Pilot-channel-aided successive interference cancellation for uplink WCDMA systems

Pilot-Channel-Aided Successive Interference

Cancellation for Uplink WCDMA Systems

Chih-Hsuan Tang, Student Member, IEEE, and Che-Ho Wei, Fellow, IEEE

Abstract—In this paper, a pilot-channel-aided

successive-interference-cancellation (SIC) scheme for uplink wideband code-division multiple-access system is presented. In order to alleviate the interferences from other users’ pilot-channel signals, this scheme performs pilot-channel-signal removal (PCSR) for all users before data detection. Three ordering methods for suc-cessive-cancellation process are discussed and compared in the aspect of architecture, latency, computational complexity, and error performance. In addition, channel estimation performed independently with the pilot-channel of each user is taken into consideration in the analyses. The pilot-channel-aided SIC per-forms variously with different ordering methods, pilot-to-traffic amplitude ratios, grouping intervals, power-distribution ratios, and propagation conditions. From our analyses and simulations, the SIC with ordering based on RAKE outputs at the first stage in properly chosen grouping interval is suggested over multipath fading channels when error performance and practicability are jointly concerned.

Index Terms—Channel estimation, code-division multiple

access (CDMA), ordering, pilot-channel, successive interference cancellation (SIC), third generation (3G).

I. INTRODUCTION

D

(CDMA) technique has attracted considerable attention due to its potential of high capacity and robust performance in fading channels [1], [2]. However, the capacity of CDMA systems is primarily limited by the multiple-access interference (MAI). High-power users can seriously corrupt users with low receiving power, which is known as the “near/far” problem. Therefore, much attention has been devoted to receiver schemes that are capable of canceling multiuser interference [2]. The optimal maximum-likelihood sequence estimation was first investigated in the study in [3]. To avoid the large number of computations in the optimal detector, various subop-timum detectors with lower complexity have been proposed [2], [4]–[8]. Among the suboptimum multiuser detectors, interference-cancellation (IC) techniques, including successive IC (SIC) and parallel IC (PIC) [5], [6] are considered as viable alternatives. According to a specific order, the SIC detects user

Manuscript received August 29, 2003; revised January 4, 2005, January 19, 2006, and August 8, 2006. This work was supported in part by the National Science Council, Taiwan, R.O.C., and in part by the Lee and MTI Center for Networking Research, National Chiao Tung University. The review of this paper was coordinated by Dr. Y. Yoon.

The authors are with the Institute of Electronics, National Chiao Tung Uni-versity, Hsinchu 300, Taiwan, R.O.C. (e-mail: chihhsuan.ee86g@nctu.edu.tw; chwei@mail.nctu.edu.tw).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.897226

data and cancels MAI in a serial manner. It has the advantage of simplicity, robustness, and having superior error performance over PIC in Rayleigh fading channels [9], [10] and nonequal user-power profile. Recent work shows SIC as a practical technique where its simplified version is employed as part of a commercial device to increase the EV-DO Rev A reverse-link voice-over-IP capacity by about 15% [11].

Several techniques have been proposed to overcome the disadvantages that prevent SIC from becoming a widely used technique [12]. First, the total decoding time increases linearly with the number of users, which can now be diminished by using a pipeline scheme [13], [14]. Second, although using SIC can reduce the other-cell interference [17], [18], and in-crease the system capacity [19], [20], SIC with controlled user power distribution [12], [15], [16] is somewhat complicated. Recently, a frame-error-rate-based outer loop power control is shown to be applicable to SIC [21]. In [22], a simple iterative algorithm to achieve the optimal control power distribution is given. Third, the performance of SIC is sensitive to channel-estimation accuracy due to error propagation. Pilot-channel can be employed to reduce the channel-estimation errors. In the third-generation (3G) mobile-communication systems [23], traffic- and pilot-channel signals are transmitted simultaneously by QPSK modulation. However, the traffic-channel signals are always interfered by other users’ pilot and traffic signal. Pilot-channel-signal-removal technique combined with RAKE receiver [24], PIC [25], or SIC [26] can be used to alleviate the interference from other users’ pilot signals.

It has been shown that the ordering method has a great effect on the error performance of SIC [28]–[31]. To determine the cancellation order, Ewerbring et al. [27] use the gain-ranking list obtained from channel estimates for successive cancella-tion. In [7], the correlator outputs pass on to a selector to find the user with the strongest correlation values for decoding and cancellation in each stage. For asynchronous systems, G bits of each user are grouped into the cancellation frame, and the ranking of the users is obtained from the averages of correla-tions over G bits. In [13], the average power is used to decide the cancellation order. Another method proposed is to use the signal strength obtained at the outputs of the first stage’s RAKE bank as the basis for ranking detection order over multipath fading channels [26]. In [30], the effects of ordering methods on hard-decision-based SIC with equal received power over additive white Gaussian noise (AWGN) and flat Rayleigh fad-ing channels for an asynchronous system are examined. In this paper, with a pilot-channel-signal-removed SIC in the uplink wideband code-division multiple-access (WCDMA) systems, the architectures for three ordering methods are presented for

Fig. 1. Transmitter model for the kth user.

practical use. To minimize the complexity of the receiver, we adopt the maximum-ratio-combining RAKE receiver with hard decision for data detection and windowed moving average tech-nique for channel estimation. The implementation issues, such as reordering frequency, latency, and computational complexity, are analyzed. In addition, we examine the parameters such as pilot-to-traffic amplitude ratio, ordering method, grouping interval, received-power-distribution ratio (PDR), and channel estimation, as well as timing-estimation errors. Some prelimi-nary results with simulation over multipath fading channels are given in [31].

The rest of this paper is organized as follows. The system model is described in Section II. In Section III, the architecture of pilot-channel-aided SIC employing three ordering methods are presented and compared. SIC I decides the cancellation order according to average power as that in the study in [13]. In SIC II, the G-bit summation of the RAKE output strengths in each stage are used to find the next detected user [7]. SIC III decides the cancellation order based on the G-bit summation of the RAKE bank output strengths at the first stage. The bit-error-rate (BER) analysis considering channel-estimation errors over AWGN channel in asynchronous systems is presented in Section VI. In Section V, we examine the analytical and simulated results for the three SICs in AWGN and multi-path Rayleigh fading channels. A brief conclusion is given in Section VI.

II. SYSTEMMODEL

Consider an asynchronous CDMA system with QPSK mod-ulation. The transmitter model for the kth user is shown in Fig. 1. The equivalent complex baseband representation of the transmitted signal at the mobile station is given by

sk(t) =

Pk{βdCO,d(t)Bk(t) + jβpCO,pApilot(t)} Ck(t). (1) Pkis the signal power, and power distribution of all users in the system follows a PDR, where PDR = Pk/Pk+1
1 for 1 ≤ k≤ K − 1 when there are K users in the system. For the sake of justice, total transmit power of all users is K with any given PDR, and PK = K(PDR− 1)/|(PDR)K− 1|.

βd and βp are the traffic- and pilot-channel gains, respec-tively. βd+ βp= 1, and βc= βp/βddenotes the pilot-to-traffic amplitude ratio.

Bk(t) = Σbk[nb]pb(t− nbTb) are the traffic-channel sig-nals, where bk is the binary data signal taking the values±1 with equal probability.

