Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
Unified Multiple-Access Performance Analysis of Several Multi-rate Multicarrier SpreadSpectrum Systems R. Nikjah, and M. Nasiri-Kenari, Member IEEE Electrical Department of Sharif University of Technology, Tehran, Iran, PO. Box: 11365-9363, Emails:
[email protected];
[email protected] Abstract—A unified multiple-access performance analysis and comparison of three multicarrier spread-spectrum multiple-access schemes, namely, MC-CDMA (multicarrier code-division multipleaccess), MC-FH (multicarrier frequency hopping), and a hybrid of the above systems, called DSMC-FH (direct-sequence MC-FH), in a multi-rate environment, where each user can have several multi-rate services, is provided. In MC-CDMA and MC-FH systems, users and their diverse services are differentiated by means of only one kind of signature code. However, in a DS-MC-FH scheme, different users and different services of the same user are distinguished through the first and second signature codes, respectively. We evaluate and compare the performance of the above systems using a unified structure in synchronous and asynchronous nonfading and synchronous correlated Rayleigh fading channels with a maximum-ratio combining (MRC) receiver. We also investigate the near-far effect on the systems’ performance. The (second) signature in MC-CDMA (DS-MC-FH) scheme is considered to be either a pseudo-noise (PN) sequence or a Walsh code. Our analyses indicate that MC-CDMA systems with Walsh codes outperform the other schemes in different synchronous and asynchronous channels. DS-MC-FH systems with Walsh codes always surpass MC-FH systems. Furthermore, all of the schemes, except synchronous MC-CDMA systems with Walsh codes, are susceptible to near-far effect with an MRC receiver. Index Terms―Unified error analysis, Code-division multiple-access, Multicarrier frequency hopping, Synchronous channel, Asynchronous channel, Correlated Rayleigh fading.
Part of this paper has been presented in IEEE MWCN 2002 Conference, Stockholm, Sweden, 911 September.
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
I. INTRODUCTION In combination with code-division multiple-access (CDMA) techniques, multicarrier modulation has attracted a lot of attention in the past decade for the future-generation wireless communications on account of countering channel frequency selectivity and removing inter-symbol interference (ISI) while supporting high-rate applications, providing frequency diversity, collecting the entire energy spread in the frequency domain, and simple implementation through fast-Fourier-transformation (FFT) techniques [1]. On the one hand, different configurations of multicarrier CDMA (MC-CDMA) schemes as combinations of direct-sequence CDMA (DS-CDMA) and orthogonal frequency-division multiplexing (OFDM) were developed after 1993 [1][2]. The performance and design of such systems have been investigated extensively in different nonfading and fading channels since then [28]. However, limited research exists in the area of asynchronous MC-CDMA (for some results, see [6-8]). In this paper, we consider MC-CDMA systems as introduced in [2]. On the other hand, frequency-hopping spread-spectrum (FH-SS) techniques in combination with OFDM or MC-CDMA received considerable attention recently, and as a result, various multicarrier frequency-hopping (MC-FH) systems were proposed [9-11][13]. MC-FH schemes, on account of fewer subcarriers transmitted in each symbol interval, have smaller peak-to-average-power ratio (PAPR) than MC-CDMA systems, making the implementation of MC-FH systems less complex than MC-CDMA schemes especially in the uplink, where linear amplification with a large dynamic range at the transmitter side is not viable. The MC-FH system studied in this paper is the one described in [13][14], wherein the frequency spacing between diversity hopping subcarriers in distinct frequency subbands is implemented in a way to diminish the correlation of fading gains on different subcarriers, while keeping the region of hopping for a single subcarrier so small that phase-shift keying (PSK) modulation and coherent detection are practically feasible [13]. This scheme was developed from a frequency-diversity spread-spectrum system, called FD-SS [12], for countering band-limited jamming interference [13]. It has been examined in a single-user fading channel [14], as well as in multiuser nonfading and fading channels with and without coding [15]. In this paper, a multi-rate environment, depicted in Fig. 1, is introduced in which each user has two signature codes, the first of which serves to distinguish the user from the other N u − 1 users, and the second discriminates ξ services of the same user. The results of exploring such environments are 1
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
also directly applicable to any system supporting independent groups of users in which different groups can be viewed as virtual users, and the users of each group can be interpreted as different services of the same virtual user. separated by the first signature code
separated by the second
user 0
user 1
user Nu − 1
service 0 service 1
service 0 service 1
service 0 service 1
service ξ - 1
service ξ - 1
service ξ - 1
signature code
Fig. 1) Distinguishing between different users and services by the first and second signature codes.
In MC-CDMA and MC-FH systems, there is only one kind of signature code serving as both the first and the second signatures. This explains the rationale for proposing a third multicarrier system as a hybrid of the previous two, called DS-MC-FH (direct-sequence MC-FH), possessing as its first and second signature codes the kind of signatures in MC-FH and MC-CDMA schemes, respectively. The first signature, as in a MC-FH system, determines the set of subcarriers allotted to a user, each belonging to one frequency subband. The second signature, the length of which is equal to the number of subcarriers in each transmission interval, modulates the subcarriers selected by the first signature, as in a MC-CDMA system, providing differentiation between services of the same user. Fig. 2 shows the transmission frequency pattern in a DS-MC-FH system. The whole frequency band is partitioned into N s subbands, each of which has exactly N h subcarriers. d i( k , q ) is the i th data bit N −1
(k , q ) s of the q th service of user k. T is the symbol duration excluding any guard interval. {h m , i }m = 0 and N −1
(k , q ) s {c m }m = 0 are the first and second signatures of the q th service of user k at transmission interval ( k ,q ) ( k ,q ) i, respectively. Contrary to hm ’s are identical for different data bit intervals. Fig. 2 is ,i ’s, c m (k , q ) simply particularized to thoroughly describe MC-CDMA systems, if we set all h m , i ’s to 0 and N h (k , q ) to 1; and MC-FH schemes, if we set all c m ’s to 1. Note that in contrast to a DS-MC-FH scheme, (k , q ) hm , i ’s are independent for different q’s in a MC-FH system. In fact in a MC-FH system, there is
only one kind of signature code utilized for differentiating users and their different services alike, (k , q ) resulting in independent h m , i ’s for different k’s and q’s. However, in a DS-MC-FH scheme, the
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
first signature is identical for all services of the same user, and the services of each user are (k , q ) distinguished by the second signature code {c m }.
second siganture code di( k ,q )c 0( k ,q )
di( k ,q )c(1k ,q )
f0
di( k ,q )c(Nk ,q−1) s
f2
f1
f0 + h 0(,ki,q ) / T
f N s −1
f1 + h1(, ki ,q ) / T
f f N s −1 + hN( k−,q1,)i / T s
first signature code Fig. 2) Transmission frequency pattern for service q of user k at the ith symbol interval.
Fig. 3 illustrates the differences between the three systems having the same bandwidth with two services for each user ( ξ = 2 ). In MC-CDMA (Fig. 3-a), all services of all users share the entire frequency band with the frequency spacing between adjacent subcarriers the same as that in OFDM. ( k ,q ) Different services are discriminated through distinct c m ’s. In MC-FH (Fig. 3-b), all the services
carried in the system are differentiated by means of assigning to each an independent set {hm( k,i,q ) } , called a hopping pattern, that select from different frequency subbands the subcarriers on which (k ) d i( k , q ) is transmitted. In the DS-MC-FH (Fig. 3-c) scheme, however, the hopping pattern {hm ,i } is
common to all services of the same user. c 0( k ,1)
c 5( k ,1)
( k ,1) c11
c 5( k , 0)
( k ,0) c11
d i( k ,1)
c 0( k ,0)
d i( k ,0)
f0
f1 f 2
f3 f4 f5 f6 f7
f 8 f 9 f 10 f 11
f
(a)
d i( k ,1) d i( k ,0)
f0
f0 + h0,( ki ,0) /T
f0 + h0,( ki ,1) /T
f1 f1 + h1,( ki ,1) /T
f2 f1 + h1,( ki ,0) /T
(b)
3
f f2+ f2+ ( k ,0) ( k ,1) h2, h2, / T i /T i
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
c 0( k ,1)
c1( k ,1)
c 2( k ,1)
c 0(k ,0)
c1( k ,0)
c 2( k ,0)
d i( k ,1) d i( k ,0)
f0
f0 + (k ) h0, i /T
f1
f1 + h1,( ki ) /T
f2
f f2+ (k ) h2, i /T
(c) Fig. 3) Transmission of two coded services over 12 /T bandwidth: a) MC-CDMA with 12 subcarriers; b) MC-FH with three frequency subbands each containing four subcarriers; c) DS-MC-FH with three frequency subbands each containing four subcarriers.
