Multirate adaptive filtering for low complexity DS/CDMA code acquisition

Multirate adaptive filtering for low complexity DS/CDMA

code acquisition

Hua-Lung Yang



, Wen-Rong Wu

Department of Communication Engineering, National Chiao-Tung University, Hsin Chu 300, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 1 June 2008 Received in revised form 27 October 2008

Accepted 27 December 2008 Available online 10 January 2009 Keywords:

Code acquisition Low complexity Adaptive filtering Multirate signal processing DS/CDMA

a b s t r a c t

Code acquisition in CDMA systems is conventionally conducted with a matched-filter based structure. However, the performance of this method degrades greatly while multiple access interference presents. Recently, an adaptive filtering scheme was proposed to solve this problem. It has been shown that the computational complexity of this approach is proportional to the delay uncertainty and inversely proportional to the required acquisition time. When propagation delay is large and the required acquisition time is short, the computational complexity of the adaptive filtering approach will become high. In this paper, we propose a multirate adaptive code acquisition approach to alleviate this problem. The proposed scheme is comprised of several acquisition units operating in different processing rates. Thanks to the decimation property in multirate processing, the overall computational complexity can be greatly reduced. Theoretical analysis of adaptive filters and mean acquisition time is also provided. Experimental results show that while the proposed scheme can have comparable performance with respect to the original adaptive acquisition scheme, its computational complexity is much lower.

&_{2009 Elsevier B.V. All rights reserved.}

1. Introduction

Code-division multiple access (CDMA) is a promising technique for wireless mobile communication. It is well known that the main performance bottleneck for a CDMA system is the multiple access interference (MAI). MAI not only affects detection, but also code synchronization. Code synchronization can be further divided into code acquisi-tion and code tracking. In this paper, we consider code acquisition with MAI. Code acquisition has been widely studied in the literature. The conventional approach to this problem is the well-known matched-filter (MF) based method[1–10,32,36](and references therein). The MF can have a serial[1], parallel[2–4], or hybrid search structure

providing an easy trade-off between hardware complexity and acquisition time. However, the MF-based method is only optimal for the single-user case. The acquisition performance degrades greatly when MAI presents, espe-cially in near–far environments [5,6]. To evaluate the performance of an acquisition scheme, a measure called acquisition-based capacity was defined in [7]. This capacity corresponds to the maximum number of users that a system can serve (with certain acquisition perfor-mance). It was shown in [7,19] that the asymptotic acquisition-based capacity for the MF is L=½2 lnðLÞ, where L is the processing gain. The quantity is less than the bit-error-rate-based capacity [8] which is proportional to L. This implies that code acquisition may become a limiting factor for a CDMA system capacity. Another discussion on the acquisition-based capacity for the MF can be found in[9].

Another category of the acquisition technique qem-ploys subspace- or matrix-based methods [11–18]. The Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/sigpro

Signal Processing

0165-1684/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.12.024

_{Corresponding author. Tel.: +886 35 731647.}

E-mail addresses: [email protected], [email protected] (H.-L. Yang),[email protected] (W.-R. Wu).

advantage of subspace-based approaches is that it does not require training sequences. However, these methods usually have to estimate, decompose, and inverse the autocorrelation matrix of the received signal vector. This often demands high computational complexity, especially at a large processing gain. The projection degree measure-ment (PDM) algorithm [11] observes two successive symbols in order to obtain the complete information of one desired symbol. As a consequence, the PDM has to estimate and inverse an autocorrelationmatrix of dimen-sion 2L-by-2L. The multiple signal classification (MUSIC) algorithm has also been applied to code acquisition [12–14]. The MUSIC algorithm has to carry out eigen-decompositions and extract eigen-vectors corresponding to noise subspace. Despite of the oversampling operation in [12], the computational complexity of the MUSIC algorithm is with

O

ðL3_{Þ. Besides, this algorithm is}

constrained under 2KoL, where K is the number of users. A matrix-based method [18] called a large sample maximum likelihood (LSML) acquisition algorithm, pro-vides excellent performance and robustness against the near-far problem. However, it requires a large amount of received bit signals and, again, pays a high computational complexity in the matrix operations. Notably, these methods are specifically designed for CDMA systems with periodic spreading codes (i.e., the spreading code repeats itself for every bit) and may not straightforwardly apply to the aperiodic-code systems (i.e., the periodicity of the spreading code is great than a bit interval).

Recently, the adaptive filter technique [19–26] was proposed to solve the acquisition problem in the presence of MAI. The method [19–24] separates the delay un-certainty into several regions, named (delay) cells. The input to the adaptive filter is the desired user’s pseudo-noise (PN) sequence with a code delay associated to a cell. Each cell is then sequentially tested and the code delay can then be estimated with the location of the maximum convergent tap-weight. This method can also have a serial or parallel searching structure trading performance with computational complexity. It was addressed in [19,20] that the adaptive filtering scheme can have a much higher acquisition-based capacity than the MF. Apart from the maximum weight testing, architectures with the thresh-old testing were also considered [21,22]. The threshold can be set for the mean-squared error (MSE) or for the maximum tap-weight (in a cell). It was found in[23]that the tap-weight testing can bring better performance than the MSE testing. The acquisition performance with fading channels was analyzed and reported in[24]. Yet, another adaptive receiver structure reported in[26]performs an exhaustive search to find the integer chip delay, and then solve aquadratic equation to find the corresponding fractional chip delay. The drawback of this approach is that its complexity is high particularly for a large processing gain.

In this paper, we propose a code acquisition algorithm using a multirate adaptive filtering technique. Similar to the original adaptive filter approach[19,20], our structure is valid for periodic as well as for aperiodic spreading codes. In fact, many commercial CDMA systems, including IS-95 standard [28], CDMA-2000 proposal[29], and 3G

CDMA-based wireless networks [30,31], adopt aperiodic codes for spreading. The fundamental structure of the proposed algorithm is similar to that in[19,20]; however, the proposed scheme contains several adaptive filters operating in different rates. The adaptive filters with low rates will search the code delay in low resolutions. The adaptive filters with higher rates will then resolve the code delay in higher resolutions. The adaptive filter with the highest rate, say the chip-rate, can finally identify the original code delay. The proposed multirate processing can have a much lower computational complexity than the conventional adaptive filtering approaches in[19,20]. This is particularly true in the applications where the processing gain as well as the delay uncertainty is large.

Throughout this paper, the notations ðÞT and Au;v

denote the transposition operator and the uvth entry of a matrix A, respectively. Also, dye indicates the smallest integer greater than or equal to the value y, whereas byc the largest integer smaller than or equal to y. Besides, z represents the unit delay operator, I the identity matrix, and Efg the statistical expectation operation. The rest of this paper is organized as follows. Section 3 reviews the conventional adaptive code acquisition scheme. Section 3 describes the proposed multirate code acquisition scheme. Section 4 analyzes the performance of the proposed scheme, and Section 5 reports simulation results. Finally, we draw conclusions in Section 6.