Apilot(t) = Σapilot[npi]ppi(t− npiTpi) are the uncoded

pi-lot signals modulated at Q-channel and have the same charac-teristic as Bk(t) but with a symbol period equal to Tpi.

Ck(t) = Σ(ck,I[n] + ck,Q[n])pc(t− ncTc) are the complex scramble sequences, where {ck,I[n]} and {ck,Q[n]} are the Gold sequences with period equal to the length of one frame and composed of N (SF) chips [23], where N is the bit number in a frame.

CO,x(t) = ΣcO,x[nc]pc(t− ncTc), where cO,x are the orthogonal-variable-spreading-factor (OVSF) codes with pe-riod equal to SF, where SF = Tb/Tcis the spreading factor for user data [23], and_nccO,x1[nc]cO,x2[nc] = 0 when x1= x2.

In the following, cO= cO,d, CO= CO,d, and cO,pis neglected, since it is an all-one sequence.

pc, pb, and ppi are unit power pulses with duration Tc, Tb,

and Tpi, respectively. SFpilot= Tpi/Tcis the spreading factor for pilot signal, and Fsf = Tpi/Tc.

After passing through a slowly fading channel, the received signal is represented as r(t) = K  k=1 P  p=1 αk,p(t)sk(t− τk,p) + n(t) (2)

where n(t) is the complex AWGN with zero mean, and one-sided power spectral density N0· τk,pand αk,p(t) are the delay and the complex channel gain of the kth user at the pth path, respectively. τk,pare uniformly distributed random variables in [0, Tb) for asynchronous systems. αk,p(t) is still a zero-mean complex-valued Gaussian random variable without loss of gen-erality when the carrier phase-shift part is absorbed in it [32]. For simplicity, we assume that the τk,pare perfectly estimated for all users, and the αk,p(t) are constant in a symbol interval. α(n)_k,p= αk,p(t)|t=τk,p+nTb, where

P p=1E[|α

(n)

k,p|2] = 1. III. PILOT-CHANNEL-AIDEDSIC

In this section, the pilot-channel-aided SICs with three order-ing methods are presented. A group of G-bit data is detected in the sequel, i.e., the nth bit of user J is detected before or after another user’s data is detected, where mG≤ n < (m + 1)G, and m is a non-negative integer. The graphical illustration is shown in Fig. 2, where 1≤ f ≤ F , and F is the number of RAKE fingers.

A. Channel Estimation

For ease of implementation, instead of using the optimal Wiener filter [33], a straightforward method to obtain the

Fig. 2. Received-signal timing and data detection group (a) τk,p;J,f ≥ 0 and (b) τk,p;J,f< 0, τk,p;J,f= τk,p− τJ,f.

channel estimate is to correlate the received signal with known pilot signal modulated at the Q-channel of user J at the f th path followed by a moving average filter to reduce the variance of the noisy channel estimates. When the channel variation is not very fast compared to WTb, the channel estimate is

ˆ α(n)_av;J,f = 1 W n+W/2−1_ nb=n−W/2 
PJα(nJ,fb)+ MAI (nb) ˆ αJ,f+ I (nb) ˆ αJ,f  . (3) W is an even integer denoting the length of the moving average filter. n is assumed to be equal to or larger than W/2. We neglect initial channel estimates, where n < W/2, since the long-term system performance measurement in a real system is negligible. The scramble sequence is reused when n + W/2 > N . MAI(n)_αˆJ,f = apilot[ nl] × 
PJ P  p=1,p=f α(n)_J,p ×  1 jβc λ(n)_J,p;J,f(τ1) + µ(n)J,p;J,f(τ1)  + K  k=1,k=J
Pk P  p=1 α(n)_k,p ×  1 jβc λ(n)_k,p;J,f(τ ) + µ(n)_k,p;J,f(τ )  (4) are MAI coming from other paths and other users. l = 1/Fsf,

and x indicates the largest integer smaller than or equal to x. τ and τ1denotes τk,p;J,f and τJ,p;J,f, respectively.

I_α(n)_ˆ J,f= 1 j2βpTb (n+1)Tb+τJ,f nTb+τJ,f Apilot(t− τJ,f)n(t)CJ∗(t− τJ,f)dt (5) is the AWGN-induced interference. (x)∗denotes the complex conjugate of x. λ(n)_k,p;J,f(τ ) and µ(n)_k,p;J,f(τ ) in (4) denote the in-terferences from other users’ traffic- and pilot-channel signals, respectively, and are defined as follows:

λ(n)_k,p;J,f(τ ) =  bk[n]ρ(n)k,p;J,f(τ ) + bk[n−1] ˙ρ(n)k,p;J,f(τ ), τ≥0 bk[n]ρ(n)k,p;J,f(τ ) + bk[n+1] ˙ρ(n)k,p;J,f(τ ), τ < 0 (6) and µ(n)_k,p;J,f(τ ) =            apilot[ nl] γ_k,p;J,f(n) (τ ) + apilot[ (n − 1)l] ˙γk,p;J,f(n) (τ ), τ≥ 0 apilot[ nl] γk,p;J,f(n) (τ ) + apilot[ (n + 1)l] ˙γk,p;J,f(n) (τ ), τ < 0 (7) where ρ(n)_k,p;J,f(τ ), ˙ρ(n)_k,p;J,f(τ ), γ_k,p;J,f(n) (τ ), and ˙γ_k,p;J,f(n) (τ ) in (A4)–(A7) are defined in the Appendix.

B. Data Detection

After obtaining the required channel estimates, if the re-ceived signal r(t) is directly sent into the correlator followed by a summation of all fingers’ outputs without performing pilot-channel-signal removal, the real part of the RAKE output is given as follows: ˆ Y_J(n)= F  f =1 Re  1 2βdTb (n+1)Tb+τJ,f nTb+τJ,f  ˆ α(n)_J,f ∗ r(t) × CO(t− τJ,f)CJ∗(t− τJ,f)dt  = F  f =1 Re  ˆ α(n)_av;J,f ∗
PJα(n)_J,fbJ[n]+ MAI_ˆ(n)_b J,f+ I (n) ˆbJ,f  (8) where MAI(n)_ˆ bJ,f =
PJ P  p=1,p=f α(n)_J,p  λ_J,p;J,f(n) (τ1)+ jβcµ (n) J,p;J,f(τ1)  + K  k=1,k=J
Pk P  p=1 α(n)_k,p  λ_k,p;J,f(n) (τ )+ jβcµ_k,p;J,f(n) (τ )  (9) and I_ˆ(n) bJ,f = 1 2βdTb (n+1)Tb+τJ,f nTb+τJ,f n(t)CO(t− τJ,f)CJ∗(t− τJ,f)dt. (10)