It is worth noting that in all three schemes, the whole modulated subcarriers of the i th data bit of all the same user’s services are added together in order to be transmitted simultaneously in one bit interval. It is also assumed that each symbol, like OFDM symbols, has head and tail guard intervals with cyclic extension in the intervals so as to avoid ISI problems. In this paper, we evaluate the multiple-access performance of the above three multicarrier schemes in synchronous and asynchronous nonfading and in synchronous correlated Rayleigh fading channels considering near-far effects, and compare the results. Even though only the use of pseudo( k ,q ) } in MC-CDMA and DS-MC-FH systems is noise (PN) and Walsh codes for the signature {c m
addressed, our analysis can be utilized to easily include any other code. The unified approach adopted for the performance analysis in this paper is based on the characteristic function method, applied previously to the performance evaluation of MC-FH, DS-CDMA, ultra-wideband impulse radio and MC-CDMA by [15-17] and [8], respectively. To the best of our knowledge, this paper is the first one considering and comparing the multiple-access performance of the two well-known schemes, namely MC-CDMA and MC-FH, and a generalization of these two schemes, called DSMC-FH, in various multi-rate channels. Our results indicate that MC-CDMA schemes with Walsh codes outperform other systems in all cases under consideration. Also, DS-MC-FH systems with Walsh codes always surpass MC-FH systems. Note that although frequency hopping schemes do not show any performance superiority over MC-CDMA systems, in some applications, such as military and jamming environments, they are much more plausible. The paper is organized as follows. Section II describes the unified structure and expressions of the above schemes. Sections III and IV provide performance analyses of the systems in nonfading 4
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
and fading channels, respectively. Numerical results are presented in Section V, and finally, Section VI presents the conclusions.
II. SYSTEM DESCRIPTION MC-CDMA, MC-FH and DS-MC-FH systems can be described in a unified way, provided that, referring to Fig. 2, the following equalities are taken into account for each system: (k , q) hm , i = 0 , for all k, q, m, and i, and N h = 1 ,
MC-CDMA:
(k , q) cm = 1 , for all k, q, and m,
MC-FH:
(k , q ) (k ) hm , i = h m, i , for all k, q, m, and i.
and DS-MC-FH:
(1-a) (1-b) (1-c)
(k , q) (k , q) Note that h m = 1 mean the lack of the frequency hopping signature and MC, i = 0 and c m
CDMA signature codes, respectively. Also it is noticeable that in a MC-CDMA scheme, each (k , q ) frequency subband is reduced to a single frequency subcarrier. h m , i ’s, for different k’s, q’s, m’s
and i’s in MC-FH, and for different k’s, m’s and i’s in DS-MC-FH, are assumed to be independent identically distributed (iid) random variables with uniform distribution over the integer interval N −1
(k , q ) s [0, N h − 1] . {c m }m = 0 ’s, in MC-CDMA and DS-MC-FH, are supposed to be either PN or Walsh (k , q ) ’s are iid random variables taking on values −1 and +1 with equal codes. In the former case, c m N −1
(k , q ) s probability. In the latter case, {c m }m = 0 ’s, for different k’s and q’s, are different Walsh codes.
Let f 0 through f N s −1 be the base frequencies of non-overlapping frequency subbands (see Fig. 2). For our analysis, we assume that the whole frequency band is contiguous; i.e,
f m 2 − f m1 = (m 2 − m1) Nh / T , where
0 ≤ m1 < m 2 ≤ N s − 1 .
(2)
The whole occupied bandwidth by the system is thus obtained as B = BW / T , where BW = N s N h is the normalized bandwidth. Hence, considering (1-a), for a constant bandwidth as it can be seen in Fig. 3, MC-CDMA systems have N h times as many transmitted subcarriers in a symbol interval as the other two systems do, where it is supposed that both the hopping schemes have N h subcarriers in each frequency subband. In all of the following analyses, without any loss of generality, we assume that each service of each user has a bit rate of 1 / T . Multi-rate services with bit rates higher than 1 / T are assigned more than one signature codes.
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
A. Transmitted Signal The baseband binary-phase-shift-keying (BPSK) transmitted signal for the q th service of user k can be expressed as follows:
s l(k , q) (t ) = d (ik , q )
Ns −1
∑∑
i m=0
(k , q) (k ) (k ) (k , q) 2S i(k ) d i(k , q)c m exp{ j [2π (t − t t(k ) − iT )( fm + h m , i / T ) + θ i ]} p(t − t t − iT ) . (3)
is the data bit of the q th service of user k at the i th symbol interval. d (ik , q ) ’s, for different k’s,
q’s and i’s, are iid random variables, and take on values −1 and +1 with the same probability. S i( k ) is the transmission power for each subcarrier of user k at symbol interval i. t t( k ) and θ i( k ) are the k th user’s transmitter time origin, and the k th user’s transmission phase at the i th interval, respectively.
p (t ) is the unit pulse in the time interval (0, T ) . B. Model of Channel We consider an additive-white-Gaussian-noise (AWGN) nonfading or a slowly correlated Rayleigh fading channel. The former case can be achieved from the latter by setting all fading coefficients to one. Therefore, we concentrate on fading channels. The length of each symbol is conventionally considered to be so much greater than the maximum delay spread of the channel that the presumption of flat fading over each subcarrier is satisfied. As a result, the m th subcarrier of the
q th service of user
k at the i th symbol interval experiences a flat fading gain
g (mk ,, qi ) = H ( k ) ( f m + h (mk ,, iq ) / T ) which is assumed to be constant over at least one symbol interval.
H ( k ) ( f ) is the baseband frequency response of the transmission channel for the k th user. If g (ik , q ) is N −1
defined as a column vector with components [ g (mk ,,iq ) ]m s= 0 , then g (ik , q ) is a proper complex Gaussian (PCG) random vector characterized by its probability-density function (pdf) f
(k , q) ( g)
gi
=
exp(− g HC −1 g )
π N s det(C )
(“H” denotes Hermitian operation, and “det” stands for determinant)
[18], where C defined as E{ g (ik , q ) g (ik , q )H } ( E{} ⋅ denotes expectation) is the frequency covariance matrix of the channel. Assuming that the channel is wide-sense stationary uncorrelated scattering (WSSUS), and the frequency band of the system is contiguous, the entries of C are obtained as C m 1 , m 2 = Φ H ( ∆f m 1 , m 2 ) = ∫
+∞
−∞
φ h (τ ) exp(− j 2 π ∆f m1 , m 2τ ) dτ with ∆f m1 , m 2 = N h (m1 − m2 ) / T , (4) 6
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
where φ h (τ ) and Φ H (f ) are the power delay profile (pdp) and the frequency autocorrelation
function of the channel, respectively. ∆f m1 , m 2 is the (averaged) frequency spacing between the m 1th and m 2th subcarriers. We adopt the following exponential model for the pdp [14]:
φ h (τ ) =
µ / Tm [exp(− µτ / Tm ) − exp(− µ )] 1 − (1 + µ ) exp(− µ )
0 < τ < Tm ,
for
(5)
where µ is a decaying factor, and Tm is the maximum delay spread of the channel. C. Received Signal, Receiver Structure, and Decision Variable
The baseband received signal can be written as r l (t ) = ∑ i
Nu −1 ξ −1 N s −1
∑ ∑ ∑
k =0 q = 0 m = 0
(k ,q ) (k , q) (k , q ) 2 R (ik ) g m c m exp{ j [2π (t − t (rk ) − i T )( f m + h(mk,,iq ) / T ) + ,i d i
,
(6)
θ (i k ) ]} p(t − t (rk ) − iT ) + z (t )
where N u is the number of active users. R (ik ) denotes the received power at each subcarrier of user k at the i th symbol interval. t r( k ) is the receiver time origin and is equal to t t( k ) + td( k ) , where td(k ) is (k , q ) 2 the reception delay for the k th user. g (mk ,,iq ) ’s are assumed to be so normalized that E{| g m ,i | } = 1 .
z (t ) is the complex zero-mean AWGN with double-sided power spectral density N 0 .
We consider a single-user maximum-ratio-combining (MRC) receiver as shown in Fig. 4 with (k , q) (k , q) * ( k , q ) wm cm , where “∗ ” stands for conjugation. Let the signal of the 0 th service of user 0 , i = gm , i
exp{− j [2 π (t − t r( k ) − iT ). (k , q) (k ) ( f 0 + h 0, i / T ) + θ i ]}
×
(k ) 1 t r + (i +1) T (⋅) dt (k ) T t r + iT
∫
(k , q) w 0, i
×
exp{− j [2 π (t − t r( k ) − iT ). (k , q)
( f 1 + h1, i rl(t )
×
(k )
/T ) +θ i
]}
(k ) 1 t r + (i +1) T (⋅) dt T t r( k ) + iT
∫
w1,( ki, q )
×
Σ
+1 ˆ Re{} ⋅ D 0 d −1
exp{− j [2 π (t − t r( k ) − iT ). (k , q) (k , q) ( f N s −1 + h N / T ) + θ i( k ) ]} w N s −1, i i 1, − s
×
(k ) 1 t r + (i +1) T (⋅) dt (k ) T t r + iT
∫
×
Fig. 4) Baseband equivalent MRC receiver for the qth service of user k at the ith symbol interval.