2. Conventional adaptive code acquisition

In this section, we briefly review the conventional adaptive code acquisition scheme[19,20].Fig. 1shows the structure of this scheme. For reference convenience, we name this scheme as a one-rate (1R) scheme since only one processing rate (i.e., chip-rate) is used. The baseband chip-rate sampled received signal can be expressed as rðnÞ ¼X K k¼1 Akxkðn 

t

kÞ þwðnÞ, (1) LMS adaptive filter Store results M_c-tap r (n) x₁ (n-qMc) e (n) + -user-1's PN sequence

Fig. 1. Conventional 1R code acquisition system, where x1ðn  qMcÞis

where K,

t

k, Ak, xkðnÞ, and wðnÞ denotes the number of

user, the code delay, the signal amplitude, the transmitted signal of user-k, and channel noise, respectively. The channel noise is assumed to be additive white Gaussian and its mean is zero. The transmitted signal of user-k can be expressed as xkðnÞ ¼ X1 j¼1 dkðjÞ XL1 l¼0 ck;jðlÞpðn  l  jLÞ; k ¼ 1; . . . ; K, (2)

where dkðjÞ denotes the jth BPSK signal of user-k and

ck;jðlÞ 2 f1; 1g corresponds to the lth chip signal in dkðjÞ.

Also, L denotes the processing gain and pðnÞ the chip-rate sampled pulse. Before proceed further, we list assump-tions to be used in the sequel:

(a) User-1’s code delay is of interest and A1¼1.

(b) The code delay is an integer multiples of the chip-duration and smaller than L.

(c) Carrier synchronization is established before code acquisition.

(d) No data are modulated for user-1’s signal in the period of code acquisition, i.e., d1ðjÞ ¼ 1.

(e) The chip-pulse is considered as a rectangular pulse with unit amplitude.

(f) The code sequence ck;jhas a period much higher than

the processing gain such that the input to the adaptive filter can be viewed as statistically white.

(g) Only the additive white Gaussian noise (AWGN) channel is considered and the summation of MAI and white Gaussian noise can be modeled as another white Gaussian noise[32].

The 1R scheme first divides L into Q ¼ dL=Mce cells,

where Mcis the length of the adaptive filter. The adaptive

filter then serially searches the code delay in these cells. The least-mean-square (LMS) algorithm is employed to minimize the MSE between the received signal rðnÞ and the adaptive filter output (see Fig. 1). The tap-weight update equations are given by

wq_{ðn þ 1Þ ¼ w}q_{ðnÞ þ}

_m

ceðnÞxqðnÞ, (3)

eðnÞ ¼ rðnÞ  ½wq_ðnÞT_xq_{ðnÞ; q 2 f0; . . . ; Q  1g, (4)}

where

m

c denotes the step size controlling the

conver-gence of the adaptive filter, wq_{ðnÞ ¼ ½w}q 0ðnÞ; w

q 1ðnÞ; . . . ;

wq_M

c1ðnÞ

T _{the filter tap-weight vector for the qth cell,}

and xq_{ðnÞ ¼ ½x}

1ðn  qMcÞ;x1ðn  qMc1Þ; . . . ; x1ðn  qMc

Mcþ1ÞT the corresponding input vector. Here, q is

sequentially increased from zero to Q  1. The tap-weight vector wq_{ðnÞ for a particular q is stored after some}

iterations, say N1chips. Then, an estimation of

t

1can be

derived with the tap-index of the maximum tap-weight (among all cells). Let the ^

Dc

th tap (0p ^

Dc

oMc) of the

adaptive filter in the ^

a

cth cell has the maximum value.

Then, we can have the delay estimation ^

t

1¼ ^

a

cMcþ ^

Dc

.

Combine wq_ðN

1Þ, q ¼ 0; 1; . . . ; Q  1 into a big vector w,

i.e., w ¼ ½½w0_ðN

1ÞT; ½w1ðN1ÞT; . . . ; ½wQ 1ðN1ÞTT. It can be

shown that the probability of acquisition error is Pe¼1  PbðwcXw_jÞ; caj; fc; jg 2 f0; 1; . . . ; L  1g, (5)

where wjdenotes the jth element of w and wc the

tap-weight corresponding to the true code delay

t

1 (i.e.,

c ¼

t

1). To evaluate (5), we need to know the stochastic

properties of the tap-weights. It has been shown in[33] that these tap-weights at convergence have Gaussian distributions with a mean vector of

mðL1Þ¼wo, (6)

and a covariance matrix of

CðLLÞ

m

c

2JminI (7)

9

s

2

wI, (8)

where wo is the optimum solution of w solved with

the Wiener equations [34], Jmin is the corresponding

minimum mean-squared error (MMSE), and

s

2 w is the

variance of each tap-weight. Let Rq¼Efxq_ðnÞ½xq_ðnÞT

gand pq_¼Efxq_{ðnÞrðnÞg. Since the input is white, R}q

¼IMcMc. It is well known that wqo¼ ðRqÞ1pq. Let

t

1¼

a

cMcþ

Dc

,

0p

Dc

oMc, and pqj is the jth entry of pq (j 2 f0; 1; . . . ;

Mc1g). It is simple to show that pq_j ¼1 when q ¼

a

cand

j ¼

Dc

, and pq_j ¼0 otherwise. This is to say that a unique peak with value one will appear in wc, and all other

weights are zeros. Thus, we can have Jmin¼Efr2ðnÞg  1.

Using Eqs. (6) and (8), we can rewrite (5) as Pe¼1  Z 1 1 1  Q wc

s

w    L1 exp ðwc1Þ 2 2

s

2 w ! dwc, (9)

where Q ðÞ denotes the Q -function[35]. It is known that an Mc-tap adaptive filter (with the LMS algorithm)

requires 2Mc multiplications per iteration. Thus, the

computational complexity is proportional to the filter size. 3. Proposed adaptive multirate code acquisition

To understand our idea easier, we start our develop-ment with a two-rate (2R) system. Then, we will extend it to a three-rate (3R) system.

3.1. 2R scheme

Following the assumptions given in Section 2, we express (1) as rðnÞ ¼X K k¼1 Akxkðn 

t

kÞ þwðnÞ ¼x1ðn 

t

1Þ þvðnÞ, (10) where vðnÞ ¼X K k¼2 Akxkðn 

t

kÞ þwðnÞ (11)

denotes the sum of MAI and white Gaussian noise. Let the variance of vðnÞ be

s

2

v. For notational simplification, we

will omit the subscripts of x1ðnÞ and

t

1 in following

derivations.Fig. 2shows the architecture of the proposed 2R acquisition system. As we can see, the system contains two units with two different processing rates. We call the unit inFig. 2(a) as a low-rate unit (LRU). In this unit, the adaptive filter updates its tap-weights with a low rate. For

this reason, we refer to the adaptive filter in this unit as a low-rate adaptive filter (LRAF). By contrast, we call the unit inFig. 2(b) a high-rate unit (HRU). The adaptive filter in this unit updates its tap-weights with a high rate. We refer to the adaptive filter in this unit as a high-rate adaptive filter (HRAF). Note that the high-rate here denotes the chip-rate. There are feedforward and feedback operations in the system. We now describe the funda-mental feedforward operation. First, consider Fig. 2(a). The system passes the received signal rðnÞ and the locally generated user-1’s signal xðnÞ through lowpass filters (LPFs) to obtain rLPFðnÞ and xLPFðnÞ, respectively. Then, it

downsamples these signals with a factor of D and feeds the resultant signals to the LRAF. Let Mp¼ dL=De. Then,

the code delay can be rewritten as

t

¼

a

D þ

D

where

a

2 f0; 1; . . . ; Mpgand D=2o

D

pD=2. Note that the ranges of

a

and

D

are defined different from that in the previous section. The LRAF will adapt to estimate a low-resolution

t

having the value in f0; D; . . . ; MpDg. Similar to the 1R

system, we select the tap-index associating with the maximum tap-weight value. Note that Mpþ1 is the filter

length of the LRAF and ðMpþ1ÞD must be great or equal to

L. Let theindex with the maximum tap-weight in the LRAF be ^

a

. The HRU inFig. 2(b) then delays xðnÞ with ^

a

D chips. We call the device to perform the delay function as the delay-tuning filter (DTF). With this operation, the HRAF adapts to refine the code-delay resolution. After conver-gence, we select the tap-index ^

D

with the maximum tap-weight. It is easy to see that the index should be in the range of D=2. Combing these two tap-weight indices, we can finally obtain a code-delay estimate. In summary, the

LRU attempts to acquire

t

in a multi-chip level (low reso-lution), while the HRU in a chip level (high resolution).

