Performance of Decorrelating Receivers in Multipath ... - IEEE Xplore
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 8, AUGUST 2006
2009
Performance of Decorrelating Receivers in Multipath Rician Fading Channels Kegen Yu, Member, IEEE, and Ian Oppermann, Senior Member, IEEE
Abstract— This letter focuses on the performance analysis of the decorrelating receiver in multipath Rician faded CDMA channels. M-ary QAM scheme is employed to improve the spectral efficiency. Approximate expressions are first derived for the two performance indexes: the average symbol error rate (SER) and the average bit error rate (BER) when the decorrelating-first receiver perfectly knows the channel information of the user of interest. To achieve desirable closed-form expressions of the SER and the BER, we exploit results in large system analysis and make assumptions of a high signalto-interference ratio (SIR) and/or a small Rician K-factor. To measure the receiver performance in the practical scenario, we further derive expressions to approximate the average SER and BER of the decorrelating-first scheme with channel uncertainty. Simulation results demonstrate that the analytical results can also be employed to evaluate the performance of the combining-first receiver.
scenarios where the signal-to-interference ratio (SIR) is high and the Rician K-factor is small. Next we consider a realistic situation where the receiver only has access to an imperfect channel estimate of the desired user. Provided that the channel estimation error vector is white and Gaussian, we are able to derive the expressions of the average error rate under channel uncertainty. II. S IGNAL M ODEL Since the receiver performance is the concern of this work, we focus on the received signal model. After down-conversion and chip-matched filtering, the received signal in symbol time can be described by1
Index Terms— Decorrelating-first/combining-first receiver, symbol/bit error rate, multipath Rician fading, M-ary QAM.
r=
Lk K
ak bk sk + n,
(1)
k=1 =1
I. I NTRODUCTION
T
HE decorrelating receiver is one of the most widely studied multiuser receivers [1]. With the knowledge of timing information, signature sequences, and channel state information (required for the combining-first scheme) of all users, the decorrelating receiver can be completely free from the interference, although it incurs some noise enhancement. The decorrelating receiver with multipath combining has been studied by [2], [3], [4], [5], [6], [7] in Rayleigh fading channels and by [8] in Nakagami fading channels for BPSK or DPSK signals. Performance evaluation of the DS-CDMA system in multipath Rician fading channels has been performed in [9], [10] for BPSK or DPSK with the conventional matched filter reception. In this paper, we present performance analysis of two decorrelating receivers, one of which is called decorrelating-first and the other is called combining-first for convenience. The M-ary QAM is employed due to its higher spectral efficiency, which has been considered in [11], [12]. With the aid of large system analysis, we derive approximate expressions of the SER and BER when perfect channel information of the desired user is available to the decorrelating-first receiver. Closed-form approximations of the average SER and BER are obtained for
where r is the column received signal vector of length N ( the processing gain), k ∈ {1, 2, . . . , K} indexes the multiple user, and Lk denotes the number of paths of user k. Also bk is the equally likely M-ary QAM data symbol of user k with E[bk ] = 0 and E[|bk |2 ] = 1, ak is the channel coefficient for path of user k with Rician-distributed amplitude, sk is the normalized random signature sequence for path of user k with ||sk ||2 = 1, and n is the complex white Gaussian noise vector with E[n] = 0 and E[nnH ] = σ 2 I. We further assume that the delay spread of the channel is much smaller compared to the symbol duration so that intersymbol interference can be neglected, and the signature sequences, data, noise, and channel processes are independent. This model is similar to that used in [7] except for the channel and the data symbol structures. Define sk = [sk1 , sk2 , . . . , skLk ] ∈ RN ×Lk , ak = [ak1 , ak2 , . . . , akLk ]T ∈ C Lk ×1 . Also define
S = [s1 , s2 , . . . , sK ] ∈ RN ×(
Lk )
,
A = [a1 , 0, . . . , 0; 0, a2 , 0, . . . , 0; . . . ; 0, . . . , 0, aK ]
K column
Manuscript received July 2004; revised April 9, 2005 and September 24, 2005; accepted September 26, 2005. The editor for coordinating the review of this letter and approving it for publication is F. Daneshgaran. Kegen Yu was with the Centre for Wireless Communications, University of Oulu, Finland. He is now with CSIRO ICT Centre, Marsfield, NSW 2122, Australia (e-mail: [email protected]). Ian Oppermann is with Nokia Networks, PO Box 365, Nokia Group, FI00045, Finland. He is also affiliated with the Centre for Wireless Communications, University of Oulu, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2006.04442.
K k=1
∈ C(
b = [b1 b2 . . . bK ]
T
∈ C
K×1
K k=1
Lk )×K
,
.
Then (1) can be written in compact form as 1 A synchronous channel model is considered. It would be desirable to consider an asynchronous model as typically the uplink channel is asynchronous; however, it is beyond the scope of the paper.
c 2006 IEEE 1536-1276/06$20.00
2010
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r = SAb + n.
(2)
For notational simplicity, we define L = L1 . III. R ECEIVER S TRUCTURES The decorrelating-first receiver performs multiuser decorrelating followed by multipath combining. When KL ≤ N and the matrix S has full column rank2, the Moore-Penrose generalized inverse of the signature matrix S is −1 T S . S+ = ST S
IV. AVERAGE E RROR R ATE WITH P ERFECT C HANNEL I NFORMATION It is rather difficult to gain insights into the performance directly from (6). However, exploiting the results in large system analysis, we can simplify the SIR expression in (6). From [14], [7] we have As N → ∞ with K = αN and α constant and αL < 1, the covariance matrix, R, converges almost surely to the deterministic matrix, R† = β −1 I where β = (1 − αL)/σ 2 . Therefore, the asymptotic SIR becomes
Multiplying both sides of (2) by S+ (decorrelating) yields ˜r = Ab + n ˜, where
γ=
γ = β|a1 |2 ,
(4)
˜, where r1 and n1 are the first L elements of ˜r and n respectively. Also n1 has the covariance matrix R equal to
−1 the first L × L sub-block of σ 2 ST S . The remaining task of the receiver is the coherent combining of the multipath signals when the channel information is known. There exist various combining techniques [13], however let us consider maximal ratio combining. The combiner first forms aH aH 1 r1 1 n1 = b + , 1 2 ||a1 || ||a1 ||2
(9)
With this simplified SIR expression, we can readily derive our expressions for the performance indexes of the decorrelatingfirst receiver. Throughout this section, γ is given by (9), and
γ¯ = βE[|a1 |2 ].
Taking user 1 as the user of interest, we extract the relevant components from (3), forming
z=
β|a1 |2 .
=1
(3)
−1 T ˜r = ST S S r, T −1 T ˜= S S n S n.
r1 = a1 b1 + n1 ,
L
(5)
and then passes this decision statistic, z, through a minimum distance detector to produce ˆb1 , the estimate of b1 . From (5), it can be shown that the SIR at the output of the receiver is L 2 1 2 |a1 | . (6) γ= H a1 Ra1 =1 In the combining-first scheme, both sides of (2) are multiplied by (SA)H (composite combining), resulting in (7) (SA)H r = (SA)H (SA)b + (SA)H n
−1 Multiplying both sides of (7) by (SA)H SA produces
−1 −1 (SA)H r = b + (SA)H SA (SA)H n (SA)H SA (8) where the matrix inverse is supposed to exist3 . The relevant components of the desired user are extracted from (8) and passed to the detector to yield the symbol/bit estimate. In the following two sections, we will present the theoretical analysis for the average error rate of the decorrelating-first receiver. 2 Thus the system loading is constrained to K ≤ N/L for the decorrelatingfirst receiver. 3 The condition for the existence of the inverse is that the matrix SA has full column rank and K ≤ N . Thus, the capacity of the combining-first receiver is L times of the decorrelating-first scheme.