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
at the 0 th symbol interval be the signal of interest. Without any loss of generality, we assume that
θ 0(0) = 0 and t r(0) = 0 . Also, we suppose that t r( k ) belongs to the interval [0, T ) . With these assumptions, the following expression for the decision variable normalized by D =d
(0, 0) 0
N s −1
∑
m =0
2R(0) 0 is attained:
2
(0,0) gm + ,0
(0, q ) (0, 0) ξ −1 N s −1 hm hm ,0 ,0 (0,0)* (0, q ) (0, q ) (0, 0) (0, q ) 1 T − Re ∑ ∑ g m ,0 g m ,0 d 0 c m c m exp[ j 2π ( )]dt + ∫ 0 T T T q =1 m = 0 N u −1 ξ −1 N s −1 N s −1 (0,0)* ( k , q ) ( k , q ) ( k , q ) (0, 0) Re ∑ ∑ ∑ ∑ R −( k1 ) / R (0) . cv 0 gv ,0 g m , −1 d −1 c m k =1 q = 0 m = 0 v = 0 (k ,q ) 0) hm h v(0, 1 t r( k ) , −1 ,0 (k ) exp{ j 2π [(f m + )(t − t r ) −(f v + ) t ] + j θ −( k1 ) }dt + T ∫0 T T (0,0)* ( k , q ) ( k , q ) ( k , q ) (0, 0) R (0k ) / R (0) c m cv . 0 gv ,0 g m ,0 d 0
, (7)
(k , q ) 0) hm h v(0, 1 T ,0 ,0 (k ) (k ) )(t − t r ) − (f v + ) t ] + j θ 0 }dt + n ( k ) exp{ j 2π [(f m + ∫ T tr T T
where Re{} ⋅ stands for the real part. Also, n represents the noise term, which, conditioned on the (0 , 0) fading coefficients g m , 0 ’s, is a zero-mean Gaussian random variable with variance equal to
σ
2
(0,0)
n| g m ,0
= ( Ns
N s −1
∑
m =0
(0, 0) gm ,0
2
) / 2γ b , where
γ b = N sγ bs
with
γ bs =
R(0) 0 T N0
.
(8)
γ bs is the received signal-to-noise ratio (SNR) per bit per diversity branch and γ b is the total received SNR per bit. In deriving (7), we assume that f m .T for any m is an integer. D. Detection and Probability of Bit Error
Detection at the receiver is accomplished by comparing the decision variable D with the threshold zero. Consequently, the bit-error rate (BER) of the system is obtained as 0) Pbe = Pr{D < 0 | d (0, = +1} = FU (0) , where 0
U
D | +1 .
(9)
FU (u ) stands for the cumulative-distribution function (cdf) of U , and D | +1 denotes the decision
variable D conditioned on transmitting +1 .
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
III. PERFORMANCE EVALUATION IN NONFADING CHANNELS A. Synchronous Systems (k , q) (k ) (k) For a synchronous nonfading system, we have g m = 0 , for any k, q, m, , i = 1 , t r = 0 , and θ i
and i. Now, from (7) and (8) with the definition of U in (9) we obtain
where MAI
(0)
=
ξ −1
∑ MAI (0, q) , MAI (k ) =
q =1
MAI ( k , q ) =
N s −1
∑
m =0
in which
MAI =
with
U = N s + MAI + n
Nu −1
∑
MAI ( k ) ,
ξ −1
∑ MAI (k , q)
for k ≥ 1 , and
q =0
( k , q ) (0, 0) ( k , q ) ( k , q ) R (0k ) / R (0) c m c m η m, 0 for any k and q, 0 d0
(k , q ) (k , q) (0, 0) ηm , i = δ [ h m, i − h m, 0 ] ;
(10-b)
(10-c)
σ n2 = N s2 /(2γ b ) .
also,
(10-a)
k =0
(10-d)
MAI ( k ) is the interference due to user k, and MAI ( k , q ) is the interference term caused by the q th service of user k. n in (10-a) is the Gaussian noise term with zero mean and variance that can be (k , q ) obtained from (10-d). δ in (10-c) indicates the Kronecker delta. As the h m , i ’s are iid random (k , q) variables with uniform distribution over the integer interval [0, N h − 1] , from (10-c) the η m , i ’s are
iid random variables with the following distribution: 0, with probability α 1, with probability β
(k , q ) ηm ,i =
1 − 1/ N h . 1/ N h
(11)
From (10), assuming that R (0k ) = R for any k, MAI is a discrete random variable with the pdf given by f MAI ( x) =
P
∑
a =− P
pMAI [a ]δ ( x − a)
with
P = N s ( Nuξ − 1) ,
where pMAI [a] is the probability function of MAI. From (9) and (10), we can write Pbe = Pr{MAI + n < − N s } .
(12)
(13)
The pdf of the “interference plus noise” is obtained as
f( MAI + n) ( x) = f MAI ( x) ∗ f n ( x) =
P
∑
a =− P
pMAI [a] f n ( x − a) ,
where f n ( x) is the pdf of n. Now, we can acquire the BER from (13) and (14) as follows: P − Ns N +a ), Pbe = ∫ f ( MAI + n) ( x) dx = ∑ pMAI [a]Q( s −∞
a =− P
9
σn
(14)
(15)
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
in which Q( x) = ∫
+∞
x
1 exp(− w2 / 2) dw is the Gaussian-tail function. In the Appendix, using the 2π
theorem proposed in [19], we prove that 1 2 L −1 M MAI (e j k π / L )e− j a k π / L pMAI [a ] = ∑ L k =1
with
L = 2P + 1 ,
(16)
k odd
where M MAI ( z ) is the moment-generating function (mgf) of MAI. The exact BER is then obtained using (15) and (16). Now for studying the near-far effect, we assume that R (0k ) = R for k ≥ 1 , and that R (0) 0 < R . In other words, all the interfering users’ signals are received with the same power which is greater than the desired signal power. From (10), supposing that A
1 R / R (0) 0 is an integer , we can follow the
same above steps to reach (15) and (16) except that P = N s [ξ − 1 + A ( Nu − 1)ξ ] . Also note that in this (k ) case MAI (0) is the same as the MAI (0) with R (0) for k ≥ 1 is A times the 0 = R , but MAI
MAI ( k ) with R (0) 0 = R. For the performance evaluation, we only need to calculate M
MAI ( k )
( z ) . As the MAI ( k ) ’s are
( z ) ’s. In the succeeding independent, M MAI ( z ) is easily acquired from the product of all M MAI ( k ) Sections, assuming that R (0) 0 = R , we evaluate M MAI ( k ) ( z ) for MC-FH and DS-MC-FH systems. M
MAI ( k )
( z ) for MC-CDMA schemes is derived from that of DS-MC-FH systems if we set N h = 1 .
A.1. MC-FH System
For this scheme, combining (1-b) and (10) results in MAI
(0)
=
ξ −1
∑ MAI
(0, q )
and
MAI
q =1
with MAI
(k , q)
= d 0( k , q )
N s −1
∑ η m(k,,0q) .
m=0
(k )
=
ξ −1
∑ MAI (k , q )
for k ≥ 1
(17)
q =0
MAI ( k , q ) conditioned on d 0( k , q ) is the sum of N s iid random
(k , q ) (k , q) variables, i.e. the d 0( k , q )η m ’s are , 0 ’s. Therefore, from (10) and (11) and that the MAI
independent, we have
1
Note that this assumption, not affecting the final result, is made only for computational simplicity, and does not impose any restriction on the subsequent analyses.