We now examine some properties of the 2R feedfor-ward operation. For low complexity consideration, we let the LPF filtered rðnÞ (in (10)) as

rLPFðnÞ ¼ X D1 j¼0 rðn  jÞ ¼ X D1 j¼0 xðn 

t

jÞ þ vLPFðnÞ, (12) where vLPFðnÞ ¼ X D1 j¼0 vðn  jÞ. (13)

It is simple to see that this is just an averaging operation with a D-tap filter (apart from a constant). InFig. 2(a), fL

indicates a vector consisting of the impulse response of the LPF. As shown, each element of fLhas the value of one.

Substituting

t

¼

a

D þ

D

, we can rewrite (12) as rLPFðnÞ ¼

X

D1

j¼0

xðn 

a

D 

D

jÞ þ vLPFðnÞ. (14)

Downsampling (14) with a factor of D, we then have rLðmÞ9rLPFðnÞjn¼mD

¼X

D1

j¼0

xððm 

a

ÞD 

D

jÞ þ vLðmÞ, (15)

where we let m ¼ bn=Dc and vLðmÞ ¼ vLPFðmDÞ. Similarly,

we can average xðnÞ to obtain xLPFðnÞ ¼

X

D1

j¼0

xðn  jÞ, (16)

and downsample xLPFðnÞ to obtain

xLðmÞ9xLPFðnÞjn¼mD

¼X

D1

j¼0

xðmD  jÞ. (17)

Let the input vector of the LRAF be xLðmÞ. Then, we have

xLðmÞ ¼ ½xLðmÞ; xLðm  1Þ; . . . ; xLðm  MpÞT. (18)

For a different value of

D

, the performance of the LRAF will be different. To evaluate the impact of

D

on LRAF, we calculate the optimal tap-weights and the corresponding steady-state MSE. We put the detailed derivation in Appendix A, and summarize the result below. For

D

X0, we have the optimal tap-weights as

wL;o;¼ 1 

r

;



¼

a

;

r

;



¼

a

þ1; 0 otherwise; 8 > < > :



2 f0; 1; . . . ; Mpg, (19) and the steady-state MSE as

JLð1Þ ¼ 1 þ

ðMpþ1Þ

m

L

2

 

JL;min. (20)

For

D

o0, we have

wL;o;¼ 1 

r

;



¼

a

;

r

;



¼

a

1; 0 otherwise; 8 > < > :



2 f0; 1; . . . ; Mpg, (21) LMS LMS Find

ˆ

Find

ˆ

Δ

( ) LPF x n xL

( )

m

ˆ

z

−Δ

( )

n

x

_{ˆ D}

z

−

w

H

( )

n

( )

L

m

w

( ) LPF r n

( )

r n

( )

r n

( )

H

n

x

adjust at every m instant

+ -- +

( )

L

r m

LPF PTF D L

f

D

( )

n

x

LRAF HRAF LPF L

f

DTF  

Fig. 2. Proposed 2R code acquisition system with (a) LRU and (b) HRU. Note that LRU and HRU interact only when n ¼ mD. The dash-lines indicate feedforward and feedback operations.

and JLð1Þ ¼ 1 þ ðMpþ1Þ

m

L 2   JL;min, (22)

where

r

¼ j

D

j=D and JL;min is defined in (A.14),

respec-tively. Using (A.14) and observing 0p

r

p1=2, we see that when

r

gets larger, JL;minwill become larger. This will also

make the steady-state MSE in (22) larger. When

D

¼0,

r

¼0 and

t

can be divided by D. The response of the LRAF can be seen as a perfectly downsampled version of the channel response. If the channel has an impulse-like response, so does the LRAF. When

D

40 (or

D

o0),

t

4

a

D (or

t

o

a

D). In both cases,

t

cannot be divided by D. The response of the LRAF cannot have an impulse-like response. From Eqs. (A.12) and (A.37), we see that a nonzero

r

(

D

¼0) will produce two nonzero weights and make the value of the peak tap-weight smaller than one. Combining these effects, we can conclude that the larger the

r

, the worse the acquisition performance. The worst case occurs when

r

¼1=2 yielding two nonzero equal weights. In what follows, we will develop a system that can null

r

.

Now, let us consider operations in the HRU. AsFig. 2(b) shows, the input to the HRAF is xðn  ^

a

DÞ. As mentioned, the optimal filter of the LRAF may have two nonzero weights with the same value. Thus, the peak position can be

a

or

a

þ1. In other words, we need at least D þ 1 taps for the HRAF. To simplify our analysis, we let ^

a

¼

a

. It is simple to see that the optimal weights of the HRAF will have a unique peak at ^

D

. Since the analysis of HRAF is straightforward, we only provide the results without detailed derivations. Let

xHðnÞ ¼ ½xðn  ^

a

D þ D=2Þ; . . . ; xðn  ^

a

DÞ; . . . ; xðn  ^

a

D  D=2ÞT

9½xH;D=2ðnÞ; . . . ; xH;0ðnÞ; . . . ; xH;D=2ðnÞT (23)

wHðnÞ9½wH;D=2ðnÞ; . . . ; wH;0ðnÞ; . . . ; wH;D=2ðnÞT, (24)

where we assume that D=2 is an integer (for notational convenience). Notice that RH9EfxHðnÞxTHðnÞg ¼ I. We then

have the optimum weights listed below:

wH;o;j¼

1; j ¼

D

; 0 otherwise; (

(25)

where wH;o;j is the jth element of wH;o, and wH;o is the

optimal solution of wHðnÞ. We then have the MMSE and

steady-state MSE as

JH;min¼Ef½rðnÞ  wTHðnÞxHðnÞ2gjwHðnÞ¼wH;o

¼Efr2_{ðnÞg  2w}T

H;oEfxHðnÞrðnÞg þ wTH;oRHwH;o

¼

s

2 v, (26) JHð1Þ ¼ 1 þ ðD þ 1Þ

m

H 2   JH;min, (27)

where

m

H is the step size used in the HRAF.

The main problem associated with the 2R scheme described above is that sampling phases for rLPFðnÞ and

xLPFðnÞ may not be synchronized (i.e.,

D

a0). As analyzed,

the acquisition performance can be greatly affected when

D

is not equal to zero. Our remedy to this problem is to adjust the sampling phase of xðnÞ during filter adaptation. This is possible if

D

estimated by the HRAF can be feedback to the LRAF. To realize this thought, we use a device, namely phase-tuning filter (PTF), to tune the input phase with ^

D

chips (see the feedback operation inFig. 2). The PTF can advance or lag the phase of its input signal. Practically, the PTF can be implemented with a buffer and a multiple-input-to-one-output selector. With this struc-ture, the sampling phases for rLPFðnÞ and xLPFðnÞ can be

synchronized and, therefore, (A.12) can have a unique peak. Note that the LRU and HRU interact only when n ¼ mD. Letting

D

¼0 (i.e.,

r

¼0) in Eqs. (A.14) and (A.25), we have JL;min¼D

s

2v, (28) Q;ðmÞ ¼

m

L 2D

s

2 v;



2 f0; . . . ; Mpg. (29)

Thus, steady-state MSEs of the LRAF and the HRAF are JLð1Þ ¼ 1 þ ðMpþ1Þ

m

L 2   D

s

2 v, (30) JHð1Þ ¼ 1 þ ðD þ 1Þ

m

H 2  

s

2 v. (31) 3.2. 3R scheme

In the previous subsection, we have proposed a 2R scheme that is able to null

r

. Since the HRAF operates in a high processing rate, it dominates the overall computa-tional complexity. This becomes an important issue when the tap-length D þ 1 is large. We can solve the problem by introducing a unit with another processing rate. We call this unit as a medium-rate unit (MRU). This unit contains a medium-rate adaptive filter (MRAF) sharing the compu-tational loading of the HRAF. As shown inFig. 3(b), the LPFs fMaverage rðnÞ and xðnÞ with a window side of DM,

and the decimators downsample the resultant signals with a factor of DM. Let the DPTF denote the device

cascading the DTF and PTF. Here, the processing rate of the MRU is D=DM times faster than that of the LRU, but DM

times slower than that of the HRU.