A. SER with Known Channel The SER of a square M-ary QAM is given by [15]
1 PsM (γ) = 4 1 − √ Q( 2gγ) M
2 1 −4 1− √ Q2 ( 2gγ), M
(10)
where Q(x) is the standard Q-function and g=
3 log2 M . 2(M − 1)
Utilizing the alternate Gaussian Q-function and the MGF (moment generating function), it can be shown that (10) becomes (see Eq. 9.21 and Table 9.1 in [16]) PsM
4 = π
1 1− √ M
π/2
f (φ)dφ 0 2 π/4
4 1 − f (φ)dφ, 1− √ π M 0
(11)
where f (φ) =
L =1
−n2 g¯ χ(φ) γ exp , g¯ γ + χ(φ) g¯ γ + χ(φ)
χ(φ) = (1 + n2 ) sin2 φ
(12) (13)
Here n2 is the Rician K-factor. To our best knowledge, there does not exist a closed form solution to (11) except for n2 = 0 which corresponds to Rayleigh fading. To obtain an approximate solution for n2 > 0, we make use of −n2 g¯ −n2 g¯ γ γ exp ≤ exp . (14) g¯ γ + χ(φ) g¯ γ + (1 + n2 )
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Then we may approximate4 the average SER by
PsM
2011
where
π/2
4 1 ≈ f1 (φ)dφ 1− √ π M 0
π/4 1 − 1− √ f1 (φ)dφ , (15) M 0
(2)
PsM
√ 2 M −1 = L M exp κ () 1 =1 L 4 1 −1 μ1 () tan − 1 , (22) × ρ π μ1 () =1
where
and ρ =
L e−κ1 () χ(φ) , f1 (φ) = g¯ γ + χ(φ) =1
κi () =
n2
γ i2 n2 g¯ . + i2 g¯ γ + 1
k=1 k=
(16)
The approximation would be expected to be more accurate as the SIR increases and/or the Rician K-factor decreases. When n2 = 0, both sides of (14) are equal to one and the approximation becomes exact. Exploiting the results in the case of i.i.d. Rayleigh fading (see Eq. 9.23 in [16], note the three minor typos), we can readily derive the average SER in i.i.d. Rician fading channels
PsM
√
L M −1 4 1 − μ1 (1) ≈ √ 2 M exp[Lκ1 (1)]
L−1 L−1+ 1 + μ1 (1) (1) × + PsM , (17) 2 =0
where 2 √ M −1 μ1 (1) π (1) PsM = − tan−1 μ1 (1) M exp[Lκ1 (1)] π 2 2
L−1 2 (1 + n ) − sin(tan−1 μ1 (1)) × [4(1 + n2 + g¯ γ1 )] =0 L−1 1 Ti [cos(tan−1 μ1 (1))]2(−i)+1 , (18) − × (1 + n2 + g¯ γ1 ) 4 i=1 4
=1
and κi ()/n2 , 2
(1 + n ) Ti = 2(−i) . {4i [1 + 2( − i)]} −i
μi () =
L
(19) (20)
In the case of distinct average path powers, employing a partial fraction expansion on (15) and using Eq. 2.562.1 in [17], the average SER becomes (see Eq. 9.26 in [16]) √ L 2 M −1 (2) PsM ≈ √ ρ (1 − μ1 ()) + PsM , L M exp =1 κ1 () =1 (21) 4 When γ ¯ takes the actual value, the right-hand-side of (15) becomes an upper bound.
γ¯ . γ¯ − γ¯k
(23)
Consider a more general situation where some paths have the same average power, while other paths have distinct average powers. In this case, partial fraction expansion can be performed first. The expression of the SER would be a hybrid form of (17) and (21). Therefore, we can have a closed form expression for the error probability under any power distribution. As shown in (17) through (23), the analytical expressions of the SER are rather complicated such that they may not be easily evaluated. To obtain simpler expressions to approximate the SER, we may further exploit 1 + n2 χ(φ) ≤ g¯ γ + χ(φ) g¯ γ + (1 + n2 )
(24)
to upper bound the first term in the integral in (11). Then, the average SER is approximated by π/2
4 1 f1 dφ PsM ≈ 1− √ π M 0
π/4 1 − 1− √ f1 dφ (25) M 0 where
L e−κ1 () (1 + n2 ) f1 = g¯ γ + (1 + n2 )
(26)
=1
is a constant independent of φ. Therefore, (25) becomes
1 PsM ≈ 1 − (27) f1 M which can be easily evaluated for either unequal or equal path (1) (2) powers. PsM in (18) and PsM in (22) are now approximated by
2 1 (i) f1 , i = 1, 2, (28) PsM ≈ − 1 − √ M (i)
which is much simpler. Note that PsM will also be exploited in section V-A. B. BER with Known Channel The bit error rate is typically of more interest in evaluating the performance of the communication systems. Usually, for M-ary QAM, the BER is approximated by first computing the SER [15]. Alternatively, we propose a method to approximate the average BER of the square M-ary QAM. Since a square M-ary QAM system can be decomposed into two independent
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000 −7d
Fig. 1.
001 −5d
011 −3d
010 −d
110 d
111 3d
101 5d
100
received signal vector
Decorrelating Reception
Multipath Combining
Decision
7d Channel Estimation
Signal-space diagram of Gray encoded 8-AM.
√ M -ary AM systems, the BER of the M-ary √ QAM can be obtained by determining the BER of the two M -ary AM √ systems. With a Gray encoded M -ary AM (Fig. 1 shows a Gray encoded 8-AM), one √ bit is different between any two adjacent symbols in the M -ary AM constellation. Then two bits are different between any two symbols which are separated by another symbol. When the incorrect decisions are made by choosing one of the two nearest symbols on either side of the correct symbol, the conditional BER of the receiver can be shown to be 4 2 (a) (Q (η) − Q (3η)) PbM (γ) = 1− √ log2 M M 8 1 3 + √ Q (η) + 1− √ log2 M M M 1 × (Q (3η) − Q (5η)) + √ Q (3η) M 4 1 = Q (η) 1− √ log2 M M
2 Q (3η) + 1− √ M
3 Q (5η) , (29) −2 1 − √ M √ where η = 2gγ. For errors of confusing the correct symbol with other incorrect symbols, it can be shown that the corresponding BER satisfies
4 3 (b) Q(5η) ≤ PbM (γ) ≤ 1− √ log2 M M √
4 log2 M 3 Q(5η), (30) 1− √ log2 M M where the two bounds come from by assuming that an√incorrect decision produces either one (minimum) or log2 M (b) (maximum) incorrect bits, respectively. Clearly, PbM (γ) is on the order of Q(5η) which decreases exponentially as (a) compared to PbM (γ) which is on the order of Q(η). Thus, (b) PbM (γ) is negligible at high or even reasonably low SNR. (b) (a) Ignoring PbM (γ) and the Q(5η) term in PbM (γ), we obtain the approximate conditional BER as 4 1 PbM (γ) ≈ Q (η) 1− √ log2 M M
2 Q (3η) . (31) + 1− √ M Following the same procedure in Section IV-A, we can obtain the approximate average BER in i.i.d. Rician fading as √ √ 4 M − 1 F1 + M − 2 F3 , PbM ≈ √ M log2 M (32)
estimated symbol
Fig. 2.
Receiver structure with channel estimation.
where
L 1 − μi (1) 2
L−1 L − 1 + 1 + μi (1)
Fi = exp[−Lκi (1)]
=0
2
. (33)
With distinct average path powers, the average BER can be approximated by √ √ 2 PbM ≈ √ M − 1 G1 + M − 2 G3 , M log2 M (34) where ⎡ ⎤ L L Gi = ρ exp ⎣− κi (j)⎦ (1 − μi ()) . =1
j=1
Here ρ is given by (23) and κi () is given by (16). Similarly, in the case of combination of equal and distinct path powers, the expression of the BER would be a hybrid form of (32) and (34). V. AVERAGE E RROR R ATE WITH I MPERFECT C HANNEL I NFORMATION In this section, we consider the more realistic situation where the receiver has access to an imperfect channel estimate. Fig. 2 shows the structure of the receiver where imperfect channel information is employed for receiver realization. We handle the situation in a similar way to [12]. Let the channel ˆ 1 (user one is the desired user) and estimate be denoted by a rewrite the received signal (see (4)) in the form ˆ1 b1 + (a1 − a ˆ 1 )b1 + n1 . r1 = a
(35)
The receiver ignores the fact that there is an estimation error and employs the detection rule specified in Section III with the true channel replaced by the available estimate. That is, the receiver forms ˆH r1 a z = 1 2 = b1 + v, ||ˆ a1 || where v=
ˆH ˆ 1 ) b1 + n1 ] a 1 [(a1 − a , ||ˆ a1 ||2
and then passes z to a minimum distance detector. Assume that the channel estimation errors of different paths are uncorrelated and the channel estimation error is independent of the channel estimate, the data bits, and the noise [7]. Then ˆ1 we see that v is a complex Gaussian conditioning on b1 and a random variable with mean zero and variance L (|b1 |2 Δ + β −1 )|ˆ a1 |2 , σv2 = =1 2 L 2 a1 | =1 |ˆ
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where Δ is the variance of the channel estimation error of path . The SIR now becomes γ = L
L a1 |2 =1 |ˆ
2 =1 (|b1 | Δ
2
+ β −1 )|ˆ a1 |2
.
