Performance Analysis of a Complex Multicode Tracking Loop in Bandlimited Rayleigh Fading Channels

DOI 10.1007/s11277-010-9924-8

Performance Analysis of a Complex Multicode Tracking

Loop in Bandlimited Rayleigh Fading Channels

Nicolas Y.-H. Hsu· Yu T. Su · Yuan-Bin Lin

Published online: 3 February 2010

© Springer Science+Business Media, LLC. 2010

Abstract This paper presents the root-mean-squared tracking error performance analysis of a class of coherent digital delay-locked loops for multicode direct sequence spread spec-trum signals in bandlimited correlated Rayleigh fading channels. In the transmit side, multiple independent PSK-modulated data streams in the in-phase and qudrature phase branches are spread by short mutually orthogonal codes before being further complex spread by a long complex PN code. We assume that the system employs a pilot code channel to assist the receiver’s synchronization and channel estimation. The proposed code tracking loop incorpo-rates the pilot-aided channel estimator and derives the timing error from all short-code-spread subchannels. Our analysis takes into account the effects of imperfect channel estimate, corre-lated frequency selective Rayleigh fading and band-limiting. Numerical results are presented to quantify the impact of the resulting multipath interference within the same code channel and amongst different code channels.

Keywords Code tracking· Complex multicode spreading · Bandlimited fading

1 Introduction

Various coherent and non-coherent pseudonoise (PN) code tracking loops for direct sequence spread spectrum (DD/SS) signal have been investigated and the related theory is well-established [3,4,6,17,18]. Most studies assume a tracking system which generates a timing error signal by comparing the early and late correlator outputs. In [2,5,9,19] use

This work was supported in part by National Science Council of Taiwan under Grant NSC-93-2218-E-009-015.

N. Y.-H. Hsu

Communication Platform R&D Department, ASUSTek Computer Inc., Taipei, Taiwan e-mail: Nicolas_Hsu@asus.com.tw

Y. T. Su (

B

Department of Communications Engineering, National Chiao Tung University, Hsinchu, Taiwan e-mail: ytsu@mail.nctu.edu.tw

various linear and nonlinear adaptive filters to estimate the timing error. A complicated multipath interference cancellation mechanism in the tracking loop analyzed in [21]. In spite of the structure differences the majority of existing literatures focus on the conventional single-code spread DS/SS or navigation systems. The design of the code tracking loop for a multicode complex spreading scheme like that used by the third generation (3G) CDMA sys-tems [15] is seldom discussed in open literature. A multicode CDMA system often employs orthogonal codes to channelize individual data streams. The composite multiple code channel signal is further spread by a long signature code. Complex code spreading is preferred to as it results in smaller peak-to-average power ratio.

When detecting a waveform that suffers from correlated frequency selective fading, a receiver needs a channel tracker to compensate for the channel distortions in various opera-tions. In particular, the estimated channel phase rotation has to be incorporated into the code tracking subsystem to de-rotate the received baseband phase so that coherent correlation and combining can be performed. Although channel estimator is a part of the code tracking system very few works investigate the effect of channel estimation error. In [14] analyzed the chan-nel estimation error effect on a single-path tracking loop but he did not taking into account the correlation of the fading process. This paper analyzes the performance of a complex delay-locked loop that is designed to take advantage of the multicode structure for tracking a bandlimited multicode DS/SS signal. Besides additive white Gaussian noise (AWGN), we also consider the effects of correlated Rayleigh fading, bandlimiting and imperfect channel estimation. Borio et al. [1] considered a tracking system structure, which is similar to [7] and what we propose here, that combines the timing estimates obtained from both data and pilot channels. Their work, however, assumes an AWGN channel and did not consider the fading and bandlimiting effects.

The rest of this paper is organized as follows. Section2introduces the transmitter and channel models, the proposed coherent code tracking loop structure and channel estimator and the associated parameters. In Sect.3, we present detailed tracking jitter performance analysis. Computer simulation results that validate our analysis is given in Sect.4. Finally, conclusion is given in Sect.5.

2 System and Channel Models

2.1 Transmitter Model

Figure1shows the transmitter model considered in this paper. Each parallel code channel car-ries a distinct data stream and is characterized by the short orthogonal spreading code used to provide the orthogonal channelization among different BPSK-modulated data streams, i.e., the PSK symbols carried by the code channels are spread by mutually orthogonal short codes. We assume that the symbol rate at each code channel is 1/T symbols/sec, although variable symbol rate transmission is feasible. By definingiM = integer part of i/M, and |i|M = i modulus M, M = T/Tc being the number of chips per short code period, we can express the orthogonal complex spread data sequence d_iMas

d_iM = d_iMI + jd_iMQ = 

k∈ΩI

Gkd_iM(k) W_|i|M(k) + j 

∈ΩQ

G_d_iM() W_|i|M() , (1)

where the subscripts are the time (chip) indices, the parenthized superscripts denote the sub-channel number,ΩI= {I1, I2, . . . , Im} and ΩQ= {Q1, Q2, . . . , Qn} are the index sets for

Σ channel I1 channel Im Σ channel Q1 channel Qn Baseband Filter channel I0 dI+jdQ Complex PN Code Generator RF Modulator and Amplifier dI dQ ( ) 1 i

G

G

G

G

Fig. 1 A complex spread multicode DS/SS transmitter

I - and Q-channels, respectively. Gl, where l∈ Ω and Ω = ΩI∪ ΩQ, are the channel gains normalized by that of Channel I0–the pilot channel, i.e., GI0 = 1. It should be noted that the

pilot does not carry any data, viz., d(I0)

iM = 1 ∀ i and uses the code of all ones.