With the additional MRU, we have three resolutions to work with. We can express the code delay as

t

¼

a

D þ

b

DMþ

d

, where

a

2 f0; 1; . . . ; Mpg;

b

2 fD=

ð2DMÞ; . . . ; 0; . . . ; D=ð2DMÞg, and

d

2 fDM=2; . . . ; 0; . . . ;

DM=2g. For convenience, again, we assume that D=ð2DMÞ

and DM=2 are integers. Then, we use the LRU, MRU and

HRU to estimate f

a

;

b

;

d

g, respectively. Note that ðD þ DMÞ=2p

b

DMþ

d

pðD þ DMÞ=2, where DMX2. In other words, the MRAF and HRAF can span a delay region greater than D þ 1. Define the tap-weight vector and the input vector of the MRAF as

wMðsÞ9½wM;D=ð2DMÞðsÞ; . . . ; wM;0ðsÞ; . . . ; wM;D=ð2DMÞðsÞ

T

, (32)

xMðsÞ9½xM;D=ð2DMÞðsÞ; . . . ; xM;0ðsÞ; . . . ; xM;D=ð2DMÞðsÞ

where s ¼ bn=DMc. The update equation for the MRAF is

given by

wMðsÞ ¼ wMðs  1Þ þ

m

MxMðsÞ½rMðsÞ  xTMðsÞwMðs  1Þ,

(34) where

m

Mis the corresponding step size and

rMðsÞ ¼ X DM1 j¼0 rðn  jÞjn¼sDM, (35) xM;ðsÞ ¼ X DM1 j¼0 xðn  ^

a

D  ^

d





DMjÞjn¼sDM,



2 fD=ð2DMÞ; . . . ; 0; . . . ; D=ð2DMÞg, (36)

where we have used ^

a

D and ^

d

obtained from other two units. The weight adaptations for the LRAF and HRAF are similar to (A.19).

We have analyzed the performance of the HRAF and LRAF in a 2R system previously. The performance of the MRU in a 3R system can be done in a similar way. We can treat the MRU as a special LRU, and replace D with DMfor

the formulas derived for the LRAF. Since this is

straight-forward, we omit the detailed results here. Note that in Fig. 3all units update parameters in their PTFs or DTFs simultaneously at n ¼ mD. Let the estimates for

a

,

b

, and

d

at the instant n ¼ mD be ^

a

ðmÞ, ^

b

ðsÞ, and ^

d

ðnÞ, respectively. When n ¼ mD, the PTF in the LRU delays xðnÞ by ^

b

ðsÞDMþ

^

d

ðnÞ chips, the DPTF in the MRU delays xðnÞ by ^

a

ðmÞD þ ^

d

ðnÞ chips, and the DPTF in the HRU delays xðnÞ by ^

a

ðmÞD þ ^

b

ðsÞDM chips. We can extend the idea to a four-rate or

higher rate system; however, the system architecture will become complex. For typical applications, a 2R or 3R system will be sufficient. As described, all the filters are adjusted using the LMS algorithm. As shown later, the tap-weight of an adaptive filter can be treated as a random variable. Thus,

a

,

b

, or

d

may be incorrectly estimated during the adaptation, which we call a decision error. Note that the LMS algorithm changes the filter-weight values slowly. For most cases, theestimation error can be corrected shortly. There are only few cases that the error will propagate between adaptive filters and the overall effect may lower the final amplitudes of the peak tap-weights. To alleviate the decision error problem, we can let the LRU operate for a short period of time without feedback at the initial. Simulations show that the error propagation effect only slightly slows the convergence.

4. Performance analysis

To compare the proposed schemes with the 1R system in Section 2, we employ some performance measures such as the required computational complexity (per iteration), acquisition error probability, and mean acquisition time.

4.1. Computational complexity

To have a fair comparison, we let N ¼ N1¼NM, where

NM denotes the iteration time of the multirate system.

Also, we let D ¼ Q such that the filter size in the 1R system is approximately equal to that of the LRAF (McMpþ1). Since the main operation in filtering is

multiplication, we only take this into account. We first calculate the total multiplications required in N iterations and then divide the result by N.

4.1.1. 1R scheme

As mentioned in Section 2, the Mc-tap adaptive filter

will require 2Mc multiplications per iteration. Then, the

computational complexity of the 1R system, denoted as C1, is 2Mc¼2dL=De.

4.1.2. 2R scheme

For the 2R scheme, we have to take both the LRAF and HRAF into account. Since the HRAF has D þ 1 taps and operates in the chip-rate, it requires 2ðD þ 1Þ multi-plications per iteration. On the other hand, the LRAF has Mpþ1 taps operating in a rate D times slower. Thus, the

required multiplications per iteration for a 2R scheme, C2, Fig. 3. Proposed 3R code acquisition system with (a) LRU, (b) MRU, and

(c) HRU. Again, all units interact only when n ¼ mD and the dash-lines are for feedforward and feedback operations.

is given by C2¼ 2ðMpþ1Þ N Dþ2ðD þ 1ÞN N (37) ¼ 2ðMpþ1Þ D þ2ðD þ 1Þ. (38) 4.1.3. 3R scheme

Similarly, we take the LRAF, MRAF, and HRAF into account. The required multiplication per iteration for a 3R system, C3, turns out to be

C3¼ 2ðMpþ1ÞN Dþ2 D þ 1 DM   N DM þ2ðDMþ1ÞN N (39) ¼ 2ðMpþ1Þ D þ 2 DM D þ 1 DM   þ2ðDMþ1Þ, (40)

where dðD þ 1Þ=DMeis the minimum required tap-length

for the MRAF.

4.2. Probability of acquisition error

For a general LMS adaptive filter with a step size

m

c, the

time to converge can be evaluated using d1=

m

ce(see p. 348

in [34]), which is called the time-constant. For the proposed 1R system, we have Q cells to adapt sequen-tially. To ensure that the steady state can be achieved closely, we let the adaptation time be four time-constants for each cell. Therefore, the overall adaptation time for the 1R system, denoted as N1, is then 4Q d1=

m

ce. Let the step

size for the adaptive filter in the 1R system and that in the HRAF be the same (i.e.,

m

c¼

m

H9

m

). For multirate systems

described in Section 3, we further let

m

L¼

m

=D and

m

M¼

m

=DM. In this way, the variances of these adaptive

filter taps are the same (see Eqs. (7) and (29)). 4.2.1. 2R scheme

An acquisition error may occur due to ^

a

a

a

, ^

D

a

D

, or both. When the phase feedback to PTF is not correct, i.e.,

^

D

a

D

, the peak magnitude of wL;aðmÞ will be reduced, and

the overall acquisition performance will be affected. If we assume that there are no decision errors, the probability of acquisition error for the time instant n, denoted as PeðnÞ,

can be written as PeðnÞ ¼ 1  PL;cðmÞPH;cðnÞ, (41) PL;cðmÞ ¼ PðwL;cðmÞXwL;jðmÞÞ; caj; fc; jg 2 f0; 1; . . . ; Mpg, (42) PH;cðnÞ ¼ PðwH;cðnÞXwH;jðnÞÞ, caj; fc; jg 2 fD=2; . . . ; 0; . . . ; D=2g, (43) where PL;cðmÞ and PH;cðnÞ denote the correct acquisition

probabilities of the LRAF and HRAF, respectively. Also, wL;cðmÞ and wH;cðnÞ denote the taps whose tap-indices

correspond to the actual code delay.