λ(n)_k,p;J,f and µ(n)_J,p;J,f have the same definition as λ(n)_k,p;J,f(τ ) in (6) and µ(n)_k,p;J,f(τ ) in (7), respectively, except that ρ(n)_k,p;J,f(τ ), ˙ρ(n)_k,p;J,f(τ ), γ_k,p;J,f(n) (τ ), and ˙γ_k,p;J,f(n) (τ ) are re-placed by ρ(n)_k,p;J,f(τ ), ˙ρ(n)_k,p;J,f(τ ), γ_k,p;J,f(n) (τ ), and ˙γ_k,p;J,f(n) (τ ), respectively [defined in (A8)–(A11) in the Appendix]. It is shown in (9) that the second and the fourth interference terms come from the pilot-channel signal of multipath and other users, respectively. To remove these MAIs, r(t) in (8) is replaced by ˆr(t) = r(t)−K_k=1Cˆpilot;k(t), where the estimated pilot

signal of user k is ˆ Cpilot;k(t) = F  p=1 jβpαˆ ( t−τk,p/Tb) av;k,p × apilot[ l(t − τk,p)/Tb] Ck(t− τk,p). (11) ˆ Y_J(n), MAI_ˆ(n) bJ,f, and I (n) ˆ bJ,f in (8) become ˆ Y(n)J , MAI (n) ˆ b  J,f, and I(n)_ˆ b  J,f, respectively, where MAI(n)_ˆ b  J,f =
PJ P  p=1,p=f α(n)_J,pλ_J,p;J,f(n) (τ1) + K  k=1,k=J
Pk P  p=1 α(n)_k,pλ_k,p;J,f(n) (τ ) + jβc 
PJ P  p=F +1 α(n)_J,pµJ,p;J,f(n) (τ1) + K  k=1,k=J
Pk P  p=F +1 α(n)_k,pµ_k,p;J,f(n) (τ )  − jβc  _F  p=1,p=f ˆ σ_J,p;J,f(n) (τ1) + K  k=1,k=J F  p=1 ˆ σ_k,p;J,f(n) (τ )  (12) with ˆ σ_k,p;J,f(n) (τ ) =           

apilot[ nl] MAI(n)_αˆav;k,pγ (n)

k,p;J,f(τ ) + apilot[ (n − 1)l] MAI(nαˆav;k,p−1) ˙γ

(n)

k,p;J,f(τ ), τ ≥ 0 apilot[ nl] MAI(n)αˆav;k,pγ

(n)

k,p;J,f(τ ) + apilot[ (n + 1)l] MAI(n+1)_αˆav;k,p˙γ

(n)

k,p;J,f(τ ), τ < 0. (13) The AWGN-induced interference I(n)_ˆ

b  J,fis expressed as I(n)_ˆ b  J,f = I_ˆ(n) bJ,f− jβc ×   K k=1,k=J F  p=1 ˆ ς_k,p;J,f(n) (τ ) + F  p=1,p=f ˆ ς_J,p;J,f(n) (τ1)   (14) where ˆ ς_k,p;J,f(n) (τ ) =            I_α(n)_ˆ av;k,pαpilot[ nl] γ (n) k,p;J,f(τ ) +I_α(n_ˆav;k,p−1)αpilot[ (n−1)l] γ (n) k,p;J,f(τ ), τ≥ 0 I_α(n)_ˆav;k,pαpilot[ nl] γ (n) k,p;J,f(τ ) +I_α(n+1)_ˆ av;k,pαpilot[ (n+1)l] γ (n) k,p;J,f(τ ), τ < 0. (15)

Both ˆσ_k,p;J,f(n) (τ ) and ˆς_k,p;J,f(n) (τ ) in (13) and (15) come from the channel-estimation errors.

In the following, SIC is performed to alleviate the MAI from other users. Generally speaking, signals are detected and canceled in order of their strength since the user with large received-signal power is more reliable but leads to serious interference to other users. However, as shown in (8), the transmitted-signal power P , channel gain α, and correlation terms, such as λand µ, all affect the received-signal strength and MAI to other users. Three ordering methods based on different consideration are described as follows.

1) SIC I—OrderingBased on Average Power: In SIC I, the cancellation order is decided by the average power mea-sured over a period much longer than 1/fd, where fd is the Doppler shift. According to computer simulations, fd/12 is large enough to be used as the reordering frequency. When the stationary channel is assumed, the reordering frequency can be much less than fd/12. To detect a group of G-bit data of all users, the architecture for SIC I is shown in Fig. 3, where the index in{·} at the output of each block denotes the processing step, andun = J denotes that the user with index J at bit index n is detected in the uth order/stage. For notational simplicity, n is omitted in the following. After performing channel estimation {1} and pilot-channel signal removing {2}, all information in Buffer A are sent to Buffer B in each G-bit interval {3}, and Buffer A can continue to collect the next G-bit information without delay. At this moment, through the buslike connection (bold line in part II), the user index and the corresponding channel estimates of the first detected user are sent to RAKE {4} followed by Decision {5}. Then, data respreading {6} and traffic-channel signal removing {7} are performed, and the remaining signal is sent back to Buffer B {8}. ˆrSIC;u(t) = ˆr(t) when u = 1. After that, according to

the cancellation order decided in part I, steps 5u− 1 to 5u + 3 are repeated for all the other users, where 2≤ u ≤ K.

2) SIC II—OrderingBased on RAKE Outputs After G-Bit Cancellation of One User: In SIC II, the G-bit sum-mation of the RAKE output strengths in each stage are used to find the next detected user. The architecture of SIC II is shown in Fig. 4. At first, we find the user with the maximum (m+1)G−1

n=mG |

ˆ

Y(n)_SICk| at the output of RAKE bank in part II {5} with the Finding Max block, and sent the user index to Buffer B {6}, whereY(n)ˆSICk=Yˆ

(n) k andYˆ (n) SICu=Yˆ (n) J with u = 1. With this index, decision making {7}, data respreading {8}, and removing from ˆr(t) {9} are performed followed by storing the remaining signal ˆrSIC;u+1(t) in Buffer B. Then, u

Fig. 3. Block diagram of SIC I with PCSR (1≤ f ≤ F ).

Fig. 4. Block diagram of SIC II with PCSR (1≤ f ≤ F ).

Fig. 5. Block diagram of SIC III with PCSR (1≤ f ≤ F ).

where 2≤ u ≤ K.Y(n)ˆ_SICkcan be obtained from (8), where J , ˆ

Y_J(n), and ˆr(t) in (8) are replaced by k,Y(n)ˆ_SICk, and ˆrSIC;u(t),

respectively. Note that the Finding Max block must suspend and wait untilY(n)ˆ_SICkare shown at the outputs of RAKE bank, where k denotes all the undetected users.

3) SIC III—OrderingBased on RAKE Outputs at First Stage in Each G-Bit Interval: In SIC III, the cancellation order is decided according to the signal strength of(m+1)G_n=mG −1|Y(n)ˆ_k | obtained in RAKE bank in part I {3}, as shown in Fig. 5. In each G-bit interval, information of all users are sent to Buffer B {4}. Then, in Finding Max block, the index of the user with maximum(m+1)G_n=mG −1|Y(n)ˆ_k | is found {6} and then sent back to Buffer B. When the first user (u = 1) is detected {8}, respread {9}, and canceled from ˆr(t) {10}, the Finding Max

block finds the user with the second largest signal strength according to the same (m+1)G_n=mG −1|Y(n)ˆ_k | at the same time {10}. Thus, the RAKE output of the second detected user can be obtained right after ˆrSIC;2(t) is sent out of Buffer B. For

2≤ u ≤ K, steps 5u + 1 to 5u + 5 are repeated.