As shown in Fig.1, the I and Q data streams are further scrambled by a long complex PN code with the same chip rate 1/Tc. After complex spreading, identical square root raised-cosine (SRRC) baseband filters with impulse and frequency responses gT(t) and GT( f ) are used to generate waveforms in the I - and Q-channels. The baseband filter and thus the power spectral density (PSD) of the transmitter signal is band-limited to the two-sided bandwidth

B= (1 + α)/Tcwhereα ∈ (0, 1) is the rolloff factor.

The equivalent baseband complex transmitted signal˜s(t) is a function of the complex-valued data sequence{d_iM} and the spreading sequence {c_iN}

˜s(t) = ∞  i=−∞

d_iMc_|i|NgT(t − iTc), (2) where Tcis the chip duration and N is the period of the long PN sequence. Since N M and I and Q branches use independent real PN codes, the complex PN sequence{ci} can be regarded as a sequence of independent and identically distributed (i.i.d.) complex random variables with the property

 c|i|Nc∗_{| j|N}  Ec|i|Nc∗_{| j|N}  = δ[i − j], (3)

where· is the time-averaging operator while E [·] represents the statistical expectation and ∗ denotes the complex conjugate. The Dirac function δ[n] is defined by

δ[n]= 

1 n= 0

0 otherwise (4)

2.2 Channel Model and Channel Estimation

We assume that the code tracking loop operates in a slow frequency-selective Rayleigh fad-ing channel, in which the received signal amplitude remains unchanged durfad-ing a short code

period. To overcome the frequency-selective fading, the so-called RAKE receiver is often employed. As in many investigations on multipath fading channels [16] and in industrial standard [15], we use the wide-sense stationary uncorrelated scattering (WSSUS) channel model with the following channel impulse response with respect to the receive timing refer-ence of a RAKE finger

¯hε(τ; t) =

L−1  l=0

ξl(t)δ[τ − (εTc+ τl)], (5) where L is the number of non-negligible paths,τlis the lth path’s relative delay andξl(t) is the corresponding complex gain. Without loss of generality, we setτ_= 0 so that −1/2 <

ε < 1/2 represents the normalized chip timing error of the tracking loop with respect to the

0th path. Letξ(k) be the random fading process with the U-shaped Jakes Doppler spectrum [8,12] S_ξ( f ) and autocorrelation function R_ξ(τ) given by

S_ξ( f ) = ρ_ξ·  1 π fd 1− ( f/fd)2−1/2 if| f | ≤ fd 0 otherwise, (6) Rξ(m) = ρξJ0(2π fd· m),

where fd is the maximum Doppler frequency andρ_ξ the average power. From (5) and (6), the space-time correlation function of the WSSUS fading process is of the form [12]

R_¯h(τ; m) =

L−1  l=0

R_ξl(m)δ(τ − τl), (7) where m indicates the time difference.

The receiver needs a channel estimator for RAKE-combining and to compensate for the phase rotation. The optimal filter for estimating the fading process{ξ(k)} with known PSD

S_ξ( f ) in the presence of background noise with PSD Sn( f ) is a Wiener filter which has the frequency response given by S_ξ( f )/[S_ξ( f ) + Sn( f )] [20]. In most case, the Doppler spectrum is unknown or may change with time. As{ξ(k)} is a low-pass random process with maximum Doppler frequency fd, the corresponding optimal Wiener filter is also a low-pass filter. Following the suggestion of [10], we use a low-pass filter with a smooth frequency response up to fd and a one-sided noise bandwidth greater than fd to produce the channel estimate.

3 Code Timing Tracking Jitter Analysis

This section provides the mean squared tracking error (jitter) analysis for the code-chan-nel-aided tracking system depicted in Fig.2. Before presenting our analysis, it is fitting to discuss the impact of the code tracking error on the bit error rate (BER) performance. 3.1 Impact of Code Timing Jitter on BER Performance

When operating in AWGN channels, the timing jitterσ_εof a PSK signal is often postulated to follow a Tikhonov probability distribution [11]

p(ε; σε) = exp cos 2πε/(2πσ_ε)2 I0 (1/2πσε)2 (8)

Σ 1 1 M M 1 1 M M * ( •) Re{ }• * ( ) • • Local PN Code Generator Chip Matched Filter Interpolator & Decimator Loop Filter ( ) MF g t ( ) y t r t( ) k e m zΔ m z Delay Tc Channel Estimator Σ Orthogonal code