Using the transient analysis of LMS algorithms in[33], we have the mean weight vector of the LRAF as

EfwLðmÞg ¼ ½I  ðI 

m

LRLÞmwL;o, (44)

and the ðMpþ1Þ-by-ðMpþ1Þ covariance matrix as

CLðmÞ ¼

m

L

D

s

2 v

2 ½I  ðI  2

m

LRLÞ

m_{. (45)}

Since RL¼DI, we can let CLðmÞ ¼

s

2wLðmÞI where

s

2

wLðmÞ

is an equivalent variance that can be derived from (45). Here, wLðmÞ and wHðnÞ are assumed to be Gaussian

distributed. Similarly, we can have the mean weight vector and the covariance matrix of wHðnÞ as

EfwHðnÞg ¼ ½I  ðI 

m

HRHÞnwH;o, (46)

and CHðnÞ ¼

m

H

s

2v

2 ½I  ðI  2

m

HRHÞn. (47)

Since RH¼I, we can let CHðnÞ ¼

s

2wHðnÞI. Similarly,

s

2

wHðnÞ is an equivalent variance that can be derived from (47). From Eqs. (45) and (47), we find that tap-weights are independent and identically distributed. As mentioned, both the HRAF and LRAF are run for N chips. For notational simplicity, we let AL as the peak in EfwLðbN=DcÞg,

s

2

L¼

s

2wLðbN=DcÞ, PL¼PL;cðbN=DcÞ, AH as the peak in EfwHðNÞg,

s

2H¼

s

2wHðNÞ, PH¼PH;cðNÞ, and Pe¼PeðNÞ. The probabilities in Eqs. (42) and (43) at n ¼ N turn out to be PL¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi 2

ps

2 L q Z 1 1 1  Q w

s

L    Mp exp ðw  ALÞ 2 2

s

2 L ! dw, (48) PH¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi 2

ps

2 H q Z 1 1 1  Q w

s

H    D exp ðw  AHÞ 2 2

s

2 H ! dw. (49) Finally, we obtain Pe¼1  PLPH. (50)

As mentioned in Section 3, incorrect decisions can occur and the error propagation between the HRAF and LRAF will lower the peak amplitudes of final tap-weights. Thus, the results in Eqs. (48)–(50) may be too optimistic. However, the exact analysis of the error propagation effect turns out to be very difficult, if not impossible. In what follows, we propose a simple approximation method to overcome the problem. We first assume that the error propagation affects the mean of a tap-weight much more serious than the variance. As a result, we only consider the variation of mean weight vectors. For an adaptation period, a decision error can occur in any instant and the error sequence can have many patterns. For simplicity, we only investigate those affecting performance most. Con-sider the LRAF. It is simple to see that if there are

k

decision errors during the adaptation period (i.e., between m ¼ 0 and bN=Dc), the error pattern corresponding to the worst performance will be the one when all errors occur between m ¼ bN=Dc 

k

þ1 and bN=Dc. In other words, once a decision error occurs, the error will continue to the end of the adaptation period. This will make the peak weight value of the LRAF decrease from m ¼ bN=Dc 

k

þ 1 monotonically. We then use this pattern to represent all possible error patterns having

k

decision errors. From (44), we have AL¼1  ð1 

m

LDÞ

bN=Dc

errors into account, we may then rewrite ALas ALð

k

Þ ¼ ½1  ð1 

m

LDÞ bN=Dck_{exp }

k

Z

L   , (51) where

Z

1

L ¼

m

L

lL

and

lL

¼D is the eigenvalue of RL[34].

We may treat

k

as a random variable with a binomial distribution as

pð

k

Þ ¼ bN=Dc

k

 

ð1  PLÞkPbN=DcL k, (52)

where PL is the correct probability in (48). We then use

Eqs. (51) and (52) to calculate the mean value of ALð

k

Þ,

denoted as ¯AL. It is given by

¯AL¼

X

bN=Dc

k¼0

ALð

k

Þpð

k

Þ. (53)

Then, the probability of correct acquisition for the LRAF, denoted as ¯PL, can be obtained by substituting ¯ALinto (48).

Similarly, we can use the same procedure to obtain the probability of correct acquisition for the HRAF, ¯PH. Finally,

the probability of acquisition error for a 2R system, denoted as PE, is obtained by

PE¼1  ¯PL¯PH. (54)

It is worth mentioning that PL and PH are the

correct acquisition probabilities without decision errors. Thus, these values essentially correspond to two upper bounds of the correct acquisition probabilities. Using these values in the calculation of pð

k

Þ (as that in (52)) will underestimate the acquisition error probability. On the other hand, we only take the worst decision error patterns into consideration and this will over-estimate the acquisition error probability. Thus, (54) is a result corresponding a compromise of these two extreme cases.

4.2.2. 3R scheme

Using the similar idea, we can have the probabi-lity of acquisition error for the decision-error-free case as PeðnÞ ¼ 1  PL;cðmÞPM;cðsÞPH;cðnÞ, (55) PL;cðmÞ ¼ PðwL;cðmÞXwL;jðmÞÞ, caj; fc; jg 2 f0; 1; . . . ; Mpg, (56) PM;cðsÞ ¼ PðwM;cðsÞXwM;jðsÞÞ, caj; fc; jg 2 fD=ð2DMÞ; . . . ; 0; . . . ; D=ð2DMÞg, (57) PH;cðnÞ ¼ PðwH;cðnÞXwH;jðnÞÞ, c_{aj; fc; jg 2 fD}M=2; . . . ; 0; . . . ; DM=2g, (58)

where PL;cðmÞ, PM;cðsÞ, and PH;cðnÞ are the correct

acquisi-tion probabilities of the LRU, MRU, and HRU, respectively; wL;cðmÞ, wM;cðsÞ, wH;cðnÞ denote the taps whose tap-indices

correspond to the actual code delay. Note that s ¼ bn=DMc.

Let PL¼PL;cðbN=DcÞ, PM¼PM;cðbN=DMcÞ, PH¼PH;cðNÞ,

and Pe¼PeðNÞ. Then PL can be calculated as that in (48),

while PMand PHare given by

PM¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 2

ps

2 M q Z1 1 1  Q w

sM

   D=DM exp ðw  AMÞ 2 2

s

2 M ! dw, PH¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffi 2

ps

2 H q Z1 1 1  Q w

sH

   DM exp ðw  AHÞ 2 2

s

2 H ! dw, (59) where AM and

s

2M can be obtained as that described in

Eqs. (44) and (45). Then, we have Pe¼1  PLPMPH. Again,

Pedoes not consider the decision error propagation effect.