For all SICs, in order to remove interferences from the pilot-channel signals of all users in the nth bit interval where mG≤ n < (m + 1)G, ˆr(t) with t > [(m + 1)G + 1]Tb in Buffer A are sent to Buffer B. After G-bit data of all users are detected, the remaining signal ˆrSIC;K+1(t) with t≥ (m + 1)GTb are cascaded with the incoming ˆr(t) for the next G-bit data detection. Ordering in SIC I only takes the transmitted power and long-term channel gain into consideration. This ordering method is often used in literature when SIC is compared with other ICs. SIC II and SIC III take advantages of instant

TABLE I

CHARACTERISTICS OFTHREESICSPERG BITS PERK USERS

received-signal strength, i.e., ordering is based on the compro-mise between reliability and channel estimates (weighting of the MAI), where SIC III is the simplified version of SIC II. Grouping interval G is used to decrease MAI due to asynchro-nous reception. Increasing grouping interval G can decrease uncanceled MAI to late detected users due to asynchronous reception. However, there is tradeoff between alleviating extra MAI and sensitivity to instant signal strength in SIC II and SIC III. For these two SICs, the early detected users may have small instantaneous SNR within a bit interval, as G increases.

In Table I, the reordering frequency, throughput, latency, and computational complexity of three SICs are summarized. The processes in part I of all SICs can be pipelined, since they do not need feedback information from part II.

For all SICs at nth bit interval, the real part of RAKE output in (8) of the uth canceled user with input ˆrSIC;u(t) is

ˆ

Y (n)

SIC
u = Pu

F  f =1  α(n) u 2 b_u[n] + F  f =1 Re  P_u  α(n)_u,f _∗ ×  MAI(n) _ˆ b_SIC
u,f + I (n)_ˆ b_SIC
u,f  +   MAI(n)_αˆav;
u,f+ I (n) ˆ αav;
u,f _∗

×P_uα(n)_u,fb_u[n] + MAI(n)_ˆ b_SIC
u,f + I(n)_ˆ b_SIC
u,f  (16) where ˆrSIC;u(t) = ˆr(t)− u−1 k=1 Cdata,k(t), and Cdata;k(t) = F  p=1 βdαˆ(av;k,p t−τk,p/Tb) ˆ bSIC;k[ (t − τk,p)/Tb] × CO(t− τk,p)Ck(t− τk,p). (17) mGTb≤ t − τk,p< (m + 1)GTb. ˆ

bSIC;k[n] is the data

decision where ˆbSIC;k[n] = sgn{Yˆ

(n)

SICk} for 1 <= k <= u − 1. It is shown in (18), shown at the bottom of the page,

MAI(n)_ˆ b SIC
_u,f = jβc   
PJ P  p=F +1 α(n)_u,pµ_u,p;u,f(n) (τ1) + K  k=1,k=J
Pk P  p=F +1 α(n)_k,pµ_k,p;u,f(n) (τ )      !

from uncanceled pilot

+  P_u P  p=1,p=f α_u,p(n) λ_u,p;u,f(n) (τ1)   !

from uncanceled multipath

+ K  k=u+1
Pk P  p=1 α(n)_k,pλ_k,p;u,f(n) (τ )   !

from uncanceled user

+ u−1_ k=1
Pk P  p=F +1 α(n)_k,pλ_k,p;u,f(n) (τ )   !

from uncanceled multipath of canceled users − jβc    F  p=1,p=f ˆ σ_u,p;u,f(n) (τ1) + K  k=1,k=J F  p=1 ˆ σ_k,p;u,f(n) (τ )      !

from imperfect pilot removal −    F  p=1,p=f ˆ ϕ (n) SIC;u,p;u,f(τ1) + u−1_ k=1 F  p=1 ˆ ϕ (n) SIC;k,p;u,f(τ )      !

from imperfect channel estimation & incorrect data decision

where τ denotes τ_k,p;u,f, and τ1 denotes τu,p;u,f. In (18) λ_k,p;u,f(n) (τ ) =            bk[n]ρ (n) k,p;u,f(τ ), n = mG bk[n]ρ (n) k,p;u,f(τ ) + bk[n− 1] ˙ρ (n) k,p;u,f(τ ), otherwise     , τ≥ 0 bk[n]ρ (n) k,p;u,f(τ ) + bk[n + 1] ˙ρ (n) k,p;u,f(τ ), τ < 0 (19) and (20), shown at the bottom of the page, with ∆(n)_k = 1 when bk[n]=

ˆ

bSIC;k[n], and ∆(n)k = 0 when bk[n] = ˆ

bSIC;k[n]. The

noise related interference is as follows: I (n)_ˆ b_SIC
u,f = I (n)_ˆ b
u,f −  _F  p=1,p=f ˆ ψ (n) SIC;u,p;u,f(τ1) + u−1_ k=1 F  p=1 ˆ ψ (n) SIC;k,p;u,f(τ )  (21)

where we have (22), shown at the bottom of the page. Ω(n)_k = 1 when bk[n]=

ˆ

bSIC;k[n], and Ω(n)k =−1 when bk[n] =

ˆ

bSIC;k[n]. The detection error is denoted as ε(n)

rˆbSIC;u= 2∆

(n)

ubu[n].

IV. PERFORMANCEANALYSIS INAWGN CHANNEL

In this section, a closed-form expression for the average BER in AWGN channel is presented. The pilot-channel-aided SICs

employing three ordering methods accompany with channel estimation are analyzed in asynchronous systems (but assumed chip synchronous, i.e., τ = 0 and τ= 0, where τ and τ are defined in the Appendix).

A. Channel Estimation

For the sake of simplicity, the long scramble sequences are viewed as random sequences, where cs are modeled as independent identically distributed random variables. The code correlations are assumed to be zero-mean complex-valued Gaussian random variables, where Var(Re[λ(n)_k;J(τk,J)]) = Var(Re[µ(n)_k;J(τk,J)]) = 1/(2SF) when chip synchronous as-sumption is made, and when chip asynchronous assump-tion is made, Var(Re[λ(n)_k;J(τk,J)]) = Var(Re[µ(n)k;J(τk,J)]) = 1/(3SF) [13]. I_αJ(n)_ˆ is also a zero-mean complex-valued Gaussian random variable. It can be shown that the mean-squared error (MSE) of the channel estimates is given by

MSE  ˆ α(n)_av;J  = 1 WE " ˆα(n) J −
PJα(n)J  2 # = 1 W $ β2 c+1 β2 c %  K k=1;k=J PK (SF)+ N0 2Tb   . (23) B. Data Detection

Without loss of generality, the time dependence is negligible in the following discussions. Thus, from (18), the decision

ˆ ϕ (n) SIC;k;u(τ ) =                            

MAI(n)_αˆav;k,p− 2√pkα(n)k,p+ MAI

(n) ˆ αav;k,p  ∆(n)_k  bk[n]ρ (n) k,p;u,f(τ ) + 

MAI(n_αˆav;k,p−1) − 2√pkα_k,p(n−1)+ MAI(n_αˆav;k,p−1)  ∆(n_k −1)  bk[n− 1] ˙ρ_k,p;u,f(n) (τ )   , τ≥ 0 

MAI(n)_αˆav;k,p− 2√pkα(n)k,p+ MAI

(n) ˆ αav;k,p  ∆(n)_k  bk[n]ρ_k,p;u,f(n) (τ ) + bk[n + 1] ˙ρ_k,p;u,f(n) (τ ), n = (m + 1)G− 1 