(36)
In the case of i.i.d. multipath channels, Δ = Δ, = 1, 2, . . . , L. Then (36) becomes γ=
L =1
β|ˆ a1 |2 . 1 + β|b1 |2 Δ
a1 |2 /(1 + β|b1 |2 Δ ). γ = β|ˆ A. SER with Channel Estimate To proceed with our analysis, let√ the positions of the (1) (2) ( M) symbols be listed as b1 , b1 , . . . , b1 and define√SI and SS as index sets for interior and side symbols in the M -ary AM constellation, respectively. In Fig. 1, 000 and 100 are side symbols while the other six symbols are interior symbols. It can be shown that conditioned on the channel estimate, the exact expression of the SER of M-ary QAM is (i) 1 Q η |b1 | 2 PsM = 1 − 1 − √ M i∈SI 2 (i) Q η |b1 | +
i∈SS
2 = √ M i∈SI i∈SS 2 (i) 1 (i) 2 Q η |b1 | + Q η |b1 | , − M (i) (i) 2 Q η |b1 | + Q η |b1 |
i∈SI
i∈SS
(38) where
L 2gβ|ˆ a1 |2 η(x) = 1 + βΔ x2
squared term. Therefore, (38) is approximated as (i) (i) 2 2 Q η |b1 | + Q η |b1 | PsM ≈ √ M i∈SI i∈SS (1)
+ PsM
E = E[|a1 |2 ] − Δ , 12 i2 gβE ξi (, x) = 2 , i gβE + (1 + n2 )(1 + βΔ x2 ) νi (, x) = n2 ξi2 (, x). Applying the results in Section IV-A, the approximate average SER in i.i.d. Rician fading becomes 2 (i) (i) (1) 2 U1 |b1 | + U1 |b1 | + PsM PsM ≈ √ M i∈SI i∈SS (41) where
L 1 − ξi (1, x) Ui (x) = exp[−L νi (1, x)] 2
L−1 L − 1 + 1 + ξi (1, x) × (42) 2 =0
For 16-QAM and 64-QAM, (41) reduces to (1)
Ps16 ≈ 2U1 (q1 ) + U1 (q3 ) + PsM |M=16 , (43) 1 Ps64 ≈ [2U1 (d1 ) + 2U1 (d3 ) + 2U1 (d5 ) + U1 (d7 )] 2 (1) (44) + PsM |M=64 , √ √ where qi = i/ 5 and di = i/ 21. With distinct average path powers, similarly we have (i) 1 (i) (2) PsM ≈ √ V1 |b1 | + V1 |b1 | + PsM 2 M i∈SI i∈SS (45) where ⎡ ⎤ L L exp ⎣− ξi (j, x)⎦ ρ (x)(1 − ξi (, x)) (46) Vi (x) = =1
.
(39)
=1
In [12] the squared term in (38) was removed for producing simple expressions of the SER. This simplification will result in relatively large error for the SER at low SNR. To improve the accuracy at low SNR, we empirically approximate it with (i) the corresponding results given by PsM , i = 1, 2. This is (i) based on the fact that the probability given by PsM , i = 1, 2 has a negligible effect on the SER at high SNR, while at low (i) SNR, approximating the squared term in (38) by PsM , i = 1, 2 could result in better results than simply removing the
j=1
and ρ (x) =
1/2
(40)
Let
(37)
When the variances of the channel estimation errors are different, (37) does not hold. For simplicity, however, we may still use (37) to approximate the SIR with Δ replaced by Δ . Based on this SIR expression, we will work out the average SER and BER with imperfect channel information. Throughout this section, γ is given by (37) and
2013
L k=1 k=
E (1 + βΔk x2 ) (E − Ek ) + β(E[|a1 |2 ]Δk − E[|a1k |2 ]Δ )x2
For 16-QAM and 64-QAM, (45) reduces to 1 (2) (47) Ps16 ≈ [2V1 (q1 ) + V1 (q3 )] + PsM |M=16 2 1 Ps64 ≈ [2V1 (d1 ) + 2V1 (d3 ) + 2V1 (d5 ) + V1 (d7 )] 4 (2) (48) + PsM |M=64 √ Since the derivation is based on the one-dimensional M ary AM constellation instead of the two-dimensional M-ary QAM constellation used in [12], the resulting expressions are simpler.
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0
10 Symbol Error Rate (SER)
64−QAM
−1
10
−2
10
approx(uneq) simul(uneq) approx simul approx:α=1/18 simul:C−F
Then, in the same way, the approximate BER in i.i.d. fading with channel estimate can be shown to be 1 PbM ≈ √ M log2 M
2
(i) (i) U1 |b1 | + U3 |b1 | i∈SII
+2
i∈SIS
4−QAM
−3
10
+ 16−QAM
−4
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 3. Approximated and simulated SER with known channel information of desired user with n2 = 4. ’approx(uneq)’ denotes results using (21) while ’approx’ denotes results using (17). ’C-F’ is for the combining-first receiver.
B. BER with Channel Estimate √ Recall the M -ary AM constellation. We further define SII and SIS as sub-index sets of the interior set SI for subinterior symbols and sub-side symbols, respectively. Note that SI = SII ∪ SIS . SS is still the index set for side symbols. For the Gray encoded 8-AM shown in Fig. 1, symbols 000 and 100 still belong to SS , symbols 001 and 101 belong to SIS , and the other four symbols belong to SII . Similarly to the results in Section IV-B, the conditional BER of M-ary QAM can be shown to be (i) 1 Q η |b1 | PbM = √ 2 M log2 M i∈SII (i) (i) +Q 3η |b1 | − 2Q 5η |b1 | (i) (i) Q η |b1 | − Q 5η |b1 | +2 (i) (i) Q η |b1 | + Q 3η |b1 |
i∈SS
! (i) −2Q 5η |b1 | + ϑ (49)
where η(x) is given by (39) and ϑ accounts for the error caused by choosing symbols that are not the two nearest symbols on either side of the correct symbol. When SNR is not small, Q(5η) term is negligible compared to Q(η) and Q(3η) terms. Removing the Q(5η) terms and ϑ (in the order of Q(5η)) from (49) yields (i) 1 Q η |b1 | 2 PbM ≈ √ M log2 M i∈SII (i) (i) +Q 3η |b1 | +2 Q η |b1 | i∈S
IS " (i) (i) Q η |b1 | + Q 3η |b1 | +
i∈SS
U1
(i) |b1 |
+ U3
"
(i) |b1 |
(51)
where Ui (x) is given by (42). For 16-QAM and 64-QAM, (51) reduces to 1 Pb16 ≈ [2U1 (q1 ) + U1 (q3 ) + U3 (q3 )] (52) 8 ] [2U1 (d1 ) + 2U1 (d3 ) + 2U1 (d5 ) + U1 (d7 ) Pb64 ≈ 24 +2U3 (d1 ) + 2U3 (d3 ) + U3 (d7 )] (53) On the other hand, with distinct path powers, the BER is 1 PbM ≈ √ M log2 M
(i) (i) V1 |b1 | + V1 |b1 | i∈SII
1 (i) V1 |b1 | + 2 i∈S
IS " 1 (i) (i) V1 |b1 | + V1 |b1 | + 2
(54)
i∈SS
where Vi (x) is given by (46). For 16-QAM and 64-QAM, (54) reduces to 1 [V1 (q1 ) + V1 (q3 ) + V3 (q3 )] Pb16 ≈ (55) 16 1 [2V1 (d1 ) + 2V1 (d3 ) + 2V1 (d5 ) + V1 (d7 ) Pb64 ≈ 48 +2V3 (d1 ) + 2V3 (d3 ) + V3 (d7 )] (56) VI. S IMULATION R ESULTS
i∈SIS
+
i∈SS
10
0
(i) U1 |b1 |
(50)
We consider a synchronous DS-CDMA system which employs random binary signature sequences with spreading gain 60. Each user employs Gray encoded 64-QAM/16-QAM/4QAM. A three path channel model is employed and in the case of distinct path powers, the three paths are assigned eighty, fifteen and five percent of the total power respectively. When perfect channel information is available to the receiver, the fading part of the channel is modelled by a complex white Gaussian random process. Simulation runs are conducted until 1000 symbol errors are observed. Figs. 3 and 4 show the SER and BER results respectively when the LOS component has a power of four times of the fading power (n2 = 4). Three signal constellation sizes are tested. The user population equals 10 leading to a system loading α = 1/6. The analytical results provide an tight upper bound for the simulated results. As SNR increases, the accuracy of the theoretical results increases. Another observation is that the SER/BER of the combiningfirst receiver at loading factor α (1/6 in this case) is very similar to the SER/BER (the dotted line) of the decorrelatingfirst receiver at loading factor α/L (1/18 in this case). This
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Bit Error Rate (BER)
10
64−QAM
−2
10
approx(uneq) simul(uneq) approx simul approx:α=1/18 simul:C−F
−3
10
16−QAM
−4
10
4−QAM
0
0
10 Symbol Error Rate (SER)
−1
−2
10
−3
10
−4
0
approx(α=0.3) simul(α=0.3) simul(α=0.9):C−F approx(α=0.25) simul(α=0.25) simul(α=0.75):C−F approx(α=1/12) simul(α=1/12) simul(α=1/4):C−F 5 10 15 Average SNR per Bit per Path (dB)
20
Fig. 6. Approximated and simulated SER with n2 = 4 and different loading factors.