We can follow the same notation definitions and proce-dures outlined in the previous subsection to obtain { ¯PL; ¯PM; ¯PH}. Finally, we have the probability of acquisition

error for the 3R system as

PE¼1  ¯PL¯PM¯PH. (60)

4.3. Mean acquisition time

Mean acquisition time analysis is generally derived with a Markov chain model [36]. Since our multirate systems is different from the MF with serial search, the commonly used model[10]cannot be applied here. Fig. 4 shows the model derived for our systems. As the figure shows, the system iterates for N chips to obtain ^

t

and the probability of acquisition error is PE.

If the acquisition fails, it will wait for a period of time Tp (chips) before the system re-starts the acquisition.

Here, Tp is generally referred to as the penalty time [32]. For our schemes, ^

t

is constructed from f ^

a

; ^

D

g or f ^

a

; ^

b

; ^

d

g at n ¼ N. If ^

t

a

t

, the receiver will re-initialize acquisition after a time interval of Tpchips. We can have

the transfer function of the Markov chain model inFig. 4 as[27,36]

HðzÞ ¼ ð1  PEÞz

N

1  PEzTpþN, (61)

where z is a delay operator and PE is the probability of

acquisition error formulated above. The mean acquisition

N chips for iteration ACQ START Tp P_Ez Tp+N P_Ez 1 − P_E ACQ START (1_−PE) zN

Fig. 4. Markov chain model for multirate code acquisition schemes. The right hand side figure illustrates an equivalent model, where z is a delay operator, PEthe probability of acquisition error, Tppenalty time, and ACQ

time can then be easily found as Tacq9 d dzHðzÞjz¼1 (62) ¼N þðTpþNÞPE ð1  PEÞ . (63)

Note that the unit of Tacqis chip.

5. Simulation results

In this section, we conduct computer simulations to demonstrate the effectiveness of the proposed algorithms. First, we investigate the computational complexity issue. Using Eqs. (38) and (40), we can evaluate the computa-tional complexity requirement per chip versus D for 1R, 2R, and 3R schemes. We list the results inTables 1–3for D ¼ 4, 8, and 16, respectively. The numbers inside the parentheses in these tables indicate the values of DMused

for the 3R system. Also, the last two rows of the tables give the complexity ratio defined as C2=C1 and C3=C1,

respectively. From these tables, we can have several observations. Firstly, the larger the processing gain, the higher efficiency the multirate system can achieve. Secondly, the 3R system is always more efficient than the 2R system. Lastly, there exists an optimum D for a given processing gain L. For example, when L ¼ 1024 and D ¼ 8, the computational complexity of the 2R system is about 20% of the 1R system. For the same processing gain with D ¼ 16, the complexity of the 3R system is only about 16% of the 1R system. These outcomes state that the

multirate system can be much more efficient than the 1R system for large L.

We set signal-to-interference-plus-noise ratio (SINRc),

which is defined 1=

s

2

v, as 13 dB (about 20 users with

equal power). Also, L ¼ 128, D ¼ 8, DM¼4,

m

¼

m

H¼

m

LD ¼

m

MD=2, and N ¼ 4Dd1=

m

e. To show how the

proposed scheme works, we let

t

¼51 and use a 3R system with above parameter setting to conduct simula-tions (100 trials).Fig. 5shows the averaged peak-weight positions associated with the LRAF, the MRAF, and the HRAF. As we can see, ^

a

¼7, ^

b

¼ 1, and ^

g

¼ 1. Thus, the code delay can be estimated as ^

t

¼ ^

a

D þ ^

b

DMþ ^

g

¼51,

which is equal to the actual delay. We then compare the probabilities of acquisition error for 1R, 2R, and 3R systems. The code delay,

t

, here is uniformly and randomly selected from ½0; LÞ. We conduct 104 indepen-dent trials and show the results inFig. 6. Also shown in the figure is the theoretical results derived in Section 4. Experimental results in Fig. 6 indicate that the perfor-mance of multirate systems are slightly better than that of the 1R system. Theoretical predictions for all systems are accurate particularly when the step size is large. For the 1R system, the deviation between experimental and theoretical values is smaller than that in 2R and 3R systems. This is not surprising, since the 1R system does not have the error propagation problem.

As mentioned, an important acquisition performance measure is the mean acquisition time. To derive the mean acquisition time, Tacq, we first set Tp¼1:28  104 chips

(100 bits) and substitute the experimental acquisition error probabilities obtained fromFig. 6 into (63).Fig. 7 shows the mean acquisition time for all systems. The lower bound in Fig. 7 corresponds to the case that no acquisition errors occur. In this case, Tacq¼N ¼ 4Dd1=

m

e

and this can serve as a performance bound for compar-ison. As we can see, ini-tially the mean acquisition time decreases when the step size increases. When the step size is larger than

m

¼5  103, the mean acquisition time starts to increase. For the setting here, the optimal step size is around

m

¼5  103. In this case, Tacq for the 1R

system is about 7500 chips, that for the 2R system is about 7150 chips, and that for the 3R system is about 7250 chips. We also examine the probability of acquisition error for various SINRc.Fig. 8shows the experimental results.

Here, we let

m

¼5  103, L ¼ 128, D ¼ 8, DM¼4, and

N ¼ 4Dd1=

m

e. We find that all systems have similar performance. Also, the higher the SINRc, the better

performance we can have. The 2R system behaves slightly better than the others. Fig. 9 shows the corresponding mean acquisition time. In terms of the mean acquisition time, we have the same conclusion that all systems have similar performance.

For all simulations shown above, we have fixed N ¼ 4Dd1=

m

e for the systems. In terms of mean acquisition time, this choice may not be optimal.Fig. 10shows the mean acquisition time for various N (SINRcis 13 dB). As

we can see, there are optimum N’s for multirate systems. For

m

¼5  103_{, we find that the mean acquisition time}

of the 3R system increases quicker than that of the 2R system when N is smaller than the optimum value. This is because the performance of low-rate units depends on N

Table 1

Computational complexity comparison for D ¼ 4.

L 128 256 512 1024 C1 64 128 256 512 C2 26.50 42.50 74.50 138.50 C3 25.5 (2) 41.5 (2) 73.5 (2) 137.5 (2) C2=C1 0.414 0.332 0.291 0.271 C3=C1 0.398 0.324 0.287 0.269 Table 2

Computational complexity comparison for D ¼ 8.

L 128 256 512 1024 C1 32 64 128 256 C2 22.25 26.25 34.25 50.25 C3 14.25 (3) 18.25 (3) 26.25 (3) 42.25 (3) C2=C1 0.695 0.410 0.268 0.196 C3=C1 0.445 0.285 0.205 0.165 Table 3

Computational complexity comparison for D ¼ 16.

L 128 256 512 1024 C1 16 32 64 128 C2 35.125 36.125 38.125 42.125 C3 13.125 (3) 14.125 (3) 16.125 (3) 20.125 (3) C2=C1 2.195 1.129 0.596 0.329 C3=C1 0.821 0.441 0.252 0.157

more strongly. When N is larger than the optimum value, the mean acquisition times of both systems approach the lower bounds. We can observe the same behaviors when

m

¼3  103_{. From the figure, we also find that the}

optimal N is about 2 and 2.5 time-constants for

m

¼ 3  103and 5  103, respectively. In these cases, Tacq¼

6  103chips (47 bits) for both step sizes.

6. Conclusions

The performance of conventional code acquisition in a CDMA system degrades greatly when MAI presents. The adaptive filtering approach proposed recently has been proven to be MAI-resistant. In this paper, we propose a multirate adaptive code acquisition scheme that can

3 3.5 4 4.5 5 5.5 6 x 10−3 10−4 10−3 10−2 10−1 step size

Probabilty of acquisiiton error

1R systems, experimental 1R systems, theoretical 2R systems, experimental 2R systems, theoretical 3R systems, experimental 3R systems, theoretical

Fig. 6. Experimental and theoretical PE((9), Eqs. (54) and (60)) versus

step sizem(D ¼ 8, DM¼4, L ¼ 128, and SINRc¼ 13 dB).