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M
ξ −1
MAI
(0) ( z ) = ∏ M
q =1
ξ −1
MAI
(0, q ) = −1}M (0,q ) ( z ) = ∏ {Pr{d 0 q =1
Pr{d0(0, q ) = +1}M
ξ −1
and in the same way,
=−1
(0, q )
=+1
MAI (0, q ) |d0
( z) + ( z )}
1 1 = ∏{ M ( z) + M ( z )} (0, q ) (0, q ) (0, ) q =−1 =+1 |d 0 MAI MAI (0, q ) |d 0 2 2 q =1 1 1 = { (α + β z −1 ) N s + (α + β z ) N s } ξ −1 2 2
1 1 ( z ) = { (α + β z −1 ) N s + (α + β z ) N s } ξ 2 2 where α = 1 − β is equal to 1 − 1/ N h . M
(0,q )
MAI (0, q ) |d0
MAI ( k )
,
for k ≥ 1 ,
(18)
A.2. DS-MC-FH System (k , q ) (k , q) In this system, from (1-c), h m , i , and thus η m, i defined in (10-c), do not depend on q, and are (k ) (k ) thereby represented by h m , i and η m , i , respectively. From (10-c), we can easily conclude that (0) ηm , i ’s are all equal to 1. Consequently, from (10) we can write
MAI
(0)
=
ξ −1
∑
q =1
and MAI
(k )
=
N s −1
∑
m =0
(k ) MAI m
=
d 0(0, q )
N s −1
ξ −1
m=0
q =0
∑
η m( k,)0
N s −1
∑
m =0
(0, 0) (0, q ) cm cm
∑ d 0(k , q)c m(0, 0)c m(k , q)
(19-a)
for k ≥ 1 .
(19-b)
PN Second-Signature Codes: In this case, MAI (0) , given in (19-a), consists of (ξ − 1) N s iid
random variables taking on values −1 and +1 with the same probability. Therefore, we have M
MAI
(0) ( z ) = (
z −1 z (ξ −1) N s + ) . 2 2
(20)
(k ) From (11) and (19-b), it is apparent that MAI m is zero with probability α = 1 − β = 1 − 1/ N h ;
otherwise, it is equal to the sum of ξ independent uniformly distributed binary-valued (with values (k ) −1 and +1 ) random variables. Also, η m , 0 ’s are independent for different m’s. Therefore, we easily
obtain z −1 z ξ N s M ( z ) = {α + β ( + ) } MAI ( k ) 2 2
for k ≥ 1 .
(21)
Walsh Second-Signature Codes: For this case, as any two different Walsh codes are orthogonal,
MAI (0) (see (19-a)) completely vanishes. Now from (19-b), we have
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MAI ( k ) =
ξ −1
∑
MAI ( k , q )
with
MAI ( k , q ) =
q =0
N s −1
∑
m=0
(0, 0) ( k , q ) ( k ) d0( k , q ) cm cm ηm, 0
for k ≥ 1 .
(22)
We consider three cases. First, if Nu = 1 , then no multiple-access interference exists, and therefore, N Pbe = Pr{U < 0} = Pr{n < − N s } = Q( s ) . (23)
σn
Second, if Nu > 1 and ξ = 1 , we only need to calculate the mgf of MAI ( k ) for k ≥ 1 , which is equal N −1
N −1
( k , 0) s (0, 0) s to MAI ( k ,0) defined in (22). Let E 1 be the set of k’s ( k ≥ 1 ) for which [c m ]m = 0 = [c m ]m = 0 ,
with the cardinality of u, and E 2 be the complement of E 1 with respect to the set {1, 2, On one hand, MAI
( k∈E 1 )
, Nu − 1} .
has a form just similar to MAI ( k , q ) for MC-FH schemes (see(17)), so
from (18) we have M
MAI
( k ∈E 1 ) ( z )
=
1 1 (α + β z −1 ) N s + (α + β z ) N s . 2 2
(24)
N −1
(0, 0) ( k , 0) s On the other hand, for k ∈ E 2 , {d 0( k , 0) c m c m }m = 0 has an equal number of −1 ’s and +1 ’s. As (k ) ηm , 0 ’s for different m’s are independent with statistics given in (11), from (22) we have
M
MAI
( k ∈E 2 ) ( z )
= (α + β z −1 ) N s / 2 (α + β z ) N s / 2 .
(25)
Therefore from (24) and (25), we obtain N
s ( N −1− u ) u 1 Ns u −1 N s 1 −1 2 M MAI ( z ) = { (α + β z ) + (α + β z ) } {(α + β z )(α + β z )} . 2 2
(26)
Now, substituting (26) in (15) and applying the result to (16) leads to Pbe | u (the BER conditioned on u). Before investigating how to determine the unconditional bit-error rate Pbe from the conditional bit-error rate Pbe | u , we first consider the remaining case, i.e. Nu > 1 and ξ > 1 . For this case, as the exact analysis is very complicated and cumbersome, we utilize a Gaussian distribution approximation for the multiple-access-interference term according to the central-limit theorem (CLT). Under this assumption, using (13) we have: Pbe = Q(
Ns
σ ( MAI + n)
)
with
2 σ (2MAI + n) = σ MAI + σ n2 .
(27)
2 2 σ MAI is the variance of MAI, and σ n2 is obtained from (10-d). In order to evaluate σ MAI , we
assume that there are u distinct pairs (k , q ) for any of which the q th service of user k has the same Walsh code as the desired service. For such pairs, the mgf of MAI ( k , q ) has a form just the same as the one given in (24). The variance of MAI ( k , q ) is then obtained, via the second derivative of the 12
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
mgf evaluated at z = 1 , as N s β ( N s β + α ) . For the other services (with a different Walsh code from that of the desired service), the mgf of MAI ( k , q ) has a form exactly similar to the mgf in (25). Then, the variance of such MAI ( k , q ) ’s is acquired as N s βα . Consequently, we can write1 2 σ MAI | u = u{N s β ( N s β + α )} + {( Nu − 1)ξ − u}( N s βα ) .
(28)
(27) and (28) lead to Pbe | u , i.e. the BER conditioned on u. Note that when taking near-far effect into account, the variance in (28) is easily multiplied by A 2 , where A = R (0k ) / R(0) 0 . For extracting Pbe from Pbe | u , we assume that the Walsh codes are assigned to different services in the system in a cyclic fashion. In other words, noting that there are totally N s distinct Walsh codes used in the system, if e defined as kξ + q is a counting index for the services in the system, the services with indices e0 and e0 + N s have the same Walsh codes, and any N s services with sequential indices have distinct Walsh codes. With this assignment, and defining r as mod( Nuξ , N s ) , where “mod” denotes the remainder operation, we can consider three cases: 1) Nuξ ≤ N s implies that u = 0 , and Pbe = Pbe | (u = 0) . 2) ( Nuξ > N s and r = 0 ) implies that Pbe = Pbe | (u = Nuξ / N s − 1) . 3) ( Nuξ > N s and r ≠ 0 ) implies that u = u 1 = [ Nuξ / N s ] − 1 with
probability 1 − r / N s , and u = u 2 = [ Nuξ / N s ] with probability r / N s , where [ x ] denotes the largest integer not greater than x; therefore, Pbe = (1 − r / N s ) Pbe | u 1 + (r / N s ) Pbe | u 2 . B. Asynchronous Systems
For an asynchronous nonfading system, using (2), (7), and the definition of U in (9), we obtain MAI =
U = N s + MAI + n , where
which
MAI
(0)
=
ξ −1
∑ MAI
(0, q )
with
MAI
(0, q )
=
q =1
N s −1
∑
m=0
MAI ( k ) = MAI −( k1) + MAI 0( k ) for k ≥ 1 , where MAI −( k1) =
ξ −1
Nu −1
∑
MAI ( k )
(29-a)
k =0 q ) (0, 0) (0, q ) (0, q ) d (0, c m c m η m, 0 . 0
∑ MAI −(k1, q)
Also,
with
q =0
Note that MAI (0) is equal to zero, and we have a total of ( Nu − 1)ξ interfering services. u services have the same code as the desired service, and ( Nu − 1)ξ − u services have different codes from the desired service. 1
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MAI −( k1, q ) =
N s −1 N s −1
∑ ∑
m=0 v =0
q) MAI −( k1,, m , v , which
q) (k ) (0) ( k , q ) ( k , q ) (0, 0) MAI −( k1,, m cv . , v = R−1 / R 0 d −1 cm (k ) m = v, and (k , q ) (k ) (k ) for ( k , q ) (0, 0) τ cos{θ −1 − 2πτ (Tf m + hm, −1 )}, hm, −1 = hv, 0 . sin{θ ( k ) − 2πτ (k ) (Tf + h(0, 0) )} − sin{θ ( k ) − 2πτ (k ) (Tf + h( k , q ) )} v m v, 0 m, −1 −1 −1 , otherwise (k , q) (0, 0) + − − N h h } π 2 {( m v ) h m, −1 v, 0 (29-b)
Also, we have MAI 0( k ) =
ξ −1
∑
q =0
MAI 0( k , q ) with MAI 0( k , q ) =
N s −1 N s −1
∑ ∑
m=0 v =0
(k , q) MAI 0, m, v , which
(k , q) (k ) (0) ( k , q ) ( k , q ) (0, 0) MAI 0, cm cv . m, v = R 0 / R 0 d 0 m = v, and (k , q) (k ) (k ) (k ) for ( k , q ) (0, 0) (1 − τ ) cos{θ0 − 2πτ (Tf m + hm, 0 )}, hm, 0 = hv, 0 . sin{θ ( k ) − 2πτ ( k ) (Tf + h( k , q ) )} − sin{θ ( k ) − 2πτ ( k ) (Tf + h(0, 0) )} m v 0 m, 0 0 v, 0 , otherwise (k , q ) (0, 0) 2 {( m − v ) N + h π h } − h m, 0 v, 0 (29-c)
τ ( k ) in (29) is equal to tr( k ) / T , and is assumed to be uniformly distributed in the interval (0,1) . Also, the τ ( k ) ’s are independent. The θ i( k ) ’s for different k’s are independent uniformly distributed random variables in the interval (−π , π ) . MAI −( k1) and MAI 0( k ) are parts of MAI ( k ) related to −1 th and 0 th transmission intervals of user k, each consisting of MAI −( k1, q ) ’s and MAI 0( k , q ) ’s q) (interference terms due to different services of user k), respectively. MAI −( k1,, m , v is the interference
caused by the m th subcarrier of the q th service of user k at the −1 th symbol interval in the v th (k , q) th subcarrier of the desired service. MAI 0, m, v denotes the same interference caused by the 0 symbol
interval of user k. Note that the variance of the noise term in this case is the same as in the case of synchronous channels (see (10-d)). As the exact BER calculation for the asynchronous case is very complicated, we derive the BER based on the Gaussian distribution for the MAI using (27). Thus, we only need to compute the variance of the MAI which is obtained as 2 σ MAI =σ
2 + MAI (0)
Nu −1 ξ −1
2 2 ∑ ∑ (σ MAI (k , q) + σ (k , q) ) . MAI
k =1 q = 0
14
−1
0
(30)
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
Now, by variable changes τ ′( k ) = 1 − τ ( k ) and θ ′0( k ) = −θ 0( k ) in (29-c), and noting that τ ′( k ) and
θ ′0( k ) have the same statistics as τ ( k ) and θ0( k ) , respectively, and assuming that R−( k1) = R (0k ) and Tf m is an integer for any m, it can be easily realized that σ 2
(k , q) MAI −1
and σ 2
(k , q)
MAI 0
are equal. Hence,
from (30), in order to derive the BER of each system using (27), we only need to calculate σ 2
MAI (0)
and σ 2
(k , q) MAI −1
for R−( k1) = R (0) 0 , which are computed in the succeeding Sections. For considering
near-far effect, it is sufficient to multiply σ 2
(k , q) MAI −1
(k ) (0) (computed for R−( k1) = R (0) 0 ) by R−1 / R 0 .