MAI(n)_αˆav;k,p− 2√pkα(n)k,p+ MAI

(n) ˆ αav;k,p  ∆(n)_k  bk[n]ρ_k,p;u,f(n) (τ ) +  MAI(n+1)_αˆk,p + 2√pkα(n+1)k,p + MAI (n+1) ˆ αav;k,p  ∆(n+1)_k  bk[n + 1] ˙ρ (n) k,p;u,f(τ ), otherwise              , τ < 0 (20) ˆ ψ (n) SIC;k,p;u,f(τ ) =−        I_α(n)_ˆ av;k,pΩ (n) k bk[n]ρ (n)

k,p;u,f(τ )+Iα(nˆav;k,p−1)Ω (n−1) k bk[n−1] ˙ρ (n) k,p;u,f(τ ), τ≥0 I_α(n+1)_ˆav;k,pΩ(n+1)_k bk[n+1] ˙ρ (n) k,p;u,f(τ ), n = (m+1)G−1 I_α(n)_ˆav;k,pΩ(n)_k bk[n]ρ (n)

k,p;u,f(τ )+Iα(n+1)ˆav;k,pΩ (n+1) k bk[n+1] ˙ρ (n) k,p;u,f(τ ), otherwise  , τ < 0 (22)

statistics at the uth stage are given as follows: ˆ

YSIC
u = Pubu[n] +

 P_u × Re&MAIav;u+ Iav;u

' b_u[n] +



P_u+ Re&MAIav;u+ Iav;u

'  × Re  MAI ˆ b_SIC
u + I ˆ b_SIC
u

+ Im&MAIav;u+ Iav;u' × Im  MAI ˆ b_SIC
u + I ˆ b_SIC
u . (24)

1) SIC I—OrderingBased on Average Power: According to the central-limit theorem, YˆSIC
u|bu are assumed as

Gaussian-distributed random variables with mean P_ub_uand variance

Var _ˆ

YSIC
u|bu



≈&P_u+ MSE(αˆav;u )' ·  Var  Re  MAI ˆ b_SIC
u  + Var  Re  I ˆ b_SIC
u  (25)

since the decision statistics are the sum of many variables. This assumption is commonly made in the case of successive cancellation [7]. As shown in the next section, it provides good approximation to the AWGN-dominant systems. The variation of signal power is neglected for simplicity, and the interference components are assumed to be uncorrelated with b_u. Accord-ing to λ(n)_k,p;u,f(τ ),ϕˆ_SIC;k;u(τ ), and ψˆ



SIC;k;u(τ ) in (19), (20), and (22), respectively, it is shown in (26), shown at the bottom of the page, where E[( ˆbSIC;k)2] = 4p(K)r,TI,k, and p

(K)

r,TI,k denotes the BER of user k [33]. The probability of τ≥ 0 or τ < 0 is equal to 1/2. In addition, only the expected values of the variance of partial code correlations are considered in (26). In addition Var  Re  I ˆ b_SIC
u  =(β 2 c + 1)N0 4Tb  1/G " 3 4· u− 1 β2 cW (SF) # + (1− 1/G) " u− 1 β2 cW (SF) # + 1 + K− 1 W (SF)  . (27)

In (26), there are terms coming from channel-estimation errors. Strictly speaking, the BER analysis must be analyzed with the method of robust statistics as far as error detection is concerned [34]. Nevertheless, for simplicity, the error probability for the uth canceled can be approximated as

p(K)_r, TI,
u ≈ Q * P2 u/Var _ˆ Y 

SIC
u|bu



(28)

where Q(x) = (1/2π)+_x∞exp(−(t2/2))dt. The average BER with K users in the system, thus, is ¯p(K)_r,TI=K_k=1p(K)_{r,T I,k}/K. 2) SIC II—OrderingBased on RAKE Outputs After G-Bit Cancellation of One User: SIC II finds the next detected user after each cancellation of the currently detected user, i.e., decision statistics of undetected users at each stage are used as ordering basis. For SIC II and SIC III, the BER analysis of the cases with G > 1 is very complicated, and thus, only the case of G = 1 is performed. Fortunately, as shown in Section V, when G increases, the BER of the three SICs are comparable in the AWGN channel. The error probability for the uth canceled user is given as follows: p(K)_r,TII,
u = (K− 1)! (K− u)! K  u=1 K 
1=1,
1=
u · · · K 
u−1=1,
u−1=
u&
1···
u−2 K 
u+1=u,
u+1=
1∼
u · · · K 
K=K−1,
K=
1∼
K−1 · ∞ 0 ∞ −∞ ∞ −∞ · · · ∞ −∞ fˆ Y  SIC
u (xu) × k=u−1_, k=1 fg
u,
k(xk+1− xk) × $ 1−Q $ P_k− xk c(k, k) % − Q $ P_k+ xk c(k, k) % % · K , k=u+1 $ Q$Pk− xu c(k, u) % − Q $ P_k+ xu c(k, u) % % × dx1dx2, . . . , dxu (29) Var  Re  MAI ˆ b_SIC
u  = 1 2SF·  1 G  3 4  _K  k=u+1 Pk+ u−1_ k=1

E&|MAIαˆav;k| 2'₊ u−1_ k=1 PkE " ( ˆbSIC;k)2 # +  1 4 u−1_ k=1 Pk   + $ 1− 1 G %   K k=u+1 Pk+ u−1_ k=1

E&|MAIαˆav;k| 2'₊ u−1_ k=1 PkE " ( ˆbSIC;k)2 #  + β2 c K  k=1,k=u

E&|MAIαˆav;k| 2'

  (26)

where c(ν1, ν2) = [Pν1+ MSE( ˆαaν;ν1)] 1/2 ·  1 2SF  3 4 K  k=1,k=ν1& 
_ν2−1
1 Pk+1 4 ν2−1 k=1 Pk +3 4 ν2−1 k=1 E " MAIαˆaν;k  2 #  + β 2 c 2SF × K  k=1,k=ν1 E " MAIαˆaν;k  2 # +(β 2 c + 1)N0 4Tb × $ 1+ K−1 W (SF)+ 3 4 ν2− 1 β2 cW (SF) % 1/2 (30) where fˆ Y  SIC
u

(x) are the probability distribution function (pdf) of Gaussian-distributed random variables YˆSIC
u with

mean−P_uand variance c(u, u) when b_u=−1. fg_ν1,ν2(x) are the pdf of Gaussian-distributed random variables gν1,ν2with

zero-mean and variance equal to 1 2SF[Pν1+ MSE ( ˆαaν;ν1)] ×3 4 " Pν2+ MSE ( ˆαaν;ν2) + (β2 c+ 1)N0 2β2 cW Tb # . (31) It is easy to show that the error probability of the case b_u= 1 is the same. Equation (29) denotes that the decision variables of the uth detected user falling on x1, x2, . . . , xu−1at the first,

second, . . ., (u− 1)th stage, respectively, are smaller than those of users1, 2, . . . , u − 1, which are decided to be detected at the first, second, . . ., (u− 1)th stage, respectively. When it comes to the uth stage, the decision variables fall on xu, since interferences from previously detected u− 1 users are canceled, and there are K− u users whose decision variables are smaller than that of useru where decision error occurs when xu≥ 0. To simplify the calculation, we assume that g_u,k(xk+1− xk)≈ 1, since Var(g_u,k) is small when it is compared to other random variables in (29). Then, the error probability is approximated as p(K)_r, TII,u= (K− 1)! (K− u)! K  u=1 K 
1=1,
1=<u · · · K 
u−1=1,
u−1=
u&
1···
u−2 × K 
u+1=u,
u+1=
1∼
u · · · K 
K=K−1,
K=
1∼
K−1 ∞ 0 f ˆ Y_SIC
u (x) · k=u_,−1 k=1 $ 1−Q $ P_k− x c(k, k) % Q $ P_k+ x c(k, k) % % × K , k=u+1 $ Q$Pk− x c(u, k) % Q $ P_k+ x c(u, k) % %dx (32)

The average BER of SIC II is given by p¯(K)_r,TII= K

k=1p

(K)

r,TII,k/K.