0 2 4 6 8 10 12 14 16 0 0.5 1 Index of tap−weights Magnitude

Averaged tap−weights of LRAF

−2 −1 0 1 2 0 0.5 1 Index of tap−weights Magnitude

Averaged tap−weights of MRAF

−2 −1 0 1 2 0 0.5 1 Index of tap−weights Magnitude

Averaged tap−weights of HRAF

Fig. 5. Averaged tap-weights for LRAF, MRAF, and HRAF for 3R systems. (L ¼ 128, D ¼ 8, DM¼4,t¼51,m¼4  103, SINRc¼ 13 dB, and 100 trials.)

3 3.5 4 4.5 5 5.5 6 x 10−3 5000 6000 7000 8000 9000 10000 11000 step size

Mean acquisition time

1R systems 2R systems 3R systems lower bound

Fig. 7. Experimental mean acquisition time Tacq versus step size m

significantly reduce the required computational complexity. We have specifically studied the 2R and 3R systems and theoretically analyzed their performance; this includes the filter convergence properties, acquisition error rate, and mean acquisition time. Experimental results show that while the proposed schemes can perform similarly with the conventional adaptive acquisition, the computational complexity is much lower. The proposed scheme is specially suitable for CDMA systems operating in large propagation delay environments. With proper choice of D or DM, the

multirate code acquisition scheme can achieve an efficient compromise between the mean acquisition time and computational complexity. The proposed scheme can also be used in a carrier-phase unsynchronized system. In this circumstance, we have to take the inphase as well as quadrature components of tap-weights into account. If the

code delay has a fractional part, the optimum tap-weights will have two peaks and this will enlarge the MMSE, which in turn affects the acquisition performance. To mitigate this problem, we can oversample the receive signal and conduct a sub-chip level acquisition. In this paper, we only consider the scenario of the AWGN channel. It is straightforward to extend the use of the proposed scheme to multipath channels. In this case, the HRAF will acquire the signal from the strongest path. The performance analysis can then serve as a potential topic for further research.

Acknowledgement

The authors would like to thank the valuable com-ments from Ren-Jr Chen who is working in the Info. & Comm. Research Lab., ITRI, Taiwan.

Appendix A

In this section, the optimal tap-weights of a LRAF with different

D

will be derived. With the results, the steady-state MSE of a LRAF can be evaluated. To simplify the notation, we let

r

¼j

D

j

D . (A.1)

A.1. The case of

D

X0

Consider the case where

D

X0 first. The element xLðm 



Þ,



2 f0; 1; . . . ; Mpgin (18) can be rewritten as xLðm 



Þ ¼ X D1 j¼0 xððm 



ÞD  jÞ ¼ X D1 j¼0 xððm 



ÞD  jÞ þX D1 j¼D xððm 



ÞD  jÞ −16 −15 −14 −13 −12 −11 −10 10−4 10−3 10−2 10−1 100

Experimental probability of acquisition error

1R systems 2R systems 3R systems

SINRc (dB)

Fig. 8. Experimental PEversus SINRcðm¼5  103_Þ.

−16 −15 −14 −13 −12 −11 −10 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 4

Mean acquisition time

1R systems 2R systems 3R systems lower bound

SINRc (dB)

Fig. 9. Experimental mean acquisition time Tacq versus SINRc

ðm¼5  103_Þ. 1.5 2 2.5 3 3.5 4 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 multiple of time−constant

Mean acquisition time

μ = 3×10−3_{, 2R systems} μ = 3×10−3_{, 3R systems} μ = 3×10−3_{, lower bound} μ = 5×10−3_{, 2R systems} μ = 5×10−3_{, 3R systems} μ = 5×10−3_{, lower bound}

Fig. 10. Experimental mean acquisition time Tacqversus N (expressed as

¼ X D1 j¼0 xððm 



ÞD  jÞ þ X DD1 j¼0 xððm 



ÞD 

D

jÞ ¼pffiffiffiffiffiffiffiD

r

1ffiffiffiffiffiffiffi D

r

p X D1 j¼0 xððm 



ÞD  jÞ 8 < : 9 = ; þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Dð1 

r

Þ p ( X DD1 j¼0 xððm 



ÞD 

D

jÞ 9 = ;. (A.2) Let

y

ðmÞ ¼ 1ffiffiffiffiffiffiffi D

r

p X D1 j¼0 xðmD  jÞ, (A.3)

f

ðmÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Dð1 

r

Þ p X DD1 j¼0 xðmD 

D

jÞ, (A.4)

H

ðmÞ ¼ ½

y

ðmÞ;

y

ðm  1Þ; . . . ;

y

ðm  MpÞT, and

U

ðmÞ ¼

½

f

ðmÞ;

f

ðm  1Þ; . . . ;

f

ðm  MpÞT. Thus, (A.2) can be

writ-ten as xLðm 



Þ ¼ ffiffiffiffiffiffiffi D

r

p

y

ðm 



Þ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 



Þ,



2 f0; 1; . . . ; Mpg, (A.5) and (18) as xLðmÞ ¼ ffiffiffiffiffiffiffi D

r

p

H

ðmÞ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

U

ðmÞ. (A.6) Note that

y

ðmÞ,

f

ðmÞ and vLðmÞ are zero mean, mutually

uncorrelated, and Ef

y

ðmÞ

y

ðm 



Þg ¼

d

ð



Þ, Ef

f

ðmÞ

f

ðm 



Þg ¼

d

ð



Þ,

EfvLðmÞvLðm 



Þg ¼D

s

2v

d

ð



Þ, (A.7)

where

d

ðÞdenotes a Kronecker Dirac delta function. Using Eqs. (A.3) and (A.4), we can also express (15) as

rLðmÞ ¼ X D1 j¼0 xððm 

a

ÞD 

D

jÞ þ vLðmÞ ¼ X DD1 j¼0 xððm 

a

ÞD 

D

jÞ þ X D1 j¼DD xððm 

a

ÞD 

D

jÞ þ vLðmÞ ¼ X DD1 j¼0 xððm 

a

ÞD 

D

jÞ þX D1 j¼0 xððm 

a

1ÞD  jÞ þ vLðmÞ ¼pDð1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

Þ

f

ðm 

a

Þ þpffiffiffiffiffiffiffiD

r

y

ðm 

a

1Þ þ vLðmÞ. (A.8)

Let the tap-weights of the LRAF be wLðmÞ ¼ ½wL;0ðmÞ; wL;1ðmÞ; . . . ; wL;MpðmÞ

T_{. Also, let the}

cor-responding optimal solution be wL;o. Using the

corre-sponding Wiener equations, we can have

wL;o¼R1L pL, (A.9)

where pL9EfxLðmÞrLðmÞg and RL9EfxLðmÞxTLðmÞg. From

Eqs. (A.6) and (A.7), it is simple to derive

RL¼DI. (A.10)

Using Eqs. (A.6) and (A.8), we can find the cross-correlation between rLðmÞ and xLðmÞ as

pL¼ 0; . . . ; 0_{|fflfflfflffl{zfflfflfflffl}} a ;D 

D

;

D

;0; . . . ; 0 2 4 3 5 T (A.11) From (A.9), we obtain

wL;o;¼ 1 

r

;



¼

a

;

r

;