+

Fig. 2 Block diagram of an all-code-channel-aided coherent PN code tracking loop

For the simplest case when only a single code channel is active, the resulting conditional BER performance, given a fixed code chip offsetε and i.i.d. data source, can be approximated by

Pb(e|ε) ≈ 1 2Q  2Eb N0 +1 2Q  2Eb N0 Rc(|ε|) M (9)

where Q(x)de f= 1/(√2)_x∞e−z2/2d z andRc(z) is the average partial autocorrelation

func-tion of the combined short and long codes.1_{The average BER is to be obtained by averaging}

Pb(e|ε) over the Tikhonov-distributed ε, i.e.,

Pb(e) = 

Pb(e|ε)p(ε; σε)dε (10) For a RAKE receiver operating in a multipath fading channel, the average partial autocor-relation function in (9) should be replaced by that of the combined multiple short codes and long complex code. Furthermore, Eb/N0should be replaced byξ2(Eb/N0), where ξ is the RAKE-combined instantaneous signal amplitude, and Pb(e) is to be obtained by taking average with respect to bothξ2andε, assuming slowly flat-fading over at least a bit (symbol) duration. The associated steady state tracking error distribution in such an environment is difficult to analyze. But for the special case when the fading amplitude can be modelled as a Gaussian process whose PSD bandwidth is much wider than that of the tracking loop, Ohlson [13] had obtained a closed-form expression. More detailed analysis and discussion are beyond the scope of this paper but it is safe to conclude that the estimated performance degradation due to code timing jitter alone is insignificant ifσ_ε< 0.05.

1_{Since the N M, the correlation is over a small portion (partial) of the long PN code period, and the}

3.2 Analysis of Various Correlator Outputs

Now consider the system shown in Fig.2in which the received signal is first down-converted to baseband and the resulting quadrature components are filtered by the chip matched filter whose impulse response is gM F(t) = (1/Tc)g∗T(−t). The filter output can be written as

˜r(t) = {˜s(τ) ⊗ ¯hε(τ; t)|τ=t+ ˜w(t)} ⊗ gM F(t) = ∞ i=−∞ d_iMc_|i|N L−1  l=0 ξl(t)g (t − [εTc+ τl] − iTc) + ˜n(t), (11) where g(t)= gT (t) ⊗ gM F(t) is the raised cosine function [16], G( f ) = (1/Tc)|GT( f )|2 and⊗ denotes the convolution operator. Since ˜ω(t)= ωc (t) + jωs(t) is a complex base-band AWGN process whose quadrature components are uncorrelated, zero mean Gaussian processes with two-sided power spectral density (PSD) S_w( f ) = N0/2P, P being the IF signal power, the PSD of the noise component, ˜n(t) = ˜ω(t) ⊗ gM F(t), becomes Sn( f ) =

N0G( f )/(2P).

We assume that code acquisition has been accomplished such that the chip timing offset is within±1/2 chip. The filter output is over-sampled by an A/D converter and the sampled sequence is forwarded to an interpolator filter. The interpolator together with the decima-tor are controlled by the chip timing error signal to generate two parallel one sample/chip streams called the middle stream and the early stream, respectively. The operation of the ideal interpolator and decimator resembles that of a down-sampler. Hence, the generation of the middle stream˜rj and the early stream˜rj+1/2is equivalent to sampling˜r(t) at tj = jTc and tj+1/2= ( j + 1/2)Tc, respectively, i.e.,

˜rk= ∞ i=−∞ d_iMc_|i|N L−1  l=0 ξl,kg([k − i − ε]Tc− τl) + ˜nk, (12) where k= j or k = j + 1/2, ξl_,kand˜nkrefer to the sampled version of the complex fading factor and the noise component. Figure2depicts a detailed block diagram of the proposed complex DS/SS digital DLL (DDLL) that is similar to an extension of that discussed in [17]. The late sample stream ˜rj_−1/2 is simply one-chip delayed version of the early sample stream˜rj_+1/2. They are complex multiplied by the conjugate of the local generated complex PN code then separated by distinct short code correlators. Only the pilot channel is used for tracking and channel estimation. The corresponding despread outputs at the early-late-dif-ference and middle branches are thus given by

˜z m = 1 M (m+1)M−1_ j=mM  ˜rj+1/2− ˜rj−1/2c∗_{| j|N}, (13) ˜zm = 1 M (m+1)M−1_ j=mM ˜rjc∗_{| j|N}. (14)

3.3 First- and Second-Order Statistics

The estimate ˆξm of the complex channel gain factor is obtained by passing the quadrature components of the despread samples˜zmthrough a pair of (identical) low-pass filters. The

resulting estimate can be expressed as

ˆξm = ˜zm⊗ he

m, (15)

where he_mis the impulse response of the channel estimation filter. The early-late difference ˜z

mis multiplied by the conjugate of the estimated phasor to obtain the error signal

λm=   ˜z m ˆξ∗ m |ˆξm|  . (16)

The sum ofλmis lowpass filtered by an integrated-and-dump filter with an integration period of K M chip intervals. The kth loop filter output ekis then used to control the interpolator filter output so that the equivalent sampling instant is updated every K M Tc seconds via