3) SIC III—OrderingBased on RAKE Outputs at First Stage in Each G-Bit Interval: In SIC III, user data is detected and canceled according to descending signal strength at the correla-tor outputs of the first stage, i.e., decision statistics of all users at the first stage are used as the ordering basis. Thus, the error probability for the uth canceled user is modified to

p(K)_r,TIII,
u = (K− 1)! (u− 1)!(K − u)! K  u=1 K 
1=1,
1=
u K 
2=2,
2=
u&
1 · · · K 
u−1=u−1,
u−1=
u&
1∼
u−2 K 
u+1=u,
u+1=
1∼
u · · · K 
K=K−1,
K=
1∼
K−1 × ∞ 0 ∞ −∞ f ˆ Y_SIC
u (x2) u−1  k=1 fg
_u,
k(x2− x1) · u_,−1 k=1 $ 1−Q $ P_k− x1 c (k, 1) % −Q $ P_k+ x1 c (k, 1) % % × K , k=u+1 $ Q$Pk− x1 c (k, 1) % −Q $ P_k+ x1 c (k, 1) % %dx1dx2. (33) Equation (33) indicates that a user with decision variables

ˆ

Y_u= c(u, 1), which have fallen on x1 at the first stage,

was decided to be canceled in the uth stage, since there are u− 1 users having larger value of decision variables and there are K− u users with smaller value of decision variables than useru. When it comes to the uth stage, the value of decision variables becomes x2 since interferences from previously

de-tected u− 1 users are canceled, and the decision error occurs when x2≥ 0 in the case of bu=−1. The decision errors of previously detected users are ignored for simplicity in both SIC II and SIC III. The average BER for a system with K users is ¯p(K)_r,TIII=K_k=1p(K)_r,TIII,k/K.

From the above analyses, channel-estimation errors are shown to lead to nonlinear influence on the decision statistics. Analytical methods used in SIC II and SIC III can be viewed as the modification of the order statistics [35], where the ordering basis of SIC II changes after each cancellation.

V. RESULTS ANDDISCUSSIONS

The general simulation parameters used are summarized in Table II according to Third-Generation Partnership Project standard [23], if they are not explicitly specified in the text. βc is chosen for the optimal BER according to the simulation results. The average SNR denotes the average of energy per bit of each user divided by noise variance. The chip resolution

TABLE II SIMULATIONPARAMETERS

in AWGN is chosen to be one chip, since chip synchronous is assumed, while the general chip resolution in multipath fading environment is 0.25 chips, where chip asynchronous is assumed.

A. Channel Estimation

Fig. 6 shows the analytical and simulated results of MSE with W = 64 and W = 128 over flat Rayleigh fading channels where Doppler shift fd= 222 Hz, corresponding to a vehicle speed of 120 km/h. The analytical MSE provides good approx-imation to the simulated MSE. It is shown that large βcresults in better channel estimates, since λ from other users in (4) is alleviated in proportion to the reciprocal of βc.

B. Data Detection

1) AWGN Channel: In Fig. 7, the influence of various G and PDR are examined in AWGN channel. The analytical results of G = 1 and G = 2400 are shown in dotted line where SIC I with G = 2400 is used to approximate SIC II and SIC III with G = 2400, and it shows good approximation to the simulated results. The BER difference of the three SICs is explicit when G is small and PDR is close to unity, and SIC II outperforms the other two SICs in this situation. As G or PDR increases, all SICs have almost the same performance. This is because the difference in the cancellation order of the three SICs is less and less as PDR or G increases.

Fig. 8(a) shows the individual BER of the uth detected user with eight active users in the system. When G = 1, we find that the late-detected users of SIC I outperform those of SIC II and SIC III. Nevertheless, the early detected users of SIC II and SIC III perform much better than those of SIC I. Thus, SIC II and SIC III still outperform SIC I after averaging, as shown in Fig. 8(b). Fig. 8(b) shows the average BER versus user number for the three SICs when G = 1 and 2400. Only SIC I with G = 2400 is presented since it is shown in Fig. 7 that the three SICs have almost the same BER. All users in SIC II in Fig. 8(a), except the last one, outperform those in SIC III, and

thus, the average BER of SIC II in Fig. 8(b) is lower than that of SIC III.

In Fig. 9, the BER with different PDRs and SNRs are exam-ined when G = 1 and G = 2400. βcis chosen for optimal BER of the corresponding SNR. In this paper, SIC without PCSR denotes that the pilot-channel signal are not removed first from the received signal, but they are removed accompanying with the data-channel signal of the corresponding user. Thus, SIC with PCSR outperforms SIC without PCSR only at the cost of demanding slightly more latency. As SNR increases, the benefit of SIC with PCSR becomes obvious, and all SICs with G = 2400 outperform SIC II with G = 1. For G = 2400, as shown in Fig. 9(d), increasing SNR also results in the increasing in PDR for the optimum BER. The reason is that MAI dominates the BER at high SNR, and increasing PDR can alleviate MAI from early detected users.

In Fig. 10, we examine the influence of pilot-to-traffic am-plitude ratio (βc). For the last two lines in Fig. 10(a) and (b), the PDRs are chosen for the minimum BER for the SIC with PCSR when G = 2400, i.e., PDR = 1.3(K = 8) and PDR = 1.1(K = 16) for all SICs. In Fig. 6, it is shown that large βc brings better channel estimates. However, large βcalso leads to poor data detection since the percentage of transmitted power for data signal decreases, and MAI from pilot signal increases. It is shown that SIC with PCSR can dramatically alleviate MAI from pilot signal, particularly when βc increases. In addition, SIC with PCSR is less sensitive to the variation of βc in both moderately loaded case in Fig. 10(a) and heavily loaded case in Fig. 10(b).

2) Multipath FadingChannels: Four cases of propagation conditions for multipath fading environments used in the fol-lowing are presented in Table III. It is assumed that all paths are tracked and combined in the RAKE receiver, i.e., F = P .

In Fig. 11, the three SICs in all considered channels perform much better than the RAKE receiver. The best BER occurs at PDR = 1.3 for SIC I in all cases, while it occurs at PDR = 1.0 for SIC II and SIC III in channel Case 1 and channel Case 2, and at PDR= 1.0 in channel Case 3 and channel Case 4. SIC II performs slightly better than SIC III when PDR is close to unity and G is small, and they both achieve better BER than SIC I. This is because SIC II and SIC III have the ability to track channel variation and SIC I does not.