0
0
64−QAM
−1
10
−2
10
approx−2(uneq) approx−1(uneq) simul(uneq) approx−2 approx−1 simul
16−QAM
−3
10
−4
10 Symbol Error Rate (SER)
10 Symbol Error Rate (SER)
−1
10
10
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 4. Approximated and simulated BER with known channel information of desired user with n2 = 4.
2015
−1
10
−2
10
−3
10
−4
10
4−QAM
10
0
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 5. Comparison of two different SER approximations and simulated results with known channel information of desired user and n2 = 4. ’approx2’ denotes results using (27).
may be related to the fact that the capacity of the combiningfirst receiver is L times of the decorrelating-first scheme. Hence, the derived analytical results could be used to evaluate the performance of the combining-first receiver. Therefore, the combining-first receiver outperforms the decorrelating-first receiver significantly in multipath fading channels. Fig. 5 shows the comparison of the two different SER approximations derived in section IV-A. The approximation using (27) results in some accuracy degradation, however, the expression of (27) is much simpler. The choice of the two approximations depends on which issue (either accuracy or simplicity) is the main concern. Fig. 6 shows the SER of the two receivers with Gray encoded 64-QAM under different loading factors. The user population is 15 and 18 for the decorrelating-first receiver while it is 15, 45, and 54 for the combining-first receiver. The results also confirm that the error rate of the combiningfirst receiver at loading factor α approximately equals the error rate of the decorrelating-first receiver at loading factor
0
approx(0) simul(0) approx(1) simul(1) approx(2) simul(2) approx(4) simul(4) approx(8) simul(8) 5 10 15 Average SNR per Bit per Path (dB)
20
Fig. 7. Approximated and simulated SER with known channel information of desired user with n2 = 0, 1, 2, 4, 8.
α/L. Fig. 7 compares the SER under five different Rician Kfactors of the decorrelating-first receiver with Gray encoded 16-QAM. Clearly, the analytical results serve as a quite tight upper bound. Let us examine the performance of the receivers and the accuracy of the theoretical results derived in Section V when imperfect channel information of the desired user is employed. The pilot-aided LMMSE channel estimator [18], [12] is supposed to be employed and channel estimation error is assumed to be Gaussian random variable with variance given by Δ=
JωN D Pf ad , JωN D + πβPf ad
(57)
where J is the pilot insertion period, ωN D is the normalized Doppler frequency, and Pf ad is the average fading power of the relevant path. The fading part of the channel is assumed to be bandlimited and has a flat spectrum up to the normalized Doppler frequency. Fig. 8 shows the SER of the decorrelating-first receiver with channel uncertainty. The Rician K-factor equals two, the pilot
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0
10 Symbol Error Rate (SER)
64−QAM
−1
10
−2
10
Even with channel estimate, the approximate expressions are quite tractable for some practical constellation sizes, e.g. the 16-QAM and the 64-QAM. In the development of the closedform expressions of the SER and BER, we made use of results in large system analysis. Although we also assumed that the SIR is high and/or the Rician K-factor is small, the approximated closed-form expressions could be treated as general for a large range of SNRs and Rician K-factors of practical interest. Although we have not derived analytical results for the combining-first receiver, the theoretical results of the decorrelating-first receiver can be readily exploited to evaluate the performance of the combining-first receiver.
approx(.02) simul(.02) approx(.005) simul(.005) approx(NCE) simul(NCE)
16−QAM
−3
10
4−QAM
−4
10
0
5 10 15 20 Average SNR per bit per Path(dB)
25
Fig. 8. Approximated and simulated SER under Gaussian random channel estimation error with n2 = 2 and normalized fading rates 0.005 and 0.02. ”NCE” denotes results for the case of no channel error.
ACKNOWLEDGEMENT The authors would like to thank the anonymous reviewers for their helpful comments, based on which the quality of the letter has been improved considerably. R EFERENCES
−1
Bit Error Rate (BER)
10
64−QAM
−2
10
approx(.02) simul(.02) approx(.005) simul(.005) approx(NCE) simul(NCE)
−3
10
16−QAM
−4
10
4−QAM
0
5
10 15 20 Average SNR per bit (dB)
25
Fig. 9. Approximated and simulated BER under Gaussian random channel estimation error with n2 = 2.
insertion period equals ten, and the normalized fading rate is chosen to be 0.005 and 0.02. Smaller fading rates produce better performance since a more accurate channel estimate can be obtained. Similar results are observed for the BER as shown in Fig. 9. Clearly, analytical results serve as a tight upper bound at low SNR and closely match the simulated results at practical range of SNR. Some performance loss would be expected when compared with the optimal receiver which accounts for the multipath and multiuser effects jointly; however, it is beyond the scope of the paper to find out the amount of the loss. VII. C ONCLUSION In this letter, we evaluated the performance of two decorrelating receivers in a multipath Rician faded CDMA channel. In the case of decorrelating-first, we first derived approximate expressions for the average SER/BER under the assumption of perfect channel information available to the receiver. We then derived approximate expressions with channel uncertainty.
[1] S. Verdu, Multiuser Detection. Cambridge, UK: Cambridge University Press, 1998. [2] A. Klein and P. W. Baier, “Linear unbaised data estimation in mobile radio system applying CDMA,” IEEE J. Select. Areas Commun., vol. 11, pp. 1058–1066, Sept. 1993. [3] T. Kawahara and T. Matsumoto, “Joint decorrelating multiuser detection and channel estimation in asynchronous CDMA mobile communications channels,” IEEE Trans. Veh. Technol., vol. 44, pp. 506–515, Aug. 1995. [4] Z. Zvonar and D. Brady, “Suboptimal multiuser detector for frequencyselective Rayleigh fading synchronous CDMA channels,” IEEE Trans. Commun., vol. 43, pp. 154–157, Feb./Mar./Apr. 1995. [5] M. K. Varanasi, “Parallel group detection for synchronous CDMA communications over frequency-selective Rayleigh fading channels,” IEEE Trans. Inform. Theory, vol. 42, pp. 116–128, Jan. 1996. [6] S. S. H. Wijayasuriya, G. H. Norton, and J. P. McGeehan, “A sliding window decorrelating receiver for multiuser DS-CDMA mobile radio networks,” IEEE Trans. Veh. Technol., vol. 45, pp. 503–521, Aug. 1996. [7] J. Evans and D. N. C. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 2059–2078, Sept. 2000. [8] E. K. Al-Hussaini and I. M. Sayed, “Performance of the decorrelating receiver for DS-CDMA mobile radio system employing RAKE and diversity through Nakagami fading channels,” IEEE Trans. Commun., vol. 50, pp. 1566–1570, Oct. 2002. [9] R. Prasad, H. S. Misser, and A. Kegel, “Performance evaluation of direct-sequence spread spectrum multiple-access for indoor wireless communication in a Rician fading channels,” IEEE Trans. Commun., vol. 43, pp. 581–592, Feb./Mar./Apr. 1995. [10] Y. Ma, T. J. Lim, and S. Pasupathy, “Error probability for coherent and differential PSK over arbitrary Rician fading channels with multiple cochannel interferers,” IEEE Trans. Commun., vol. 50, pp. 429–441, Mar. 2002. [11] K. Yu, J. S. Evans, and I. B. Collings, “Pilot symbol aided adaptive receiver for Rayleigh faded CDMA channels,” in Proc. IEEE GLOBECOM, (San Antonio, TX), pp. 753–757, 2001. [12] K. Yu, J. S. Evans, and I. B. Collings, “Performance Analysis of LMMSE receiver for M-ary QAM in Rayleigh faded CDMA channels,” IEEE Trans. Veh. Technol., vol. 52, pp. 1242–1253, Sept. 2003. [13] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [14] D. N. C. Tse and S. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, pp. 641–657, Mar. 1999. [15] J. G. Proakis, Digital Communications. McGraw-Hill, 3rd ed., 1995. [16] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels. New York: John Wiley & Sons, 2000. [17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Fifth ed.,San Diego: Academic Press, 1994. [18] J. S. Evans, “Optimal resource allocation for pilot symbol aided multiuser receivers in Rayleigh faded CDMA channels,” IEEE Trans. Commun., vol. 50, pp. 1316–1325, Aug. 2002.