εk+1= εk+ μek, whereμ is the sensitivity of the loop filter. The average loop error characteristic is defined by

η(ε)= E {ek|εk = ε ∀ k} , (17)

which is related to the normalized S-curve of the loop via S(ε) = 1_Aη(ε), where A = dη(ε)

dε _ε=0, the slope atε = 0 of the loop error characteristic, is used for normalization such that d S_d(ε)_ε _

ε=0 = 1. For the tracking loop shown in Fig.2, we define the noise sequence

{Nk}, Nk = −μ[ek − η(εk)], so that the loop update equation can be expressed as εk+1=

εk− μAS(εk) + Nk. Invoking the linear approximation at neighborhood ofε = 0, S(ε) ≈ ε, we obtainεk+1= (1−μA)εk+ Nk. Letting a= 1−μA < 1, we rewrite the update equation asεk+1= aεk+ Nk.

From (17), we haveE{Nk} = −μ E{ek− η(εk)} = 0. To simply the analysis, we assume that{Nk} is a sequence of uncorrelated and identical distributed random variables, i.e.,

E{NkNk+n} = 0 if n= 0,

(18) V ar{Nk} =E{N_k2}.

After some algebra, we obtain E{εk} =E  akε0+ ak−1N0+ ak−2N1+ · · · + Nk−1  = ak_ε 0 (19) and  E{ε_k2}= a2E{ε_k−12 }+ 2a E{εk−1Nk−1} +E{N_k−12 } = a2k_ε2 0+  E{N_k−12 } 1 − a 2k 1− a2. (20)

As expected,εkis asymptotically independent ofε0, andεkandεk+nare almost uncorrelated if|n| 1. (19) and the properties of{Nk} imply that the steady-state timing error is given by σ2 ε =  E{N_k2}_ε=0 1− a2 , (21)

where the average power of the zero-mean loop noise is 

To evaluateE{e2_k}we first notice that both˜z _m and ˆξm are complex Gaussian distributed. Define

˜z m = x1+ jx2= r1ejφ1, ˆξm= x3+ jx4= r2ejφ2. (23) Applying the general Rayleigh fading process assumption, i.e., the in-phase and quadrature phase processes are mutually independent and identically distributed (i.i.d.), we have

E{x1x2} = E{x1x2} = 0 (24)

E{x12} = E{x22} = E{|˜z _m|2}/2= σ1 2 (25)

E{x32} = E{x42} = E{|˜ξ_m |2}/2= σ2 2 (26)

E{x1x3} = E{x2x4}= μ1 (27)

E{x1x4} = −E{x2x3}= μ2 . (28)

By transforming the rectangular coordinate (x1, x2, x3, x4) into the polar coordinate

(r1, r2, φ1, φ2) and making the change of variables, ψ = φ1− φ2, we obtain

p(r1, r2, ψ) = r1r2 2πσ₁2σ₂2(1 − ρ2₎ · exp  − 1 2(1 − ρ2₎  r₁2 σ2 1 + r22 σ2 2 − 2r1r2 σ1σ2(ρ1 cosψ − ρ2sinψ)  , (29) whereρ1= μ1 /(σ1σ2), ρ2= μ2 /(σ1σ2), and ρ=  ρ2 1 + ρ22.

Substituting (23) into (16), we obtainλm= r1cosψ. In “Appendix A” we prove that the corresponding first and second moments are

E{λm} = π  −π ∞  0 ∞  0 (r1cosψ)p(r1, r2, ψ)dr1dr2dψ = μ1 σ2  π 2, (30) E{λ2_m} = π  −π ∞  0 ∞  0 (r1cosψ)2p(r1, r2, ψ)dr1dr2dψ = σ12(1 + ρ12− ρ22). (31) The detail expression forσ12, σ22andμ1are given in “Appendix B”.

Assumingλm’s are i.i.d. random variables with first and second moments given by

E{ek} = E{λm}, V ar{ek} = 1

KV ar{λm}, (32)

where K is the accumulation interval, and substituting (32) into (22), we obtain  E{N_k2}= μ 2 K σ 2 1  1+ ρ₁2  1−π 2  . (33)

Substituting it into (21), replacing a by 1− μA, and taking the square root, we then obtain the chip timing jitter

σε= 1 A  2BLTc K σ 2 1  1+ ρ₁2  1−π 2  , (34)

where the normalized loop bandwidth is

BLTc= μA

2(2 − μA). (35)

4 Simulation Results and Discussion

This section reports some numerical behaviors of the proposed combined channel estimate and coherent code tracking subsystem. These results are obtained by using both the derived analytic expressions and computer simulations based on the following system parameters:

0 200 400 600 800 1000 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Channel Impulse Response and Estimation

Simulation Time (T=MT_c) In-Phase Component E c/N0 = 0 dB f d = 108.3 Hz

True fading process Estimated fading process

Fig. 3 Comparison of the trajectories of the true and estimated in-phase amplitude fading processes

0 200 400 600 800 1000 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Channel Impulse Response and Estimation