In Fig. 12, the BER is investigated with channel estimation where W = 128. The timing-estimation errors are modeled as Gaussian-distributed random variables with zero-mean and variance 0.0625 (two samples) at 1/32 chips resolution, where 95% of the probability mass is concentrated within±4 samples [37]. The simulated results are similar to those in Fig. 11 except that the best BER occurs at PDR = 1.0 for SIC II and SIC III in all channel cases.

Fig. 13 indicates the BER of individual user in different cancellation order with the PDR for the optimum BER in each SIC in channel Case 3. For SIC I in Fig. 13(a), influences of G are obvious only when G is small. For SIC II in Fig. 13(b) and SIC III in Fig. 13(c), the early canceled users have smaller BER than that of the late canceled users for small G. As G > 1000, their BER becomes worse than SIC I where the early canceled users have larger BER than that of the late canceled

Fig. 6. MSE of channel estimates with various SNRs, flat Rayleigh fading channel, PDR = 1.0, and G = 1.

Fig. 7. BER versus PDR with different grouping interval G for (a) SIC I, (b) SIC II, and (c) SIC III. AWGN, known channel parameters, with PCSR.

Fig. 9. BER comparison with different PDRs and SNRs with/without PCSR for (a) SIC I with G = 1, (b) SIC II with G = 1, (c) SIC III with G = 1, and (d) SIC I with G = 2400. AWGN, channel estimation with W = 128.

Fig. 10. BER versus βcwith/without PCSR when there are (a) eight users and (b) 16 users in the system. AWGN, channel estimation with W = 128. TABLE III

PROPAGATIONCONDITIONS FORMULTIPATH FADINGENVIRONMENTS[36]

users. In addition, SIC II slightly outperforms SIC III when G is small.

The BER versus βc over all channel cases are given in Fig. 14, where G and PDR are selected for the best BER, i.e., G = 2400 in channel Case 1 and channel Case 2, G = 32 in channel Case 3, and G = 16 in channel Case 4 for all SICs. Similar to the results shown in Fig. 10 for AWGN channel, the PCSR helps to improve BER and increase βcfor the optimum BER. SIC II and SIC III still perform much better than SIC I. Moreover, SIC II and SIC III have similar performance.

In Fig. 15, the BER for various channel cases are examined. According to the results in Fig. 11, the PDRs for the minimum BER of each SIC are chosen. In channel Case 1 and channel Case 2, the one-frame long grouping interval, i.e., G = 2400, results in the best BER for all SICs. However, in channel Case 3 and channel Case 4, the optimum BER for SIC II and SIC III occur at about G = 100 and G = 50, respectively. Among all

Fig. 11. BER versus PDR for the three SICs in multipath fading channels. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. With PCSR, known channel parameters.

Fig. 12. BER versus PDR for the three SICs in multipath fading channels. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. Channel estimation with W = 128 and timing-estimation error with variance two samples at 1/32 chips resolution.

channel cases, SIC II and SIC III outperform SIC I, and except in channel Case 3 and channel Case 4 when G is small than about 20 with channel estimation, these two SICs almost have the same BER. It is worthy to note that SIC II and SIC III, with properly chosen G as PDR = 1, are suitable for fast fading channels such as channel Case 4.

From the above simulations and analyses, we find the following.

1) The SIC with PCSR can achieve better BER than that without PCSR when βc is large in both AWGN and selective fading channels. The optimal βc for SIC with PCSR is larger than that without PCSR. Larger βc also

Fig. 13. BER versus grouping interval G for user in different detection order for (a) SIC I with PDR = 1.3, (b) SIC II with PDR = 1.0, and (c) SIC III with PDR = 1.0. Channel Case 3, known channel parameter, with PCSR.

Fig. 14. BER versus βcfor multipath fading channels. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. Channel estimation with W = 128.

implies less timing-estimation error, which is critical in most communication systems.

2) The benefit of detecting and canceling a group of G-bit in AWGN channel becomes obvious when noise and MAI decrease. The reason has been mentioned in Section III-B. The relationship between fd and the op-timal G of SIC III-like SIC in multirate systems in multipath fading channels is examined in the study in [38], where the optimal G is shown inversely proportional to fdin fading environment. In addition, larger G results

in better BER in SIC I in all fading channel cases since the detection order is not affected by G.

3) When the noise (including estimation errors) and MAI increase, the PDR for the optimum BER in AWGN channel becomes closer to one as explained in Fig. 9. For SIC I over fading channels, it is shown that the optimum BER always occurs at PDR= 1, since detection order is fixed. As for SIC II and SIC III, whether PDR of the optimum BER is equal to unity depends on the channel condition.

Fig. 15. BER versus grouping interval G for multipath fading channels. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. With PCSR.

4) The BER of SIC I is inferior to the other two SICs when G is small. For SIC II and SIC III, they perform almost the same in both AWGN channel and fading channel when G or PDR is larger than one. In the view of BER, SIC II would be a good choice when it is applied to a heavily loaded system over AWGN channel with G = 1 or when G for the optimum BER is smaller than 161over multipath fading channels.

VI. CONCLUSION

In this paper, we investigate a pilot-channel-aided SIC scheme for uplink WCDMA systems. The scheme alleviates the interference from traffic-channel as well as pilot-channel signals of other users. We discuss SIC with three ordering methods and compare their corresponding architectures as well as processing delays and computational complexities. In addi-tion to showing the superiority over the convenaddi-tional RAKE receiver, the influence of ordering method, pilot-to-traffic am-plitude ratio, grouping interval, PDR, and channel- and timing-estimation errors are jointly examined and discussed. It is found out that ordering based on average power (SIC I) requires the least computational complexity at the expense of BER when grouping interval is small, and it seems to be suitable for AWGN channels with moderate loading. Ordering based on RAKE outputs after each cancellation of grouping-interval bits of one user (SIC II) outperforms ordering based on RAKE outputs at initial stage in each grouping interval (SIC III) when grouping interval is small. But SIC II has the highest compu-tational complexity and the largest processing delay among all SICs. SIC III is proposed to be a better choice over multipath

1_{With the simulation parameters used in this paper, G = 16 when the vehicle}

speed is at about 250 km/hr.

fading channels when BER and computational complexity are jointly concerned.

APPENDIX

We are to derive MAI(n)_αJ,fˆ in (4) by first considering the following expression: 1 2Tb (n+1)T b+τJ nTb+τJ bk(t− τk,p)CO(t− τk,p) × Ck(t− τk,p)C_J∗(t− τJ,f)dt. (A1) After taking definitions of bk(t), CO(t), and Ck(t) into (A1), it becomes 1 2Tb  n  m  q bk[n]cO[q]ck[q]c∗_J[m] · (n+1)SFT c+τJ,f nSFTc+τJ,f p(t− qTc− τk,p) × p (t − qTc− τk,p)p (t− mTc− τJ,p)dt where n, m, and q are all nonzero integers. With the timing-delay illustration shown in Fig. 2, we define τk,p = qTc+ τ_k,p , τ_k,p < Tc and τJ,f = mTc+ τJ,f , τJ,f < Tc for q≥ 0, m≥ 0, and τ_k,p − τ_J,f = τ_k,p;J,f . In the following, we take τ = τk,p;J,f and τ= τk,p;J,f for notational simplicity.