B.1. MC-CDMA System
For this scheme, applying (1-a), (29-b) simplifies to q) MAI −( k1,, m ,v
τ ( k ) cos(θ ( k ) − 2πτ ( k )Tf ), for m = v m −1 (k ) (k ) (k ) − 2πτ Tf v ) − sin(θ −1 − 2πτ Tf m ) , otherwise 2π (m − v)
( k , q ) (0, 0) cv . sin(θ ( k ) = d −( k1, q ) cm −1
(31)
PN Signature Codes: Just similar to the synchronous case, MAI (0) in this case is equal to the sum of
(ξ − 1) N s independent uniformly distributed binary-valued random variables, and therefore, we have
σ
2 MAI (0)
q) = (ξ − 1) N s . From (31), it can simply be realized that MAI −( k1,, m , v ’s for different m’s and v’s
are uncorrelated, and therefore, from (29), σ 2
(k , q) MAI −1
From (31), after some simple statistical calculations, σ 2
is easily obtained as
σ
(k , q)
MAI −1, m, v
∑ ∑
m =0 v =0 (k , q)
MAI −1, m, v
2
N s −1 N s −1
σ
2
(k , q)
MAI −1, m, v
.
is acquired as
for m = v 1/ 6, 1 1 = , otherwise . 4π 2 (m − v)2
(32)
Walsh Signature Codes: In this case, just the same as the synchronous case, MAI (0) vanishes, and
σ
2 MAI (0)
q) = 0 . In contrast to the PN-signature case, MAI −( k1,, m , v ’s in (31) for different m’s and v’s are
correlated. After some lengthy calculations which are dropped here for briefness, the following result is easily attained:
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σ
2
(k , q)
MAI −1
=
N −1 N −1
( k , q ) (k , q) (0, 0) (0, 0) (0, 0) (0, 0) s Ns cm cv (2cm cv − 1) − cm cv +1 1 s + + ∑ ∑ 6 4π 2 m = 0 v = 0 (m − v)2 v≠m
1
N s −1 N s −1 N s −1
∑ ∑ ∑
( k , q ) (0, 0) ( k , q ) (0, 0) cv(0, 0) cv(0, 0) + cv( k , q ) cv( k , q ) + 4cm cm cv cv . 1
2
1
2
8π 2 m = 0 v 1 = 0 v 2 = 0
1
2
(33)
(m − v 1 )(m − v 2 )
v1 ≠ m v 2 ≠ m v 2 ≠ v1
B.2. Hopping Systems (MC-FH and DS-MC-FH)
In hopping systems, contrary to MC-CDMA schemes, the frequency spacing between neighboring subcarriers is so large1 that the subcarrier spectra can well be assumed to be disjoint. In other words, q) MAI −( k1,, m , v can well be assumed to be zero for m ≠ v . For shortness, hereafter we denote q) (k , q) MAI −( k1,, m , m by MAI −1, m . Now, from (29-b) we have
MAI −( k1, q )
with
=
N s −1
∑
m=0
q) MAI −( k1,, m
(34-a)
q) ( k , q ) (0, 0) MAI −(k1,, m = d −( k1, q ) cm cm . (k , q) (0, 0) τ ( k ) cos{θ ( k ) − 2πτ ( k ) (Tf + h( k , q ) )}, for hm = hm m 1 m , 1 , 1 ,0 − − − . sin{θ (k ) − 2πτ ( k ) (Tf + h(0, 0) )} − sin{θ ( k ) − 2πτ ( k ) (Tf + h(k , q ) )} m m m, 0 m, −1 −1 −1 , otherwise (k , q ) (0, 0) 2π (hm, −1 − hm, 0 ) (34-b)
DS-MC-FH with PN Second-Signature Codes: For this scheme, MAI (0) is just the same as in MC-
CDMA systems with PN codes, and therefore, σ 2
MAI
are uncorrelated, σ 2
(k , q) MAI −1
(0)
q) is obtained as (ξ − 1) N s . As the MAI −( k1,, m ’s
is equal to the sum of the σ 2
(k , q)
MAI −1, m
’s over all m’s. Knowing the
(k , q) (k , q) ’s, τ ( k ) ’s, and θ i( k ) ’s, and noting that hm statistics of the di( k , q ) ’s, h(mk,,iq ) ’s, cm , −1 is equal to (0, 0) hm , 0 with probability 1/ N h , it can easily be shown that
σ From (35), σ 2
(k , q) MAI −1, m
1
2
N −1 N −1
(k , q)
MAI −1, m
h h 1 1 1 }. = β ( ) + α{ ∑ ∑ 2 6 4π 2 N h a = 0 b = 0 (a − b) 2
(35)
b≠a
does not depend on m, so σ 2
(k , q) MAI −1
is obtained as N s .σ 2
(k , q)
MAI −1, m
This spacing in hopping systems on average is N h times greater than that in MC-CDMA schemes.
16
.
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DS-MC-FH with Walsh Second-Signature Codes: In this case, MAI (0) vanishes like the
synchronous case, and we only need to compute σ 2
(k , q)
MAI −1
σ
σ
2
2
(k , q) MAI −1
=
N s −1
∑
m=0
σ
2
(k , q) MAI −1, m
N s −1 N s −1
+
∑ ∑
m1 = 0 m 2 = 0 m 2 ≠ m1
, which from (34-a) is acquired as
q) q) E{MAI −( k1,, m .MAI −( k1,, m } . 1
(36)
2
is obtained just the same as in (35). However, in contrast to the PN-signature case,
(k , q)
MAI −1, m
q) MAI −( k1,, m ’s for different m’s are correlated, and from (34-b), after some simple statistical
calculations, we obtain q) q) E{MAI −( k1,, m .MAI −( k1,, m }= 1 2
( k , q ) (0, 0) ( k , q ) (0, 0) β cm cm cm cm 1
1
2
2
2
4π N h3 N h −1 N h −1 N h −1
{
N h −1 N h −1
∑ ∑
a =0 b=0
β Nh
{(m 1 − m 2 ) N h + a − b}2
+
α
∑ ∑ ∑ {(m − m ) N + a − b }{(m − m ) N + a − b }} 1 2 h 1 1 2 h 2 a = 0 b1 = 0 b 2 = 0
. (37)
b 2 ≠ b1
MC-FH: In this case, MAI (0) is the same as the corresponding term in synchronous MC-FH, and
thus, σ 2
MAI
(0)
is obtained as (ξ − 1) N s β ( N s β + α ) from (18). Applying (1-b) in (34-b), σ 2
(k , q)
MAI −1
is
(k , q) attained via (35)-(37) with cm ’s all set to one.