Simulation Time (T=MTc) Quadrature Component E c/N0 = 0 dB f d = 108.3 Hz

True fading process Estimated fading process

0 200 400 600 800 1000 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Phase of Channel Impulse Response

Simulation Time (T=MT_c)

E

c/N0 = 0 dB

f

d = 108.3 Hz

True fading process Estimated fading process

Fig. 5 Trajectories of the true and estimated phases of the correlated fading process

-20 -15 -10 -5 0 5 10 0.01 0.1 B_LT_c=0.01 B LTc=0.1 N o rm al iz ed C h ip Ti m ing Ji tt er σε E_c/N₀ Simulation Analysis

Fig. 6 Normalized timing jitter performance predicted by analysis and simulation in a bandlimited correlated

fading channel; BLTc= 0.1 and BLTc= 0.01 • The generator polynomial for the long code:

pL(x) = 1 + x2+ x3+ x5+ x6+ x7+ x10+ x16+ x17+ x18+ x19+ x21 +x22_{+ x}25_{+ x}26_{+ x}27_{+ x}31_{+ x}33_{+ x}35_{+ x}42

• Chip rate: Rc= 1/Tc= 1.2288 Mcps • Spreading factor: M = 16

• Maximum Doppler frequency: fd = 108.3 Hz

• Ec/N0: Pilot channel’s chip energy to noise power level ratio

• Number of code channels: 3 channels for the I -branch and 2 channels for the Q-branch • Power vector P: P= [P0I P1I P2I P1Q P2Q] = [1 1 1 2 5]

-20 -15 -10 -5 0 5 10 0.01 K=192 K=96 N o rm al iz ed C h ip Ti m ing Ji tt er σε E_c/N₀ B_LT_c = 0.023 Simulation Analysis

Fig. 7 Effect of the filter window size on the normalized timing jitter performance predicted by analysis and

simulation in a bandlimited correlated fading channel

-20 -15 -10 -5 0 5 10 0.01 P= [1 0 0 0 0 ] P= [1 1 1 2 5 ] Nor maliz ed RMS Chip Timing Jitter σε E_c/N₀ B LTc= 0 .0 2 3 S im u la tio n A n a lysis

Fig. 8 Effect of self-interference on the code tracking performance in a correlated fading channel

We use a two-ray Rayleigh fading model with the fading process of each ray derived from Jakes’ two-dimensional isotropic scattering model. To estimate the channel gain fac-tor, a forth-order Butterworth low-pass filter with sampling frequency 76.8 kHz and cutoff frequency 1 kHz is used.

Figures3 and4compare the estimated trajectories of the magnitudes of in-phase and quadrature-phase channel fading processes. Figure5compares the true and estimated fading phase trajectories and indicates that there is a delay between simulated and estimated phase trajectories which is caused by the low-pass filtering process and is inverse proportional to the filter bandwidth. The estimate phase delay will impact the tracking performance.

Figures6and7compare the simulation and analytical results for various normalized loop bandwidths, BLTc, and filter window size K . As is expected, the jitter performance is improved by decreasing the loop bandwidth or increasing the filter window size. The increase of the window size enhances the correlation of the estimated error indicator samplesλmwhile for a fixed Doppler shift the increase of the normalized loop bandwidth tends to decrease the

correlation. The increased correlation onλmleads to our overestimate on the timing jitter especially when Ec/N0is high.

The existence of a floor in each jitter performance curve is due to the self-interference caused by data channels. Bandlimiting leads to a non-negligible chip pulse extension in the time domain and thus the orthogonality among various orthogonal codes is destroyed, creating non-zero cross-correlation between the pilot code and other channelization codes. Figure8examines the effect of self-interference. We compare the timing jitter performance for the case when five code channels are all activated with that when only the pilot chan-nel is used. Imperfect chanchan-nel estimation also enhances the self-interference effect which dominates the jitter performance at higher Ec/N0.

5 Conclusion

We have investigated the behavior of a coherent digital delay-locked loop for tracking band-limited complex DS/SS signals in slow frequency-selective Rayleigh fading channels and analyzed the corresponding root mean square tracking jitter performance. The code tracking loop uses both pilot and data channels and incorporates a suboptimal channel estimator to compensate for the fading effect.

Simulation results indicate that our analysis does offer accurate performance prediction. Both our analysis and simulation results pointed out that, for multicode complex PN code tracking loop, the bandlimiting and multipath effects destroy the orthogonality of the short orthogonal spreading codes and induces self-interference among code channels, resulting in an irreducible tracking jitter. Interference cancellation is therefore, called for to remove this performance lower bound.