2009
Performance of Decorrelating Receivers in Multipath Rician Fading Channels Kegen Yu, Member, IEEE, and Ian Oppermann, Senior Member, IEEE
Abstract— This letter focuses on the performance analysis of the decorrelating receiver in multipath Rician faded CDMA channels. M-ary QAM scheme is employed to improve the spectral efficiency. Approximate expressions are first derived for the two performance indexes: the average symbol error rate (SER) and the average bit error rate (BER) when the decorrelating-first receiver perfectly knows the channel information of the user of interest. To achieve desirable closed-form expressions of the SER and the BER, we exploit results in large system analysis and make assumptions of a high signalto-interference ratio (SIR) and/or a small Rician K-factor. To measure the receiver performance in the practical scenario, we further derive expressions to approximate the average SER and BER of the decorrelating-first scheme with channel uncertainty. Simulation results demonstrate that the analytical results can also be employed to evaluate the performance of the combining-first receiver.
scenarios where the signal-to-interference ratio (SIR) is high and the Rician K-factor is small. Next we consider a realistic situation where the receiver only has access to an imperfect channel estimate of the desired user. Provided that the channel estimation error vector is white and Gaussian, we are able to derive the expressions of the average error rate under channel uncertainty. II. S IGNAL M ODEL Since the receiver performance is the concern of this work, we focus on the received signal model. After down-conversion and chip-matched filtering, the received signal in symbol time can be described by1
Index Terms— Decorrelating-first/combining-first receiver, symbol/bit error rate, multipath Rician fading, M-ary QAM.
r=
Lk K
ak bk sk + n,
(1)
k=1 =1
I. I NTRODUCTION
T
HE decorrelating receiver is one of the most widely studied multiuser receivers [1]. With the knowledge of timing information, signature sequences, and channel state information (required for the combining-first scheme) of all users, the decorrelating receiver can be completely free from the interference, although it incurs some noise enhancement. The decorrelating receiver with multipath combining has been studied by [2], [3], [4], [5], [6], [7] in Rayleigh fading channels and by [8] in Nakagami fading channels for BPSK or DPSK signals. Performance evaluation of the DS-CDMA system in multipath Rician fading channels has been performed in [9], [10] for BPSK or DPSK with the conventional matched filter reception. In this paper, we present performance analysis of two decorrelating receivers, one of which is called decorrelating-first and the other is called combining-first for convenience. The M-ary QAM is employed due to its higher spectral efficiency, which has been considered in [11], [12]. With the aid of large system analysis, we derive approximate expressions of the SER and BER when perfect channel information of the desired user is available to the decorrelating-first receiver. Closed-form approximations of the average SER and BER are obtained for
where r is the column received signal vector of length N ( the processing gain), k ∈ {1, 2, . . . , K} indexes the multiple user, and Lk denotes the number of paths of user k. Also bk is the equally likely M-ary QAM data symbol of user k with E[bk ] = 0 and E[|bk |2 ] = 1, ak is the channel coefficient for path of user k with Rician-distributed amplitude, sk is the normalized random signature sequence for path of user k with ||sk ||2 = 1, and n is the complex white Gaussian noise vector with E[n] = 0 and E[nnH ] = σ 2 I. We further assume that the delay spread of the channel is much smaller compared to the symbol duration so that intersymbol interference can be neglected, and the signature sequences, data, noise, and channel processes are independent. This model is similar to that used in [7] except for the channel and the data symbol structures. Define sk = [sk1 , sk2 , . . . , skLk ] ∈ RN ×Lk , ak = [ak1 , ak2 , . . . , akLk ]T ∈ C Lk ×1 . Also define
S = [s1 , s2 , . . . , sK ] ∈ RN ×(
Lk )
,
A = [a1 , 0, . . . , 0; 0, a2 , 0, . . . , 0; . . . ; 0, . . . , 0, aK ]
K column
Manuscript received July 2004; revised April 9, 2005 and September 24, 2005; accepted September 26, 2005. The editor for coordinating the review of this letter and approving it for publication is F. Daneshgaran. Kegen Yu was with the Centre for Wireless Communications, University of Oulu, Finland. He is now with CSIRO ICT Centre, Marsfield, NSW 2122, Australia (e-mail: [email protected]). Ian Oppermann is with Nokia Networks, PO Box 365, Nokia Group, FI00045, Finland. He is also affiliated with the Centre for Wireless Communications, University of Oulu, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2006.04442.
K k=1
∈ C(
b = [b1 b2 . . . bK ]
T
∈ C
K×1
K k=1
Lk )×K
,
.
Then (1) can be written in compact form as 1 A synchronous channel model is considered. It would be desirable to consider an asynchronous model as typically the uplink channel is asynchronous; however, it is beyond the scope of the paper.
c 2006 IEEE 1536-1276/06$20.00
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r = SAb + n.
(2)
For notational simplicity, we define L = L1 . III. R ECEIVER S TRUCTURES The decorrelating-first receiver performs multiuser decorrelating followed by multipath combining. When KL ≤ N and the matrix S has full column rank2, the Moore-Penrose generalized inverse of the signature matrix S is −1 T S . S+ = ST S
IV. AVERAGE E RROR R ATE WITH P ERFECT C HANNEL I NFORMATION It is rather difficult to gain insights into the performance directly from (6). However, exploiting the results in large system analysis, we can simplify the SIR expression in (6). From [14], [7] we have As N → ∞ with K = αN and α constant and αL < 1, the covariance matrix, R, converges almost surely to the deterministic matrix, R† = β −1 I where β = (1 − αL)/σ 2 . Therefore, the asymptotic SIR becomes
Multiplying both sides of (2) by S+ (decorrelating) yields ˜r = Ab + n ˜, where
γ=
γ = β|a1 |2 ,
(4)
˜, where r1 and n1 are the first L elements of ˜r and n respectively. Also n1 has the covariance matrix R equal to
−1 the first L × L sub-block of σ 2 ST S . The remaining task of the receiver is the coherent combining of the multipath signals when the channel information is known. There exist various combining techniques [13], however let us consider maximal ratio combining. The combiner first forms aH aH 1 r1 1 n1 = b + , 1 2 ||a1 || ||a1 ||2
(9)
With this simplified SIR expression, we can readily derive our expressions for the performance indexes of the decorrelatingfirst receiver. Throughout this section, γ is given by (9), and
γ¯ = βE[|a1 |2 ].
Taking user 1 as the user of interest, we extract the relevant components from (3), forming
z=
β|a1 |2 .
=1
(3)
−1 T ˜r = ST S S r, T −1 T ˜= S S n S n.
r1 = a1 b1 + n1 ,
L
(5)
and then passes this decision statistic, z, through a minimum distance detector to produce ˆb1 , the estimate of b1 . From (5), it can be shown that the SIR at the output of the receiver is L 2 1 2 |a1 | . (6) γ= H a1 Ra1 =1 In the combining-first scheme, both sides of (2) are multiplied by (SA)H (composite combining), resulting in (7) (SA)H r = (SA)H (SA)b + (SA)H n
−1 Multiplying both sides of (7) by (SA)H SA produces
−1 −1 (SA)H r = b + (SA)H SA (SA)H n (SA)H SA (8) where the matrix inverse is supposed to exist3 . The relevant components of the desired user are extracted from (8) and passed to the detector to yield the symbol/bit estimate. In the following two sections, we will present the theoretical analysis for the average error rate of the decorrelating-first receiver. 2 Thus the system loading is constrained to K ≤ N/L for the decorrelatingfirst receiver. 3 The condition for the existence of the inverse is that the matrix SA has full column rank and K ≤ N . Thus, the capacity of the combining-first receiver is L times of the decorrelating-first scheme.