IV. PERFORMANCE EVALUATION IN FADING CHANNELS For the performance evaluation in fading channels, we consider synchronous single-service ( ξ = 1 ) systems. From (7) and (8) with R (0k ) = R (0) 0 for any k, we can write U = S + MAI + n ,
where
S=g
(0)H (0)
g
σ
and
2 n| g (0)
,
MAI = Re{I } ,
I=g
(0)H
Nu −1
∑
d ( k )V ( k ) g ( k ) ,
k =1
N = s g (0)H g (0) , 2γb
(38-a)
g (k )
with
V
σ
(k )
diag{v 0( k ) , v 1( k ) ,
[ g0(k ) g1(k )
, v (Nk )−1} , s
(k ) T gN ] , −1 s
and
v(mk )
(0) ( k ) ( k ) cm c m ηm .
(38-b)
2 is the variance of n conditioned on g (0) . In (38) and throughout this Section, as n| g (0)
ξ is equal
to 1 and all the parameters are related to 0 th symbol intervals, we have dropped the indices q and i. 17
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(k ) Note that ηm has the same definition and statistics as given in (10-c) and (11) with q = 0 and i = 0 . (k ) From (38), U conditioned on g (0) and vm ’s is a Gaussian random variable with the following mgf:
M
(k ) U | g (0) , vm
(s) = M
S | g (0)
( s) M
(k ) MAI | g (0) , vm
( s) M n (s ) .
(39)
Using the mgf of a Gaussian variable, from (38), (39), and that MAI is a zero-mean variable, we have (0)H [s I Ns + ( k ) ( s ) = exp{ g
M
U | g (0) , vm
s2 Ns s2 I N s ] g (0) + σ 2 (k ) } , 4 γb 2 MAI | g (0) , vm
wherein I N s is the identity matrix of order N s . σ 2
(k )
MAI | g (0) , vm
(40)
is obviously half of the variance of
(k ) I | g (0) , vm , defined in (38-a). Now, using the fact that g ( k ) ’s are independent, we can write
σ
N −1
2
(k )
MAI | g (0) , vm
where
Χ =
u 1 1 = E{I I H | g (0) ,V ( k ) } = g (0) [ ∑ d ( k )V ( k ) E{ g ( k ) g ( k )H }V ( k ) d ( k ) ] g (0)H 2 2 k =1 Nu − 1 1 , = g (0) ( ∑ V ( k )C V ( k ) ) g (0)H 2 k =1 1 (0) = g ( Χ i C ) g (0)H 2
Nu −1
∑
Χ (k )
and
k =1
Χ ( k ) [v0( k ) v1( k )
v(Nk )−1 ]T [v0( k ) v1( k ) s
v(Nk )−1 ] . s
(41) C and V ( k ) are defined in (4) and (38-b), respectively. “ i ” is the matrix-matrix dot product
operator, and the superscript “T” stands for transposition. Note that Χ ( k ) i C is in fact the covariance matrix of V ( k ) g ( k ) , conditioned on V ( k ) . As for any given V ( k ) , V ( k ) g ( k ) is a Gaussian random vector, we well approximate the unconditional distribution of V ( k ) g ( k ) as Gaussian distribution. (k ) Hence, removing conditionality on vm from (40), for calculating M (0) ( s ) , we only need to U|g
compute σ 2
MAI | g (0)
. From (41) we easily obtain
σ
1 (0) 2 (E{Χ } i C ) g (0)H . (0) = g MAI | g 2
(42)
From the definition of V ( k ) and Χ in (38-b) and (41), respectively, it is easy to show that E{Χ } = ( Nu − 1) αβ I N s + β 2 E{L LT } ,
(43)
(k ) where β = 1 − α = 1/ N h , and L is an N s × ( Nu − 1) matrix with its entry (m, k ) equal to c(0) m cm . (k ) From (40) (after removing conditionality on vm ), (42), and (43), we obtain
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M
U|g
(0)H Q ( s ) g (0) } with Q(s) = s I N + (0) ( s ) = exp{ g s
N s2 2 {β E{L LT }i C + [( Nu −1)αβ + s ] I Ns } . 4 γb (44)
As g (0) is a PCG random vector, taking the expectation of (44) results in [20] MU ( s ) = det{I N s − Q ( s )C }−1 .
(45)
In the cases that (45) leads to an explicit expression for the pdf of U, the BER can be computed by evaluating the cdf of U at zero. Otherwise, we make use of the Beaulieu series [21] in the following manner in order to obtain the BER directly from the mgf of U: Pbe = Pr{U < 0} = lim L → ∞ FU (0; L, N ) , where FU (0; L, N ) = N →∞
N 1 2 2 nπ Im{MU ( j )} . (46) − ∑ 2 n =1 nπ L n odd
Im{} ⋅ denotes the imaginary part. From the error bounds given in [21], FU (0; L,100) with
L = µU + 20σU , where µ U and σU are the mean and standard deviation of U respectively, is a very good approximation of Pbe . In the subsequent Sections, we evaluate the mgf of U for the hopping schemes. The pertinent expressions for MC-CDMA are derived from those of DS-MC-FH systems by substituting N h = 1 .
A. DS-MC-FH (k ) PN Signature Code: In this case, from the statistics of c m , it can simply be shown that
E{LLT } = ( Nu − 1) I N s . Therefore, from (44) and (45) we have
1 MU ( s) = det{I N s − ( s + [ β ( Nu − 1) + N s / γ b ]s 2 )C}−1 . 4
(47)
Assuming that the eigenvalues of C are distinct, after some simplifications we obtain MU ( s) = M U− ( s ) + M U+ ( s) , where MU− ( s ) =
MU+ ( s ) =
N s −1
∏
(
m=0 N s −1
∏
m=0
4λ m−1
β ( Nu − 1) +
(
Ns
γb
4λ m−1
β ( Nu − 1) +
Ns
γb
)
N s −1
∑
m=0
)
am, −1
N s −1
∑
m=0
am, +1 pm, +1 − s
λ m ’s are eigenvalues of C . pm, −1 , pm, +1 , am, −1 and am, +1 are given by
19
and
s − pm, −1
.
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
pm, ±1 = 2(1 ± 1 + {β ( Nu − 1) +
Ns
γb
}/ λ m )−1
and
N −1
s 1 1 . am, ±1 = ∏ pm, +1 − pm, −1 n = 0 ( pm, ±1 − pn, −1 )( pn, +1 − pm, ±1 )
(48)
n≠m
As C is positive definite, λ m ’s are all positive. Therefore, pm, −1 ’s are all negative, and pm, +1 ’s are all positive. Since the region of convergence of MU ( s) includes the axis s = jω , M U− ( s ) and M U+ ( s ) are the right-sided and left-sided parts of MU ( s ) , respectively, and therefore, the BER is equal to M U− (0) . After some algebraic manipulations, the BER is obtained as 1 N −1
N −1
s s − N 1 Pbe = {1 − ∑ {1 + [ β ( Nu − 1) + s ] / λm } 2 ∏ (1 − λ n / λ m )− 1} . γb 2 m=0 n=0
(49)
n≠m
For N h = 1 and Nu = 1 , the result in (49) is in compliance with the one obtained in [3].
Walsh Signature Code: In this case, L is deterministic and E{LLT } = LLT . An explicit expression for the BER can not be derived, so we exploit the Beaulieu series (see (46)) for extracting the probability of error.
B. MC-FH For this system, all the elements of L are equal to 1, so from (44) and (45) we obtain
N 1 β2 MU ( s ) = det{I N s − [ s + (αβ ( Nu − 1) + s ) s 2 ]C − ( Nu − 1) s 2C 2 }−1 , 4 γb 4
(50)
which, following similar steps from (47) to (49), gives the BER as Pbe =
N s −1
4
m =0
λ m [αβ ( Nu − 1) + N s / γ b + λ m β 2 ( Nu − 1)]
∏
N s −1
∑
m =0
(−
am, −1 pm, −1
),
where am, −1 is as given in (48) with the parameters pm, −1 and pm, +1 defined as: pm , ±1 =
2
λm
{1 ± 1 + β 2 ( Nu − 1) + λ −m1[αβ ( Nu − 1) +
Ns
γb
]}−1 .
(51) For studying near-far effect, we only need to multiply the variance of the multiple-access interference by R / R (0) (like the analyses in nonfading channels), and modify (38)-(51) accordingly.