Appendix A Derivation of the First Two Moments ofλm Invoking (29) and the changes of variables r₁ = r1

σ1 √ 1−ρ2 and r  2=_σ r2 2 √ 1−ρ2, we obtain E{λm} = π  −π ∞  0 ∞  0 (r1cosψ)p(r1, r2, ψ)dr1dr2dψ = σ1(1 − ρ2)3/2 2π ∞  0 ∞  0 π  −π r₁2r2cosψ · exp  −1 2 r₁2+ r₂2− 2r1r2(ρ1cosψ − ρ2sinψ)  dψdr1dr2 (A.1)

The substitutions x= r1cosψ and y = r1sinψ then lead to

E{λm} = σ1(1 − ρ 2₎3/2 2π ∞  0 ∞  −∞ ∞  −∞ xr2 · exp  −1 2 x2+ y2+ r₂2− 2ρ1r2x+ 2ρ2r2y  d yd xdr2

= σ1(1 − ρ2₎3/2 ∞  0 r₂2exp  −1 2(1 − ρ 2_)r2 2  dr2 = μ1 σ2  π 2 (A.2)

Similarly, we can shown

E{λ2_m} = π  −π ∞  0 ∞  0 (r1cosψ)2p(r1, r2, ψ)dr1dr2dψ = 1− ρ2 2π ∞  0 ∞  0 π  −π (σ1  1− ρ2_r 1cosψ) 2 r1r2 · exp  −1 2 r₁2+ r₂2− 2r1r2(ρ1cosψ − ρ2sinψ)  dψdr1dr2 = σ12(1 − ρ_√ 2)2 2π ∞  0 ∞  −∞ x2r2exp  −1 2(x − ρ1r2) 2  exp  −1− ρ2 2 r 2 2  d xdr2 = σ12(1 − ρ_√ 2)2 2π ⎧ ⎨ ⎩ 2√π 4(1₂)3/2 ∞  0 r2exp  −1− ρ2 2 r 2 2  dr2+ ρ12 √ 2π × ∞  0 r₂3exp  −1− ρ2 2 r 2 2  dr2 ⎫ ⎬ ⎭ = σ2 1(1 + ρ12− ρ22). (A.3)

Appendix B The Derivation of the Average Autocorrelation Function

Rewrite (14) as˜z _m= E[˜z _m] + {˜z_m }= ˜u _m+ ˜v _m, From (12), we have

˜u m= 1 M (m+1)M−1_ j=mM ∞  i=−∞ d_iMc_|i|Nc∗_{| j|N} L−1  l=0 ξl,mfl( j − i − ε, 1/2) (B.1) ˜v m = 1 M (m+1)M−1_ j=mM  ˜nj+1/2− ˜nj−1/2c∗_{| j|N} (B.2) where

fl(i, j)= g([i + j]Tc − τl) − g([i − j]Tc− τl). (B.3) In the slow fading channel, the discrete fading factor,ξl,m, remains unchanged over an orthog-onal code period. To deriveσ₁2, σ₂2 andμ1, we ought to calculate the correlation of signal components˜u _m, ˜umand noise components˜v m, ˜vm.

 E[u _mu ∗_m−k] = 1 M2 & E ⎡ ⎢ ⎢ ⎣ (m+1)M−1_ j=mM ∞  i=−∞ d_iMc_|i|Nc∗_{| j|N} L−1  l=0 ξl,mfl( j − i − ε, 1/2) · ⎛ ⎜ ⎝ (m−k+1)M−1_ j=(m−k)M ∞  i=−∞ d_iMc_|i|Nc∗_{| j|N} L−1  l=0 ξl,m−kfl( j − i − ε, 1/2) ⎞ ⎟ ⎠ ∗⎤ ⎥ ⎥ ⎦ 3 = 1 M2 (m+1)M−1_ j=mM ∞  i=−∞ (m−k+1)M−1_ q=(m−k)M ∞  p=−∞ & E  d_iMd_pM∗ c_|i|Nc_{| j|N}∗ c_|p|N∗ c_|q|N · L−1  l=0 L−1  l=0 ξl,mξl∗,m−kfl( j − i − ε, 1/2) fl(q − p − ε, 1/2) 3 (B.4)

From (6) and (7), we obtain the autocorrelation of the fading gain factor:

E _L−1  l=0 L−1  l=0 ξl,mξl∗,m−k  = L−1  l=0 ρξlJ0(2π fdkT). (B.5) Substituting above equation into (B.4) and after some algebra, we obtain

 E[u _mu ∗_m−k] = 1 M2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ M−1_ j=0 (−k+1)M−1_ q=−kM Rd( j − q) L−1  l=0 ρξlJ0(2π fdkT) f 2 l (−ε, 1/2) +M Rd(0)δ[k] ∞  i=−∞ i=0 L−1  l=0 ρξlJ0(2π fdkT) f 2 l (−i − ε, 1/2) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (B.6)

Similarly, we can show that  E[umu∗_m−k] = 1 M2 & E ⎡ ⎢ ⎢ ⎣ (m+1)M−1_ j=mM ∞  i=−∞ d_iMc_|i|Nc_{| j|N}∗ L−1  l=0 ξl,mg([ j − i − ε]Tc− τl) · ⎛ ⎜ ⎝ (m−k+1)M−1_ q=(m−k)M ∞  p=−∞ d_pM∗ c_|p|N∗ c_|q|N L−1  l=0 ξl,m−kg([p − q − ε]Tc− τl) ⎞ ⎟ ⎠ ∗⎤ ⎥ ⎥ ⎦ 3