A. SER with Known Channel The SER of a square M-ary QAM is given by [15]
1 PsM (γ) = 4 1 − √ Q( 2gγ) M
2 1 −4 1− √ Q2 ( 2gγ), M
(10)
where Q(x) is the standard Q-function and g=
3 log2 M . 2(M − 1)
Utilizing the alternate Gaussian Q-function and the MGF (moment generating function), it can be shown that (10) becomes (see Eq. 9.21 and Table 9.1 in [16]) PsM
4 = π
1 1− √ M
π/2
f (φ)dφ 0 2 π/4
4 1 − f (φ)dφ, 1− √ π M 0
(11)
where f (φ) =
L =1
−n2 g¯ χ(φ) γ exp , g¯ γ + χ(φ) g¯ γ + χ(φ)
χ(φ) = (1 + n2 ) sin2 φ
(12) (13)
Here n2 is the Rician K-factor. To our best knowledge, there does not exist a closed form solution to (11) except for n2 = 0 which corresponds to Rayleigh fading. To obtain an approximate solution for n2 > 0, we make use of −n2 g¯ −n2 g¯ γ γ exp ≤ exp . (14) g¯ γ + χ(φ) g¯ γ + (1 + n2 )
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Then we may approximate4 the average SER by
PsM
2011
where
π/2
4 1 ≈ f1 (φ)dφ 1− √ π M 0
π/4 1 − 1− √ f1 (φ)dφ , (15) M 0
(2)
PsM
√ 2 M −1 = L M exp κ () 1 =1 L 4 1 −1 μ1 () tan − 1 , (22) × ρ π μ1 () =1
where
and ρ =
L e−κ1 () χ(φ) , f1 (φ) = g¯ γ + χ(φ) =1
κi () =
n2
γ i2 n2 g¯ . + i2 g¯ γ + 1
k=1 k=
(16)
The approximation would be expected to be more accurate as the SIR increases and/or the Rician K-factor decreases. When n2 = 0, both sides of (14) are equal to one and the approximation becomes exact. Exploiting the results in the case of i.i.d. Rayleigh fading (see Eq. 9.23 in [16], note the three minor typos), we can readily derive the average SER in i.i.d. Rician fading channels
PsM
√
L M −1 4 1 − μ1 (1) ≈ √ 2 M exp[Lκ1 (1)]
L−1 L−1+ 1 + μ1 (1) (1) × + PsM , (17) 2 =0
where 2 √ M −1 μ1 (1) π (1) PsM = − tan−1 μ1 (1) M exp[Lκ1 (1)] π 2 2
L−1 2 (1 + n ) − sin(tan−1 μ1 (1)) × [4(1 + n2 + g¯ γ1 )] =0 L−1 1 Ti [cos(tan−1 μ1 (1))]2(−i)+1 , (18) − × (1 + n2 + g¯ γ1 ) 4 i=1 4
=1
and κi ()/n2 , 2
(1 + n ) Ti = 2(−i) . {4i [1 + 2( − i)]} −i
μi () =
L
(19) (20)
In the case of distinct average path powers, employing a partial fraction expansion on (15) and using Eq. 2.562.1 in [17], the average SER becomes (see Eq. 9.26 in [16]) √ L 2 M −1 (2) PsM ≈ √ ρ (1 − μ1 ()) + PsM , L M exp =1 κ1 () =1 (21) 4 When γ ¯ takes the actual value, the right-hand-side of (15) becomes an upper bound.
γ¯ . γ¯ − γ¯k
(23)
Consider a more general situation where some paths have the same average power, while other paths have distinct average powers. In this case, partial fraction expansion can be performed first. The expression of the SER would be a hybrid form of (17) and (21). Therefore, we can have a closed form expression for the error probability under any power distribution. As shown in (17) through (23), the analytical expressions of the SER are rather complicated such that they may not be easily evaluated. To obtain simpler expressions to approximate the SER, we may further exploit 1 + n2 χ(φ) ≤ g¯ γ + χ(φ) g¯ γ + (1 + n2 )
(24)
to upper bound the first term in the integral in (11). Then, the average SER is approximated by π/2
4 1 f1 dφ PsM ≈ 1− √ π M 0
π/4 1 − 1− √ f1 dφ (25) M 0 where
L e−κ1 () (1 + n2 ) f1 = g¯ γ + (1 + n2 )
(26)
=1
is a constant independent of φ. Therefore, (25) becomes
1 PsM ≈ 1 − (27) f1 M which can be easily evaluated for either unequal or equal path (1) (2) powers. PsM in (18) and PsM in (22) are now approximated by
2 1 (i) f1 , i = 1, 2, (28) PsM ≈ − 1 − √ M (i)
which is much simpler. Note that PsM will also be exploited in section V-A. B. BER with Known Channel The bit error rate is typically of more interest in evaluating the performance of the communication systems. Usually, for M-ary QAM, the BER is approximated by first computing the SER [15]. Alternatively, we propose a method to approximate the average BER of the square M-ary QAM. Since a square M-ary QAM system can be decomposed into two independent
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000 −7d
Fig. 1.
001 −5d
011 −3d
010 −d
110 d
111 3d
101 5d
100
received signal vector
Decorrelating Reception
Multipath Combining
Decision
7d Channel Estimation
Signal-space diagram of Gray encoded 8-AM.
√ M -ary AM systems, the BER of the M-ary √ QAM can be obtained by determining the BER of the two M -ary AM √ systems. With a Gray encoded M -ary AM (Fig. 1 shows a Gray encoded 8-AM), one √ bit is different between any two adjacent symbols in the M -ary AM constellation. Then two bits are different between any two symbols which are separated by another symbol. When the incorrect decisions are made by choosing one of the two nearest symbols on either side of the correct symbol, the conditional BER of the receiver can be shown to be 4 2 (a) (Q (η) − Q (3η)) PbM (γ) = 1− √ log2 M M 8 1 3 + √ Q (η) + 1− √ log2 M M M 1 × (Q (3η) − Q (5η)) + √ Q (3η) M 4 1 = Q (η) 1− √ log2 M M
2 Q (3η) + 1− √ M
3 Q (5η) , (29) −2 1 − √ M √ where η = 2gγ. For errors of confusing the correct symbol with other incorrect symbols, it can be shown that the corresponding BER satisfies
4 3 (b) Q(5η) ≤ PbM (γ) ≤ 1− √ log2 M M √
4 log2 M 3 Q(5η), (30) 1− √ log2 M M where the two bounds come from by assuming that an√incorrect decision produces either one (minimum) or log2 M (b) (maximum) incorrect bits, respectively. Clearly, PbM (γ) is on the order of Q(5η) which decreases exponentially as (a) compared to PbM (γ) which is on the order of Q(η). Thus, (b) PbM (γ) is negligible at high or even reasonably low SNR. (b) (a) Ignoring PbM (γ) and the Q(5η) term in PbM (γ), we obtain the approximate conditional BER as 4 1 PbM (γ) ≈ Q (η) 1− √ log2 M M
2 Q (3η) . (31) + 1− √ M Following the same procedure in Section IV-A, we can obtain the approximate average BER in i.i.d. Rician fading as √ √ 4 M − 1 F1 + M − 2 F3 , PbM ≈ √ M log2 M (32)
estimated symbol
Fig. 2.
Receiver structure with channel estimation.
where
L 1 − μi (1) 2
L−1 L − 1 + 1 + μi (1)
Fi = exp[−Lκi (1)]
=0
2
. (33)
With distinct average path powers, the average BER can be approximated by √ √ 2 PbM ≈ √ M − 1 G1 + M − 2 G3 , M log2 M (34) where ⎡ ⎤ L L Gi = ρ exp ⎣− κi (j)⎦ (1 − μi ()) . =1
j=1
Here ρ is given by (23) and κi () is given by (16). Similarly, in the case of combination of equal and distinct path powers, the expression of the BER would be a hybrid form of (32) and (34). V. AVERAGE E RROR R ATE WITH I MPERFECT C HANNEL I NFORMATION In this section, we consider the more realistic situation where the receiver has access to an imperfect channel estimate. Fig. 2 shows the structure of the receiver where imperfect channel information is employed for receiver realization. We handle the situation in a similar way to [12]. Let the channel ˆ 1 (user one is the desired user) and estimate be denoted by a rewrite the received signal (see (4)) in the form ˆ1 b1 + (a1 − a ˆ 1 )b1 + n1 . r1 = a
(35)
The receiver ignores the fact that there is an estimation error and employs the detection rule specified in Section III with the true channel replaced by the available estimate. That is, the receiver forms ˆH r1 a z = 1 2 = b1 + v, ||ˆ a1 || where v=
ˆH ˆ 1 ) b1 + n1 ] a 1 [(a1 − a , ||ˆ a1 ||2
and then passes z to a minimum distance detector. Assume that the channel estimation errors of different paths are uncorrelated and the channel estimation error is independent of the channel estimate, the data bits, and the noise [7]. Then ˆ1 we see that v is a complex Gaussian conditioning on b1 and a random variable with mean zero and variance L (|b1 |2 Δ + β −1 )|ˆ a1 |2 , σv2 = =1 2 L 2 a1 | =1 |ˆ
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where Δ is the variance of the channel estimation error of path . The SIR now becomes γ = L
L a1 |2 =1 |ˆ
2 =1 (|b1 | Δ
2
+ β −1 )|ˆ a1 |2
.