V. NUMERICAL RESULTS In this Section, we provide some numerical results to compare the three multicarrier schemes, namely, MC-CDMA, MC-FH, and DS-MC-FH in nonfading and fading channels. Comparisons are made within a constant bandwidth. In all the graphs presented (Figs. 5-13), N s , N h , Nu , ξ , BW, 20
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
and SNR signify the number of frequency subbands, the number of subcarriers in each frequency subband, the number of users, the number of services of each user, the normalized bandwidth (product of N s and N h ), and γ b (as defined in (8)), respectively. Also, Tr , defined as Tm / T ( Tm is the maximum delay spread used in (5)), denotes the relative delay spread of the channel. µ is the decaying factor in (5). (P) and (W) stand for the PN and Walsh schemes, respectively. The pair ( x, y ) after the name of a hopping system indicates N s and N h for that system ( N s = x and
N h = y ). In the cases where Walsh codes are utilized, we have used the cyclic assignment of Walsh
codes as described in the Subsection A.2 of Section III. Figs. 5-7 present plots of the BER versus the number of active users in different systems separately. For comparison, in synchronous cases we have also included the plots of the BER based on the Gaussian distribution assumption for the multiple-access interference using (27). The exact and Gaussian plots for MC-CDMA and DS-MC-FH systems with PN codes almost coincide. This uniformity is justified when noting that the multiple-access interference term generally contains sums of iid random variables, and the CLT is thus applicable. The same is true for asynchronous environments, and therefore, the validity of the Gaussian approximation is retained. From these figures, except for the case of MC-CDMA with Walsh codes, where the synchronous system, as a result of a complete orthogonality between signatures, outperforms the asynchronous system, in the -1
10 -2 10 10
BER
10 10 10 10 10 10
MC-CDMA BW = 256 SNR = 15 dB ξ=4
-4
-6
-8
(P), Sync. (Exact) (P), Sync. (Gaussian) (P), Async. (W), Sync. (W), Async.
-10
-12
-14
-16
1
2
3
4
6
8 10
14
20
30 40 50 64 75
Nu Fig. 5) Performance of synchronous and asynchronous MC-CDMA systems in nonfading channels versus N u with constant BW, SNR and ξ .
21
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
10 10
BER
10 10 10 10 10
-1
-2
MC-FH(16, 16) SNR = 15 dB ξ= 4
-3
-4
Sync. (Exact) Sync. (Gaussian) Async.
-5
-6
-7
1
2
3
4
6
8
Nu
12
20
32
50
80
Fig. 6) Performance of synchronous and asynchronous MC-FH systems in nonfading channels versus N u with constant N s , N h , SNR and ξ .
10
BER
10
10
10
10
0
-5
(P), Sync. (Exact) (P), Sync. (Gaussian) (P), Async. (W), Sync. (W), Async.
-10
DS-MC-FH(16, 16) SNR = 15 dB ξ=4
-15
-20
1
2
3
4
6
8 10
Nu
20
40
60
82
Fig. 7) Performance of synchronous and asynchronous DS-MC-FH systems in nonfading channels versus N u with constant N s , N h , SNR and ξ .
other cases, an asynchronous scheme, because of the lower variance of the multiple-access interference term, has a better performance up to two orders of magnitude than the synchronous scheme. Also, from Fig. 7, both the synchronous and asynchronous DS-MC-FH systems with PN codes perform much worse than the DS-MC-FH systems with Walsh codes, verifying the inappropriateness of PN codes as the second signature in DS-MC-FH schemes.
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
In Fig. 8, the performance of MC-FH schemes versus N s is evaluated when the bandwidth is constant. The phenomenon of having an optimum value for N s is observed for synchronous and asynchronous cases alike. In fact, an increase in N s , on the one hand, would enhance the frequency diversity of the system, and on the other hand, it would augment the variance of the multiple-access interference dramatically, as a result of modulating all the transmitted subcarriers with an identical uncoded bit, as shown in Fig. 2, thus causing a great correlation among the signals carried by the 10
10
BER
10
10
10
10
0
-2
-4
-6
Sync., BW=128 Sync., BW=256 Sync., BW=512 Async., BW=128 Async., BW=256 Async., BW=512
MC-FH SNR = 15 dB Nu = 6
-8
ξ=2
-10
2
4
8
16
Ns
32
64
128
256
Fig. 8) Performance of synchronous and asynchronous MC-FH systems in nonfading channels versus N s with constant BW, SNR, N u and ξ .
transmitted subcarriers. For example, in the synchronous case according to (18), the variance induced due to a single interfering service is equal to N s β (N s β + α ) , which is a fourth-degree function of N s when taking the bandwidth constant. As a result of the above trade-off, the existence of an
optimum N s is foreseeable. Similar results in fading environments, dropped here due to space limitation, retain this phenomenon. An optimum N s , however, will not be obtained for the other schemes, and the performance is constantly improved by increasing N s . In the other schemes, in contrast to MC-FH systems, the data stream is coded by a signature before being sent over transmitted subcarriers. In Figs. 9 and 10, we have compared the three schemes in asynchronous nonfading and synchronous fading channels, respectively. From these figures, it can be realized that the systems with Walsh codes perform much better than the systems with PN codes. Also, MC-CDMA schemes 23
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
with PN codes outperform MC-FH schemes. MC-CDMA systems with PN codes always perform better than DS-MC-FH schemes with PN codes. However, the performance of the latter obviously approaches that of the former as N s increases for a given bandwidth. This can be seen in Fig. 10, where the DS-MC-FH system with PN codes almost has the same performance as the MC-CDMA system with PN codes. For studying the near-far effect, Fig. 11 presents the plots of the BER versus R (0k ) / R (0) 0 in dB for -1
10-2 10 10 10
BER
10 10 10 10 10
-4
-6
Async BW = 256 SNR = 15 dB ξ= 2
-8
-10
MC-CDMA(P) MC-CDMA(W) MC-FH(8, 32) DS-MC-FH(P)(32, 8) DS-MC-FH(W)(32, 8)
-12
-14
-16
1
2
4
8
16
Nu
32
64
128 175
Fig. 9) Performance of different asynchronous systems in nonfading channels versus N u with constant BW, N s , N h , SNR and ξ .
10
MC-CDMA(P), DS -MC-FH(P)(32,4) MC-CDMA(W) MC-FH(16,8) MC-FH(32,4) DS -MC-FH(W)(32,4) DS -MC-FH(W)(64,2)
-3
BER
10
-2
10
10
-4
Fading BW = 128 SNR = 15 dB Tr = .03 µ = .5
-5
1
2
3
4
6
8 10
Nu
14
20
30
40 50
Fig. 10) Performance of different synchronous systems in fading channels versus N u with constant BW, N s , N h , SNR, T r and µ .
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
different systems in a synchronous fading environment. From this figure, it is apparent that for the single-user MRC detector under consideration, except for MC-CDMA schemes with Walsh codes where the orthogonality among the codes is retained under the near-far effect but not in fading environments, the other systems are not near-far resistant. Note that in DS-MC-FH schemes with Walsh codes, the hopping itself rather distorts the orthogonality among the codes. 10
10
BER
10
10
10
10
-1
-2
Fading BW = 256 SNR = 15 dB Nu = 8 Tr = .03
-3
µ = .5
-4
MC-CDMA(P),DS -MC-FH(32,8) MC-CDMA(W) MC-FH(32,8) MC-FH(64,4) DS -MC-FH(W)(64,4) DS -MC-FH(W)(128,2)
-5
-6
1
2
3
4
5
(k) (0) R0 /R0 ,
dB
6
7
8
9
10
Fig. 11) Performance of different synchronous systems in fading channels considering the near-far effect with constant BW, N s , N h , SNR, N u , T r and µ .
In Figs. 12 and 13, we compare the performances of different systems versus SNR and T r in nonfading and fading channels, respectively. The performance priority order of different schemes is 10
10
BER
10
10
10
10
-2
-4
-6
Async BW = 256 Nu = 4
-8
-10
MC-CDMA(P) MC-CDMA(W) MC-FH(16, 16) DS-MC-FH(P)(32, 8) DS-MC-FH(W)(32, 8)
ξ=4
-12
10
11
12
13
14
15
SNR, dB
16
17
18
19
20
Fig. 12) Performance of different asynchronous systems in nonfading channels versus SNR with constant BW, N s , N h , N u and µ .