= 1 M2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ M_−1 j=0 (−k+1)M−1_ q=−kM Rd( j − q) L−1  l=0 ρξlJ0(2π fdkT)g 2_(−εT c− τl) +M Rd(0)δ[k] ∞ i=−∞ i=0 L−1  l=0 ρξlJ0(2π fdkT)g 2_{([−i − ε]T} c− τl) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (B.7) and  E[u _mu∗_m−k] = 1 M2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ M−1_ j=0 (−k+1)M−1_ q=−kM Rd( j − q) L−1  l=0 ρξlJ0(2π fdkT) fl(−ε, 1/2)g(−εTc− τl) +M Rd(0)δ[k] ∞  i=−∞ i=0 L−1  l=0 ρξlJ0(2π fdkT) fl(−i − ε, 1/2)g([−i − ε]Tc− τl) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (B.8)

The average autocorrelation function of the noise component is derived as follows.  E[v_m v_m−k∗ ] = 1 M2 & E ⎡ ⎣(m+1)M−1 j=mM (˜nj+1/2− ˜nj−1/2)c∗| j|N (m−k+1)M−1_ j=(m−k)M (˜nj+1/2− ˜nj−1/2)∗c| j|N ⎤ ⎦ 3 = 1 M2 &_(m+1)M−1  j=mM (m−k+1)M−1_ q=(m−k)M E (˜n_j+1/2− ˜n_j−1/2)(˜nq+1/2− ˜nq−1/2)∗ c∗_{| j|N}c_|q|N 3 = 1 M2 (m+1)M−1_ j=mM (m−k+1)M−1_ q=(m−k)M  4σ_n2δ[ j − q] − 2σ_n2δ[ j − q − 1] − 2σ_n2δ[ j − q − 1] ×c∗_{| j|N}c_|q|N  = 1 M2 (m+1)M − 1_ j=mM (m − k+1)M − 1_ q=(m − k)M  4σ_n2δ[ j − q] − 2σ_n2δ[ j − q − 1] − 2σ_n2δ[ j − q−1]δ[ j − q] = 4 Mσ 2 nδ[k] (B.9)

Similarly, we can verify thatE[vmv∗_m−k]= _M2σ_n2δ[k] andE[v_m v∗_m−k]= 0. Hence σ₁2, σ₂2

σ2 1 = 1 2  E{|˜z_m |2} = 1 2M2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ L−1  l=0 ρξlJ0(0) M−1_ j=0 M−1_ q=0 Rd( j − q) + Rd(0)M L−1  l=0 ρξlJ0(0) × ∞  i=−∞ i=0 f_l2(−i − ε, 1/2) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭+ 2 Mσ 2 n. (B.10) σ2 2 = 1 2  E{|ˆξm|2}  = 1 2 ∞  α=−∞ 5 E[um_−αu∗_m] + E[vm_−αv_m∗]6 ∞  β=−∞ he[α + β]he[β] = 1 2 ∞  α=−∞ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 M2 M−1_ j=0 (α+1)M−1_ q=αM Rd( j − q) L−1  l=0 ρξlJ0(−2π fdαT )g 2_(−εT c− τl) +δ[α]Rd(0)1 M ∞  i=−∞ i=0 L−1  l=0 ρξlJ0(−2π fdαT )g 2_{([−i − ε]T} c− τl) + δ[α] 2 Mσ 2 n ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ × ∞ β=−∞ he[α + β]he[β]. (B.11) and μ1 = 1 2  E  ˜z mˆξm∗  = 1 2 ∞  α=−∞ he[α]5E[u _mu∗_m−α]+E[v_m v∗_m−α]6 = 1 2M2 ∞  α=−∞ he[α] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ M−1_ j=0 (−α+1)M−1_ q=−αM Rd( j − q) × L−1  l=0 ρξlJ0(2π fdαT ) fl(−ε, 1/2)g(−εTc− τl) + δ[α]Rd(0)M × L−1  l=0 ρξlJ0(2π fdαT ) ∞  i=−∞ i=0 fl(−i − ε, 1/2)g([−i − ε]Tc− τl) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭. (B.12)

References

1. Borio, D., Mongrédien, C., & & Lachapelle, G. (2009). Collaborative code tracking of composite GNSS signals. IEEE Journal of Selected Signal Processing, 3(4), 613–626.

2. El-Tarhuni, M., & Ghrayeb, A. (2004). A robust PN code tracking algorithm for frequency selective Rayleigh-fading channels. IEEE Transactions on Wireless Communication, 3(4), 1018–1023. 3. Gaudenzi, R. D. (1999). Direct-sequence spread-spectrum chip tracking in the presence of unresolvable

multipath components. IEEE Transactions on Vehicular Technology, 48(5), 1573–1583.

4. Gaudenzi, M. D., Luise, M., & Viola, R. (1993). A digital chip timing recovery loop for band-limited direct-sequence spread-spectrum signals. IEEE Transactions on Communication, 41(11), 1760–1769. 5. Gustafson, D. E., Dowdle, J. R., Elwell, J. M., & Flueckiger, K. W. (2009). A nonlinear code tracking

filter for GPS-based navigation. IEEE Journal of Selected Signal Processing, 3(4), 627–638. 6. Holmes, J. K. (1982). Coherent spread spectrum systems. New York: Wiely.