If τ = iTc+ τ for 0≤ τk,p;J,f < Tc, where i is a nonzero integer, when τ≥ 0, (A1) become

1 2Tb bk[n]  m  q cO[q]ck[q]c∗J[m] × (n+1)SFT c nSFTc+τ p (λ− (q − m + i)Tc− τ) p(λ)dλ + 1 2Tb bk[n− 1] m  q cO[q]ck[q]c∗_J[m] × nSFT c+τ nSFTc p (λ− (q − m + i)Tc− τ) p(λ)dλ (A2)

where λ = t− mTc. We can find that|(q − m + i)Tc+ τ| < Tc, such that the integral is nonzero, i.e.,−τ/Tc+ m− i − 1 < q <−τ/Tc+ m− i + 1. Thus, (A2) becomes

1 2SF Tc− τ Tc   bk[n] (n+1)SF_−1 m=nSF+i

cO[m− i]ck[m− i]c∗J[m]

+ bk[n− 1] nSF+i−1_ m=nSF cO[m− i]ck[m− i]c∗J[m]    when q = m− i, and 1 2SF τ Tc   bk[n] (n+1)SF_−1 m=nSF+i+1 cO[m− i − 1]ck[m− i − 1]c∗J[m] + bk[n− 1] nSF+i_ m=nSF cO[m− i − 1]ck[m− i − 1]c∗_J[m]   

when q = m− i − 1. Similarly, when τ < 0, τ = −iTc+ τ, where 0≥ τ>−Tcand (A1) become

1 2Tbbk[n]  m  q cO[q]ck[q]c∗J[m] × (n+1)SFT c+τ nSFTc p (λ− (q − m − i)Tc− τ) p(λ)dλ + 1 2Tbbk[n + 1]  m  q cO[q]ck[q]c∗J[m] × (n+1)SFT c (n+1)SFTc+τ p (λ− (q − m − i)Tc− τ) p(λ)dλ (A3)

where λ = t− mTc. We can find that|(q− m− i)Tc+ τ|<Tc, such that the integral is nonzero, i.e., −τ/Tc+ m + i− 1 < q <−τ/Tc+ m + i + 1. When q = m + i, (A3) becomes

1 2SF Tc− τ Tc   bk[n] (n+1)SF_−i−1 m=nSF cO[m + i]ck[m + i]c∗J[m] + bk[n + 1] (n+1)SF_−1 m=(n+1)SF−i cO[m + i]ck[m + i]c∗J[m]    when q = m + i + 1, (A3) becomes

1 2SF τ Tc   bk[n] (n+1)SF_−i−2 m=nSF−1 cO[m + i + 1] × ck[m + i + 1]c∗J[m] + bk[n + 1] × (n+1)SF_−1 (n+1)SF−i−1 cO[m + i + 1]ck[m + i + 1]c∗J[m]   . ρ(n)_k,p;J,f(τ ) =            1 2SF  (n+1)SF_ −1 m=nSF+i Tc−τ

Tc cO[m− i]ck[m− i]c

∗ J[m]+ (n+1)SF_ −1 m=nSF+i+1 τ TccO[m− i − 1]ck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF−i−1 m=nSF Tc−τ

Tc cO[m+ i]ck[m + i]c

∗ J[m]+ (n+1)SF−i−2 m=nSF τ TccO[m + i + 1]ck[m+ i + 1]c ∗ J[m]  , τ < 0 (A4) ˙ ρ(n)_k,p;J,f(τ ) =                    1 2SF _nSF+i−1 m=nSF Tc−τ Tc cO[m− i]ck[m− i]c ∗ J[m] + nSF+i m=nSF τ TccO[m− i − 1]ck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF −1 m=(n+1)SF−i Tc−τ

Tc cO[m + i]ck[m + i]c

∗ J[m] + (n+1)SF −1 m=(n+1)SF−i−1 τ TccO[m + i + 1]ck[m + i + 1]c ∗ J[m]  , τ < 0 (A5)

γ_k,p;J,f(n) (τ ) =            1 2SF  (n+1)SF −1 m=nSF+i Tc−τ Tc ck[m− i]c ∗ J[m] + (n+1)SF −1 m=nSF+i+1 τ Tcck[m− i − 1]c ∗ J[m]  , τ ≥ 0 1 2SF  (n+1)SF_−i−1 m=nSF Tc−τ Tc ck[m + i]c ∗ J[m] + (n+1)SF_−i−2 m=nSF τ Tcck[m + i + 1]c ∗ J[m]  , τ < 0 (A6) ˙γ_k,p;J,f(n) (τ ) =          1 2SF _nSF+i−1 m=nSF Tc−τ Tc ck[m− i]c ∗ J[m] + nSF+i m=nSF τ Tcck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF −1 m=(n+1)SF−i Tc−τ Tc ck[m + i]c ∗ J[m] + (n+1)SF −1 m=(n+1)SF−i−1 τ Tcck[m + i + 1]c ∗ J[m]  , τ < 0 (A7) ρ(n)_k,p;J,f(τ ) =                                1 2SF  (n+1)SF −1 m=nSF+i Tc−τ Tc cO[m− i]cO[m]ck[m− i]c ∗ J[m] + (n+1)SF_ −1 m=nSF+i+1 τ TccO[m− i − 1]cO[m]ck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF−i−1 m=nSF Tc−τ

Tc cO[m + i]cO[m]ck[m + i]c

∗ J[m] + (n+1)SF−i−2 m=nSF τ TccO[m + i + 1]cO[m]ck[m + i + 1]c ∗ J[m]  , τ < 0 (A8) ˙ ρ(n)_k,p;J,f(τ ) =                              1 2SF _nSF+i−1 m=nSF Tc−τ

Tc cO[m− i]cO[m]ck[m− i]c

∗ J[m] + nSF+i m=nSF τ TccO[m− i − 1]cO[m]ck[m− i − 1]c ∗ J[m]  , τ ≥ 0 1 2SF  (n+1)SF −1 m=(n+1)SF−i Tc−τ Tc cO[m + i]cO[m]ck[m + i]c ∗ J[m] + (n+1)SF_ −1 m=(n+1)SF−i−1 τ TccO[m + i + 1]cO[m]ck[m + i + 1]c ∗ J[m]  , τ < 0 (A9) γ_k,p;J,f(n) (τ ) =            1 2SF  (n+1)SF_ −1 m=nSF+i Tc−τ Tc cO[m]ck[m− i]c ∗ J[m] + (n+1)SF_ −1 m=nSF+i+1 τ TccO[m]ck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF_−i−1 m=nSF Tc−τ Tc cO[m]ck[m + i]c ∗ J[m] + (n+1)SF_−i−2 m=nSF τ TccO[m]ck[m + i + 1]c ∗ J[m]  , τ < 0 (A10) and ˙γ_k,p;J,f(n) (τ ) =          1 2SF _nSF+i−1 m=nSF Tc−τ Tc cO[m]ck[m− i]c ∗ J[m] + nSF+i m=nSF τ TccO[m]ck[m− i − 1]c ∗ J[m]  , τ≥ 0 1 2SF  (n+1)SF −1 m=(n+1)SF−i Tc−τ Tc cO[m]ck[m + i]c ∗ J[m] + (n+1) rmSF −1 m=(n+1)SF−i−1 τ TccO[m]ck[m + i + 1]c ∗ J[m]  , τ < 0 (A11)

We rearrange the above results and obtain (A4) and (A5), shown at the bottom of the previous page. In addition, we have (A6)–(A11), shown at the top of the page.

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