(36)
In the case of i.i.d. multipath channels, Δ = Δ, = 1, 2, . . . , L. Then (36) becomes γ=
L =1
β|ˆ a1 |2 . 1 + β|b1 |2 Δ
a1 |2 /(1 + β|b1 |2 Δ ). γ = β|ˆ A. SER with Channel Estimate To proceed with our analysis, let√ the positions of the (1) (2) ( M) symbols be listed as b1 , b1 , . . . , b1 and define√SI and SS as index sets for interior and side symbols in the M -ary AM constellation, respectively. In Fig. 1, 000 and 100 are side symbols while the other six symbols are interior symbols. It can be shown that conditioned on the channel estimate, the exact expression of the SER of M-ary QAM is (i) 1 Q η |b1 | 2 PsM = 1 − 1 − √ M i∈SI 2 (i) Q η |b1 | +
i∈SS
2 = √ M i∈SI i∈SS 2 (i) 1 (i) 2 Q η |b1 | + Q η |b1 | , − M (i) (i) 2 Q η |b1 | + Q η |b1 |
i∈SI
i∈SS
(38) where
L 2gβ|ˆ a1 |2 η(x) = 1 + βΔ x2
squared term. Therefore, (38) is approximated as (i) (i) 2 2 Q η |b1 | + Q η |b1 | PsM ≈ √ M i∈SI i∈SS (1)
+ PsM
E = E[|a1 |2 ] − Δ , 12 i2 gβE ξi (, x) = 2 , i gβE + (1 + n2 )(1 + βΔ x2 ) νi (, x) = n2 ξi2 (, x). Applying the results in Section IV-A, the approximate average SER in i.i.d. Rician fading becomes 2 (i) (i) (1) 2 U1 |b1 | + U1 |b1 | + PsM PsM ≈ √ M i∈SI i∈SS (41) where
L 1 − ξi (1, x) Ui (x) = exp[−L νi (1, x)] 2
L−1 L − 1 + 1 + ξi (1, x) × (42) 2 =0
For 16-QAM and 64-QAM, (41) reduces to (1)
Ps16 ≈ 2U1 (q1 ) + U1 (q3 ) + PsM |M=16 , (43) 1 Ps64 ≈ [2U1 (d1 ) + 2U1 (d3 ) + 2U1 (d5 ) + U1 (d7 )] 2 (1) (44) + PsM |M=64 , √ √ where qi = i/ 5 and di = i/ 21. With distinct average path powers, similarly we have (i) 1 (i) (2) PsM ≈ √ V1 |b1 | + V1 |b1 | + PsM 2 M i∈SI i∈SS (45) where ⎡ ⎤ L L exp ⎣− ξi (j, x)⎦ ρ (x)(1 − ξi (, x)) (46) Vi (x) = =1
.
(39)
=1
In [12] the squared term in (38) was removed for producing simple expressions of the SER. This simplification will result in relatively large error for the SER at low SNR. To improve the accuracy at low SNR, we empirically approximate it with (i) the corresponding results given by PsM , i = 1, 2. This is (i) based on the fact that the probability given by PsM , i = 1, 2 has a negligible effect on the SER at high SNR, while at low (i) SNR, approximating the squared term in (38) by PsM , i = 1, 2 could result in better results than simply removing the
j=1
and ρ (x) =
1/2
(40)
Let
(37)
When the variances of the channel estimation errors are different, (37) does not hold. For simplicity, however, we may still use (37) to approximate the SIR with Δ replaced by Δ . Based on this SIR expression, we will work out the average SER and BER with imperfect channel information. Throughout this section, γ is given by (37) and
2013
L k=1 k=
E (1 + βΔk x2 ) (E − Ek ) + β(E[|a1 |2 ]Δk − E[|a1k |2 ]Δ )x2
For 16-QAM and 64-QAM, (45) reduces to 1 (2) (47) Ps16 ≈ [2V1 (q1 ) + V1 (q3 )] + PsM |M=16 2 1 Ps64 ≈ [2V1 (d1 ) + 2V1 (d3 ) + 2V1 (d5 ) + V1 (d7 )] 4 (2) (48) + PsM |M=64 √ Since the derivation is based on the one-dimensional M ary AM constellation instead of the two-dimensional M-ary QAM constellation used in [12], the resulting expressions are simpler.
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10 Symbol Error Rate (SER)
64−QAM
−1
10
−2
10
approx(uneq) simul(uneq) approx simul approx:α=1/18 simul:C−F
Then, in the same way, the approximate BER in i.i.d. fading with channel estimate can be shown to be 1 PbM ≈ √ M log2 M
2
(i) (i) U1 |b1 | + U3 |b1 | i∈SII
+2
i∈SIS
4−QAM
−3
10
+ 16−QAM
−4
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 3. Approximated and simulated SER with known channel information of desired user with n2 = 4. ’approx(uneq)’ denotes results using (21) while ’approx’ denotes results using (17). ’C-F’ is for the combining-first receiver.
B. BER with Channel Estimate √ Recall the M -ary AM constellation. We further define SII and SIS as sub-index sets of the interior set SI for subinterior symbols and sub-side symbols, respectively. Note that SI = SII ∪ SIS . SS is still the index set for side symbols. For the Gray encoded 8-AM shown in Fig. 1, symbols 000 and 100 still belong to SS , symbols 001 and 101 belong to SIS , and the other four symbols belong to SII . Similarly to the results in Section IV-B, the conditional BER of M-ary QAM can be shown to be (i) 1 Q η |b1 | PbM = √ 2 M log2 M i∈SII (i) (i) +Q 3η |b1 | − 2Q 5η |b1 | (i) (i) Q η |b1 | − Q 5η |b1 | +2 (i) (i) Q η |b1 | + Q 3η |b1 |
i∈SS
! (i) −2Q 5η |b1 | + ϑ (49)
where η(x) is given by (39) and ϑ accounts for the error caused by choosing symbols that are not the two nearest symbols on either side of the correct symbol. When SNR is not small, Q(5η) term is negligible compared to Q(η) and Q(3η) terms. Removing the Q(5η) terms and ϑ (in the order of Q(5η)) from (49) yields (i) 1 Q η |b1 | 2 PbM ≈ √ M log2 M i∈SII (i) (i) +Q 3η |b1 | +2 Q η |b1 | i∈S
IS " (i) (i) Q η |b1 | + Q 3η |b1 | +
i∈SS
U1
(i) |b1 |
+ U3
"
(i) |b1 |
(51)
where Ui (x) is given by (42). For 16-QAM and 64-QAM, (51) reduces to 1 Pb16 ≈ [2U1 (q1 ) + U1 (q3 ) + U3 (q3 )] (52) 8 ] [2U1 (d1 ) + 2U1 (d3 ) + 2U1 (d5 ) + U1 (d7 ) Pb64 ≈ 24 +2U3 (d1 ) + 2U3 (d3 ) + U3 (d7 )] (53) On the other hand, with distinct path powers, the BER is 1 PbM ≈ √ M log2 M
(i) (i) V1 |b1 | + V1 |b1 | i∈SII
1 (i) V1 |b1 | + 2 i∈S
IS " 1 (i) (i) V1 |b1 | + V1 |b1 | + 2
(54)
i∈SS
where Vi (x) is given by (46). For 16-QAM and 64-QAM, (54) reduces to 1 [V1 (q1 ) + V1 (q3 ) + V3 (q3 )] Pb16 ≈ (55) 16 1 [2V1 (d1 ) + 2V1 (d3 ) + 2V1 (d5 ) + V1 (d7 ) Pb64 ≈ 48 +2V3 (d1 ) + 2V3 (d3 ) + V3 (d7 )] (56) VI. S IMULATION R ESULTS
i∈SIS
+
i∈SS
10
0
(i) U1 |b1 |
(50)
We consider a synchronous DS-CDMA system which employs random binary signature sequences with spreading gain 60. Each user employs Gray encoded 64-QAM/16-QAM/4QAM. A three path channel model is employed and in the case of distinct path powers, the three paths are assigned eighty, fifteen and five percent of the total power respectively. When perfect channel information is available to the receiver, the fading part of the channel is modelled by a complex white Gaussian random process. Simulation runs are conducted until 1000 symbol errors are observed. Figs. 3 and 4 show the SER and BER results respectively when the LOS component has a power of four times of the fading power (n2 = 4). Three signal constellation sizes are tested. The user population equals 10 leading to a system loading α = 1/6. The analytical results provide an tight upper bound for the simulated results. As SNR increases, the accuracy of the theoretical results increases. Another observation is that the SER/BER of the combiningfirst receiver at loading factor α (1/6 in this case) is very similar to the SER/BER (the dotted line) of the decorrelatingfirst receiver at loading factor α/L (1/18 in this case). This
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 8, AUGUST 2006
Bit Error Rate (BER)
10
64−QAM
−2
10
approx(uneq) simul(uneq) approx simul approx:α=1/18 simul:C−F
−3
10
16−QAM
−4
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4−QAM
0
0
10 Symbol Error Rate (SER)
−1
−2
10
−3
10
−4
0
approx(α=0.3) simul(α=0.3) simul(α=0.9):C−F approx(α=0.25) simul(α=0.25) simul(α=0.75):C−F approx(α=1/12) simul(α=1/12) simul(α=1/4):C−F 5 10 15 Average SNR per Bit per Path (dB)
20
Fig. 6. Approximated and simulated SER with n2 = 4 and different loading factors.