25
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
10
BER
10
10
10
-1
-2
-3
-4
Fading BW = 128 mu = 10 SNR = 18 dB Nu = 6 ξ=2
0.001
0.002
MC-CDMA(P) MC-CDMA(W) MC-FH(4,32) MC-FH(8,16) MC-FH-CDMA(P)(32,4) MC-FH-CDMA(P)(64,2) MC-FH-CDMA(W)(32,4) MC-FH-CDMA(W)(64,2)
0.004
0.008
0.016
0.032
0.072
Tr
Fig. 13) Performance of different systems in fading channels versus T r with constant BW, µ , SNR, N u and ξ .
much clearer in these figures. These figures show a noticeable improvement in performance by augmenting SNR or T r . In DS-MC-FH schemes with PN codes, the multiple-access interference term is dominant in relation to the noise term. The variance of the MAI term that is obtained from (30) as N s (ξ − 1) + 2( Nu − 1) N sξσ 2 ( k , q ) with σ 2 ( k , q ) given in (35), is quite large with respect MAI −1, m MAI −1, m to the variance of the noise term which is equal to N s2 /(2γ b ) (see (10-d)). Therefore, as can be seen in Fig. 12, the performance improvement with SNR in the DS-MC-FH system is not as evident as that of the other schemes. In Fig. 13, an increase in T r is simply translated into narrowing the spectrum of the pdp of the channel described in (5), thereby diminishing the correlation between the fading gains at adjacent transmitted subcarriers.
VI. CONCLUSION A unified multiple-access performance analysis of several multimedia multicarrier CDMA systems in synchronous and asynchronous nonfading and synchronous correlated Rayleigh fading channels considering near-far effect was provided. Our analysis has first indicated that MC-CDMA systems have the best performance. Furthermore, it has been found that the schemes with Walsh codes outperform the other systems in both nonfading and fading channels. Also, it has been realized that for the single-user MRC detector considered in this paper, MC-CDMA schemes with Walsh
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Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
codes have considerable robustness against near-far effect in synchronous channels. However, the other systems are vulnerable to near-far effect.
APPENDIX In this Appendix, we derive the result given in (16). According to [19], if X is an integer-valued random variable with cdf FX [n] = E{ X ≤ n} and mgf M X ( z ) = E{z X } , then
FX [ n] = lim L →∞ FX [ n; L ]
with
FX [n; L] =
where,
1 2 L −1 ∑ µL[k ]M X (e j k π / L )e − j n k π / L , 2 L k =0 k =0
L ,
µ L [k ] = 1 + j cot[kπ /(2 L)], k odd
otherwise 0, and furthermore, we have the following error bounds [19]
,
(A-2)
− Pr{ X > L + n} ≤ FX [ n] − FX [ n; L] ≤ Pr{ X ≤ − L + n} . Now,
if
p X [ n]
is
the
probability
function
of
X
(A-1)
( p X [n] = Pr{ X = n} ),
(A-3) from
p X [n ] = FX [n ] − FX [n − 1] and (A-1)–(A-3) we easily conclude that p X [n ] = lim L →∞ p X [ n; L ] ,
where
p X [n; L] =
1 2 L −1 M X (e j k π / L ) e − j n k π / L ∑ L k =1
(A-4)
k odd
with the following error bounds
− Pr{ X − n > L} ≤ p X [n] − p X [n; L] ≤ Pr{ X − n ≥ L} . (A-5) Therefore, if p X [n] is nonzero for a finite set of integers, then choosing a sufficiently large L results in p X [n; L] = p X [n] . Now, as pMAI [a] is zero for a greater than P = N s ( Nuξ − 1) (see (12)), we have pMAI [a] = pMAI [a; 2 P + 1] for any a with a ≤ P ; thus, (16) is derived.
REFERENCES [1] S. Hara and R. Prasad, “Overview of Multicarrier CDMA,” IEEE Commun. Mag., pp. 126-33, Dec. 1997. [2] N. Yee, J-P. Linnartz, and G. Fettweis, “Multi-carrier CDMA in Indoor Wireless Radio Networks,” Proc. of IEEE PIMRC 1993, Yokohama, Japan, pp. 109-13, Sep. 1993. [3] S. Hara and R. Prasad, “Design and Performance of Multicarrier CDMA System in Frequency-Selective Rayleigh Fading Channels,” IEEE Trans. on Vehicular Tech., Vol. 48, No. 5, pp. 1584-94, Sep. 1999. [4] H. Schulze, “The Performance of Multicarrier CDMA for the Correlated Rayleigh Fading Channel,” AEŰ Int. J. Electron. Commun., 2000, no. X, 1-8. [5] S. Tsumura and S. Hara, “Design and Performance of Quasi-Synchronous Multi-Carrier CDMA system,” IEEE VTC 2001, vol.: 2, pp: 843-7.
27
Scientia Iranica (International Journal of Science & Technology, Sharif Univ. Technol., Tehran, Iran), vol. 13, no. 1, pp. 33–49, Spring 2006.
[6] K-W Yip and T-S Ng, “Tight Error Bounds for Asynchronous Multicarrier CDMA and Their Application,” IEEE Communications Letters, vol. 2, no. 11, Nov. 1998, pp. 295-297. [7] X. Gui and T-S Ng, “Performance of Asynchronous Orthogonal Multicarrier CDMA System in Frequency Selective Fading Channel,” IEEE Trans. Commun., vol. 47, no. 7, Jul. 1999, pp. 1084-91. [8] B. Smida, C. L. Despins, and G. Y. Delisle, “MC-CDMA Performance Evaluation Over a Multipath Fading Channel Using the Characteristic Function Method,” IEEE Trans. Commun., vol. 49, no. 8, Aug. 2001, pp. 1325-8. [9] M. Ohkawa, R. Kohno, and H. Imai, “Orthogonal Multi-Carrier FH-CDMA Scheme for Frequency Selective Fading,” Proc. ICCS Conf., Singapore, 14-18 Nov., 1994, vol. 2, pp. 612-9. [10] C. T. Ong and C. Leung, “Code Diversity Transmission in a Slow-Frequency-Hopped Spread Spectrum Multiple-Access Communication System,” IEEE Trans. Commun., vol. 43, no. 12 Dec. 1995, pp. 28979. [11] M. Elkashlan and C. Leung, “Performance of Frequency-Hopping Multicarrier CDMA in Rayleigh Fading,” IEEE 56th VTC, Sep. 2002, Vancouver, vol. 1, pp. 341-5. [12] G. K. Kaleh, “Frequency-Diversity Spread-Spectrum Communication System to Counter Bandlimited Gaussian Interference,” IEEE Trans. Commun., vol. 44, no. 7, Jul. 1996. [13] E. Lance and G. K. Kaleh, “A Diversity Scheme for a Phase-Coherent Frequency-Hopping SpreadSpectrum System,” IEEE Trans. Commun., vol. 45, no. 9, pp. 1123-9, Sep. 1997. [14] O-S. Shin, K. Bok, “Performance Comparison of FFH and MCFH Spread-Spectrum Systems with Optimum Diversity Combining in Frequency-Selective Rayleigh Fading Channels,” IEEE Trans. Commun., vol. 49, no. 3, pp. 409-16, Mar. 2001. [15] M. Ebrahimi and M. Nasiri-Kenari, “Performance Analysis of Multicarrier Frequency-Hopping (MC-FH) Code Division Multiple-Access Systems: Uncoded and Coded Schemes,” IEEE 55th VTC, spring 2002, vol. 2, pp. 630-4; and under revision in IEEE Trans. on Vehicular Technology. [16] D. Liu, C. L. Despins, and W. A. Krzymien, “Efficient and Accurate DS-SSMA Deterministic Signature Sequence Performance Evaluation over Wireless Fading Channels,” Proc. IEEE ICC’97, Montreal, PQ, Canada, Jun. 1997. [17] A. R. Forouzan, M. Nasiri-Kenari, and J. A. Salehi, “Performance Analysis of Ultrawideband TimeHopping Code Division Multiple Access Systems: Uncoded and Coded Schemes,” Proc. IEEE Intl. Conf. Communications, pp. 3017–3021, Jun. 11-14, 2001. [18] F. D. Neeser and J. L. Massey, “Proper Complex Random Processes with Applications to Information Theory,” IEEE Trans. Information Theory, vol. 39, no. 4, pp. 1293-302, Jul. 1993. [19] J. A. Gubner and M. M. Hayat, “A Method to Recover Counting Distributions from their Characteristic Functions,” IEEE Signal Processing Letters, vol. 3, no. 6, pp. 184-6, Jun. 1996. [20] G. L. Turin, “The Characteristic Function of Hermitian Quadratic Forms in Complex Normal Variables,” Biometrika, vol. 47, no. 1/2, Jun. 1960, pp. 199-201. [21] N. C. Beaulieu, “An Infinite Series for the Computation of the Complementary Probability Distribution Function of a Sum of Independent Random Variables and Its Application to the Sum of Rayleigh Random Variables,” IEEE Trans. Commun., vol. 38, no. 9, Sep. 1990, pp. 1463-74.
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