code Q4, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation). 100

PO

M=1

10-1

10-2 10-5

M=4

10-4

10-3

10-2

10-1

Pb

Figure 16: Outage probability vs system BER (P b ) for code Q16, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation). 29

100

M=1

PO

10-1

10-2

M=4 10-3 10-5

10-4

10-3

10-2

10-1

BER

Figure 17: Outage probability vs system BER (P b ) for code Q16, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation). 100

PO

M=1

10-1

10-2 10-5

M=4

10-4

10-3

10-2

10-1

Pb

Figure 18: Outage probability vs system BER (P b ) for code Q64, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 3. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation). 30

100

M=1

PO

10-1

10-2

M=4 10-3 10-5

10-4

10-3

10-2

10-1

BER

Figure 19: Outage probability vs system BER (P b ) for code Q64, dierent diversity orders (M = 1; 2; 3; 4) and Kf = 9. Solid curves: dierential detection (single interferer). Dotted: coherent detection. Dot-dashed: dierential detection (Gaussian approximation).

31