0
0
64−QAM
−1
10
−2
10
approx−2(uneq) approx−1(uneq) simul(uneq) approx−2 approx−1 simul
16−QAM
−3
10
−4
10 Symbol Error Rate (SER)
10 Symbol Error Rate (SER)
−1
10
10
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 4. Approximated and simulated BER with known channel information of desired user with n2 = 4.
2015
−1
10
−2
10
−3
10
−4
10
4−QAM
10
0
5 10 15 20 Average SNR per Bit per Path (dB)
Fig. 5. Comparison of two different SER approximations and simulated results with known channel information of desired user and n2 = 4. ’approx2’ denotes results using (27).
may be related to the fact that the capacity of the combiningfirst receiver is L times of the decorrelating-first scheme. Hence, the derived analytical results could be used to evaluate the performance of the combining-first receiver. Therefore, the combining-first receiver outperforms the decorrelating-first receiver significantly in multipath fading channels. Fig. 5 shows the comparison of the two different SER approximations derived in section IV-A. The approximation using (27) results in some accuracy degradation, however, the expression of (27) is much simpler. The choice of the two approximations depends on which issue (either accuracy or simplicity) is the main concern. Fig. 6 shows the SER of the two receivers with Gray encoded 64-QAM under different loading factors. The user population is 15 and 18 for the decorrelating-first receiver while it is 15, 45, and 54 for the combining-first receiver. The results also confirm that the error rate of the combiningfirst receiver at loading factor α approximately equals the error rate of the decorrelating-first receiver at loading factor
0
approx(0) simul(0) approx(1) simul(1) approx(2) simul(2) approx(4) simul(4) approx(8) simul(8) 5 10 15 Average SNR per Bit per Path (dB)
20
Fig. 7. Approximated and simulated SER with known channel information of desired user with n2 = 0, 1, 2, 4, 8.
α/L. Fig. 7 compares the SER under five different Rician Kfactors of the decorrelating-first receiver with Gray encoded 16-QAM. Clearly, the analytical results serve as a quite tight upper bound. Let us examine the performance of the receivers and the accuracy of the theoretical results derived in Section V when imperfect channel information of the desired user is employed. The pilot-aided LMMSE channel estimator [18], [12] is supposed to be employed and channel estimation error is assumed to be Gaussian random variable with variance given by Δ=
JωN D Pf ad , JωN D + πβPf ad
(57)
where J is the pilot insertion period, ωN D is the normalized Doppler frequency, and Pf ad is the average fading power of the relevant path. The fading part of the channel is assumed to be bandlimited and has a flat spectrum up to the normalized Doppler frequency. Fig. 8 shows the SER of the decorrelating-first receiver with channel uncertainty. The Rician K-factor equals two, the pilot
2016
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 8, AUGUST 2006
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10 Symbol Error Rate (SER)
64−QAM
−1
10
−2
10
Even with channel estimate, the approximate expressions are quite tractable for some practical constellation sizes, e.g. the 16-QAM and the 64-QAM. In the development of the closedform expressions of the SER and BER, we made use of results in large system analysis. Although we also assumed that the SIR is high and/or the Rician K-factor is small, the approximated closed-form expressions could be treated as general for a large range of SNRs and Rician K-factors of practical interest. Although we have not derived analytical results for the combining-first receiver, the theoretical results of the decorrelating-first receiver can be readily exploited to evaluate the performance of the combining-first receiver.
approx(.02) simul(.02) approx(.005) simul(.005) approx(NCE) simul(NCE)
16−QAM
−3
10
4−QAM
−4
10
0
5 10 15 20 Average SNR per bit per Path(dB)
25
Fig. 8. Approximated and simulated SER under Gaussian random channel estimation error with n2 = 2 and normalized fading rates 0.005 and 0.02. ”NCE” denotes results for the case of no channel error.
ACKNOWLEDGEMENT The authors would like to thank the anonymous reviewers for their helpful comments, based on which the quality of the letter has been improved considerably. R EFERENCES
−1
Bit Error Rate (BER)
10
64−QAM
−2
10
approx(.02) simul(.02) approx(.005) simul(.005) approx(NCE) simul(NCE)
−3
10
16−QAM
−4
10
4−QAM
0
5
10 15 20 Average SNR per bit (dB)
25
Fig. 9. Approximated and simulated BER under Gaussian random channel estimation error with n2 = 2.
insertion period equals ten, and the normalized fading rate is chosen to be 0.005 and 0.02. Smaller fading rates produce better performance since a more accurate channel estimate can be obtained. Similar results are observed for the BER as shown in Fig. 9. Clearly, analytical results serve as a tight upper bound at low SNR and closely match the simulated results at practical range of SNR. Some performance loss would be expected when compared with the optimal receiver which accounts for the multipath and multiuser effects jointly; however, it is beyond the scope of the paper to find out the amount of the loss. VII. C ONCLUSION In this letter, we evaluated the performance of two decorrelating receivers in a multipath Rician faded CDMA channel. In the case of decorrelating-first, we first derived approximate expressions for the average SER/BER under the assumption of perfect channel information available to the receiver. We then derived approximate expressions with channel uncertainty.
[1] S. Verdu, Multiuser Detection. Cambridge, UK: Cambridge University Press, 1998. [2] A. Klein and P. W. Baier, “Linear unbaised data estimation in mobile radio system applying CDMA,” IEEE J. Select. Areas Commun., vol. 11, pp. 1058–1066, Sept. 1993. [3] T. Kawahara and T. Matsumoto, “Joint decorrelating multiuser detection and channel estimation in asynchronous CDMA mobile communications channels,” IEEE Trans. Veh. Technol., vol. 44, pp. 506–515, Aug. 1995. [4] Z. Zvonar and D. Brady, “Suboptimal multiuser detector for frequencyselective Rayleigh fading synchronous CDMA channels,” IEEE Trans. Commun., vol. 43, pp. 154–157, Feb./Mar./Apr. 1995. [5] M. K. Varanasi, “Parallel group detection for synchronous CDMA communications over frequency-selective Rayleigh fading channels,” IEEE Trans. Inform. Theory, vol. 42, pp. 116–128, Jan. 1996. [6] S. S. H. Wijayasuriya, G. H. Norton, and J. P. McGeehan, “A sliding window decorrelating receiver for multiuser DS-CDMA mobile radio networks,” IEEE Trans. Veh. Technol., vol. 45, pp. 503–521, Aug. 1996. [7] J. Evans and D. N. C. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,” IEEE Trans. Inform. Theory, vol. 46, pp. 2059–2078, Sept. 2000. [8] E. K. Al-Hussaini and I. M. Sayed, “Performance of the decorrelating receiver for DS-CDMA mobile radio system employing RAKE and diversity through Nakagami fading channels,” IEEE Trans. Commun., vol. 50, pp. 1566–1570, Oct. 2002. [9] R. Prasad, H. S. Misser, and A. Kegel, “Performance evaluation of direct-sequence spread spectrum multiple-access for indoor wireless communication in a Rician fading channels,” IEEE Trans. Commun., vol. 43, pp. 581–592, Feb./Mar./Apr. 1995. [10] Y. Ma, T. J. Lim, and S. Pasupathy, “Error probability for coherent and differential PSK over arbitrary Rician fading channels with multiple cochannel interferers,” IEEE Trans. Commun., vol. 50, pp. 429–441, Mar. 2002. [11] K. Yu, J. S. Evans, and I. B. Collings, “Pilot symbol aided adaptive receiver for Rayleigh faded CDMA channels,” in Proc. IEEE GLOBECOM, (San Antonio, TX), pp. 753–757, 2001. [12] K. Yu, J. S. Evans, and I. B. Collings, “Performance Analysis of LMMSE receiver for M-ary QAM in Rayleigh faded CDMA channels,” IEEE Trans. Veh. Technol., vol. 52, pp. 1242–1253, Sept. 2003. [13] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [14] D. N. C. Tse and S. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, pp. 641–657, Mar. 1999. [15] J. G. Proakis, Digital Communications. McGraw-Hill, 3rd ed., 1995. [16] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels. New York: John Wiley & Sons, 2000. [17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. Fifth ed.,San Diego: Academic Press, 1994. [18] J. S. Evans, “Optimal resource allocation for pilot symbol aided multiuser receivers in Rayleigh faded CDMA channels,” IEEE Trans. Commun., vol. 50, pp. 1316–1325, Aug. 2002.