7. Hsu, Y. -H. (2000). Analysis of complex code tracking with channel estimation in bandlimited

Ray-leigh fading channels, Master thesis, Dept. Commun. Eng., National Chiao Tung University, Hsinchu,

Taiwan, July.

8. Jakes, W. C., Jr. (1974). Microwave mobile communications. New York: Wiely.

9. Li, H., Wang, R., & Amleh, K. (2006). Blind cod-timing estimation for CDMA systems with bandlimited chip waveforms in multipath fading channels. IEEE Transactions on Communication, 54(1), 141–149. 10. Ling, F. (1999). Optimum reception, performance bound, and cutoff rate analysis of references-assisted coherent CDMA communications with applications. IEEE Transactions on Communication,

47(10), 1583–1592.

11. Lindsey, W. C., & Simon, M. K. (1973). Telecommunication systems engineering, Chap. 9, Prentice-Hall. 12. Meyr, H., Moeneclaey, M., & Fechtel, S. A. (1997). Digital communication receivers: Synchronization,

channel estimation, and signal processing. New York: Wiely.

13. Ohlson, J. E. (1978). Statistics of the first-order phase-locked loop with fluctuating signal ampli-tude. IEEE Transactions on Communication, COM-26, 1472–1474.

14. Park, H. -R. (2006). Performance analysis of a decision-feedback coherent code tracking loop for pilot-symbol-aided DS/SS system. IEEE Transactions on Vehicular Technology, 55(4), 1249–1258. 15. Physical Layer Standard for cdma2000 Spread Spectrum System, 3GPP2, C.S0002 Version 3.0, June

15, 2001,www.3gpp2.org.

16. Proakis, J. G. (1995). Digital communications (3rd ed.). New York: McGraw-Hill.

17. Sawahashi, M., Adachi, F., & Yamamoto, H. (1998). Coherent delay-locked code tracking loop using time-multiplexed pilot for DS-CDMA mobile radio. IEICE Transactions on Communication,

E81-B(7), 1426–1432.

18. Simon, M. K., Omura, J. K., Scholtz, R. A., & Levitt, B. K. (1985). Spread spectrum communications (Vol. III). Rockville, MD: Computer Science Press.

19. Ueng, F. -B., Chen, J. -D., & Tsai, S. -C. (2008). Adaptive DS-CDMA receiver with code tracking in phase unknown environments. IEEE Transactions on Wireless Communication, 7(4), 1227–1235. 20. Van Trees, H. L. (1968). Detection, estimation, and modulation theory—Part I. New York: Wiely. 21. Wu, T. -M., & Tsai, T. -H. (2007). Digital code tracking loops over frequency-selective fading

channels, In Proceedings of ICC2007, ICC 2009, Glasgow, Scotland, 24–28 June (pp. 5246–5251). Author Biographies

Nicolas Y.-H. Hsu received the B.S. degree in electrical engineering

from National Taiwan University of Science and Technology, Taipei, Taiwan in 1998. In 2000 he received the M.S. degree in communica-tions engineering from the National Chiao Tung University, Hsinchu, Taiwan. Since 2000, he joined ASUSTeK Computer Incorporation as an R&D engineer in the Communication Software Department of the Personal Mobile Devices Business Unit.

Yu T. Su received the B.S. and Ph.D. degrees in electrical

engineer-ing from Tatung Institute of Technology, Taipei, Taiwan and the Uni-versity of Southern California, Los Angeles, USA, in 1974 and 1983, respectively. From 1983 to 1989, he was with LinCom Corporation, Los Angeles, USA, where his was a Corporate Scientist involved in the design of various measurement and digital satellite communica-tion systems. Since September 1989, he has been with the Nacommunica-tional Chiao Tung University, Hsinchu, Taiwan where he is a Professor at the Department of Electrical Engineering. He was an Associative Dean of the College of Electrical and Computer Science from 2004 to 2007, and was the Head of the Communications Engineering Department from 2001 to 2003. He is also affiliated with the Microelectronic and Information Systems Research Center of the same University and served as a Deputy Director from 1997 to 2000. From 2005 to 2008, he the Area Coordinator of Taiwan National Science Council’s Telecom-munications Programmer. His main research interests include commu-nication theory and statistical signal processing.

Yuan-Bin Lin received his B.S. degree and M.S. degree in

communi-cations engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 1998 and 2000, respectively. His primary research inter-ests include all aspects of wireless communication with emphases on multiple access techniques and radio resource management. From 2003 to 2005, he was a lecturer in the Ta Hwa Institute of Technology, Hsinchu, Taiwan. In 2009 he received the Ph.D. degree in commu-nications engineering from National Chiao Tung University, Hsinchu, Taiwan. He is currently a Post doctoral Research Fellow in the Department of Communications Engineering, National Chiao Tung University. His research interests include communication theory, radio resource management, convex optimization and cross-layer optimiza-tion for efficient transmissions on next generaoptimiza-tion wireless networks.