¼

a

þ1; 0 otherwise; 8 > < > :



2 f0; 1; . . . ; Mpg, (A.12) where wL;o;is the



th element of wL;o. Let the MSE that the

Wiener filter minimizes be JLðmÞ. Then,

JLðmÞ ¼ Ef½rLðmÞ  wTLðmÞxLðmÞ2g

¼Efr2

LðmÞg  2wTLðmÞpLþwTLðmÞRLwLðmÞ, (A.13)

where Efr2

LðmÞg ¼ DEfr2ðnÞg ¼ Dð

s

2vþ1Þ. Replacing wLðmÞ

with wL;o, we can obtain the corresponding MMSE, JL;min,

as

JL;min¼DEfr2ðnÞg  D½ð1 

r

Þ2þ

r

2

¼D½

s

2

vþ2

r

ð1 

r

Þ. (A.14)

From (A.14), we can see that a nonzero

r

will produce an extra term in the MMSE. We now proceed to find the MSE yielded by the LMS algorithm. Using Eqs. (A.6) and (A.12), we derive xT LðmÞwL;o¼ ð1 

r

Þf ffiffiffiffiffiffiffi D

r

p

y

ðm 

a

Þ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 

a

Þg þ

r

fpffiffiffiffiffiffiffiD

r

y

ðm 

a

1Þ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 

a

1Þg (A.15) ¼pDffiffiffiffiffiffiffi

r

y

ðm 

a

Þ þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 

a

Þ 

r

pffiffiffiffiffiffiffiD

r

y

ðm 

a

Þ 

r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 

a

Þ þ

r

pffiffiffiffiffiffiffiD

r

y

ðm 

a

1Þ þ

r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

f

ðm 

a

1Þ. (A.16) Substituting (A.8) into (A.16), we obtain

xT LðmÞwL;o¼rLðmÞ  vLðmÞ þ ð1 

r

Þ ffiffiffiffiffiffiffi D

r

p f

y

ðm 

a

Þ 

y

ðm 

a

1Þg 

r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þf

f

ðm 

a

Þ 

f

ðm 

a

1Þg. (A.17) Rewriting (A.17), we have

rLðmÞ ¼xTLðmÞwL;oþvLðmÞ ð1 

r

ÞpffiffiffiffiffiffiffiD

r

½

y

ðm 

a

Þ 

y

ðm 

a

1Þ þ

r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ½

f

ðm 

a

Þ 

f

ðm 

a

1Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 9xðmÞ , (A.18)

where

x

ðmÞ is zero mean and its variance is

s

2

x¼2D

r

ð1 

r

Þ. The LMS tap-weight update equation

for the LRAF is given by

wLðmÞ ¼ wLðm  1Þ þ

m

LxLðmÞ½rLðmÞ  xTLðmÞwLðm  1Þ,

(A.19) where

m

L is the step size. Substituting (A.18) into (A.19),

we have

wLðmÞ ¼ wLðm  1Þ þ

m

LxLðmÞ½xTLðmÞwL;oþvLðmÞ þ

x

ðmÞ

xT

LðmÞwLðm  1Þ. (A.20)

Subtracting wL;o on both sides of (A.20) and letting

D

wLðmÞ ¼ wLðmÞ  wL;o, we can rewrite (A.20) as

D

wLðmÞ ¼

D

wLðm  1Þ 

m

LxLðmÞxTLðmÞ

D

wLðm  1Þ þ

m

LxLðmÞvLðmÞ þ

m

LxLðmÞ

x

ðmÞ ¼ ½I 

m

LxLðmÞxTLðmÞ

D

wLðm  1Þ þ

m

LxLðmÞvLðmÞ þ

m

LxLðmÞ

x

ðmÞ. (A.21) Let Q ðmÞ ¼ Ef

D

wLðmÞ

D

wTLðmÞg. Then, Q ðmÞ ¼ Ef½I 

m

LxLðmÞxTLðmÞ

D

wLðm  1Þ 

D

wT Lðm  1Þ½I 

m

LxLðmÞxTLðmÞTg þ

m

2 LEfv2LðmÞxLðmÞxTLðmÞg þ

m

2 LEf

x

2 ðmÞxLðmÞxTLðmÞg. (A.22)

Eq. (A.22) can be written as Q ðmÞ ¼ ðI 

m

LRLÞQ ðm  1ÞðI 

m

LRLÞ þ

m

2 LD

s

2 vRLþ

m

2L

s

2 xRL. (A.23)

Note that in (A.23) we implicitly assume that

x

2ðmÞ and xLðmÞxTLðmÞ are uncorrelated. The



th entry on the main

diagonal of Q ðmÞ is Q_;ðmÞ ¼ ð1 

m

LDÞ2Q;ðm  1Þ þ

m

2LD 2

_s

2 vþ

m

2 LD

s

2 x. (A.24) When m!1, we have the asymptotic result as Q;ðmÞ 

m

₂LðD

s

2vþ

s

2xÞ

¼

m

LD 2 ½

s

2

vþ2

r

ð1 

r

Þ



2 f0; 1; . . . ; Mpg. (A.25)

Using Eqs. (A.14) and (A.25), we can have the MSE for the LMS algorithm in steady-state[34]as

JLð1Þ ¼JL;minþ ðMpþ1Þ

m

LD 2 ½

s

2 vþ2

r

ð1 

r

Þ ¼ 1 þðMpþ1Þ

m

L 2   JL;min. (A.26)

A.2. The case of

D

o0

Next, let us consider the case where

D

o0. We define a new set of

y

ðmÞ and

f

ðmÞ as

y

ðmÞ9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Dð1 

r

Þ p X DjDj1 j¼0 xðmD  jÞ, (A.27)

f

ðmÞ9 1ffiffiffiffiffiffiffi D

r

p X jDj1 j¼0 xððm  1ÞD þ j

D

j jÞ. (A.28)

Then, we can have xLðm 



Þas

xLðm 



Þ ¼ X D1 j¼0 xððm 



ÞD  jÞ (A.29) ¼ X DjDj1 j¼0 xððm 



ÞD  jÞ þ X D1 j¼DjDj xððm 



ÞD  jÞ (A.30) ¼ X DjDj1 j¼0 xððm 



ÞD  jÞ þX jDj1 j¼0 xððm 



1ÞD þ j

D

j jÞ (A.31) ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDð1 

r

Þ

y

ðm 



Þ þpffiffiffiffiffiffiffiD

r

f

ðm 



Þ, (A.32) and (15) as rLðmÞ ¼ X D1 j¼0 xððm 

a

ÞD þ j

D

j jÞ þ vLðmÞ (A.33) ¼ X jDj1 j¼0 xððm 

a

ÞD þ j

D

j jÞ þX D1 j¼jDj xððm 

a

ÞD þ j

D

j jÞ þ vLðmÞ (A.34) ¼ X jDj1 j¼0 xððm 

a

ÞD þ j

D

j jÞ þ X DjDj1 j¼0 xððm 

a

ÞD  jÞ þ vLðmÞ (A.35) ¼pffiffiffiffiffiffiffiD

r

f

ðm 

a

þ1Þ þpDð1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

Þ

y

ðm 

a

Þ þvLðmÞ. (A.36) Following the similar procedure for the case that

D

X0, we can derive wL;o;¼ 1 

r

;



¼

a

;

r

;



¼

a

1; 0 otherwise; 8 > < > :



2 f0; 1; . . . ; Mpg, (A.37) and JLð1Þ ¼JL;minþ ðMpþ1Þ

m

LD 2 ½

s

2 vþ2

r

ð1 

r

Þ ¼ 1 þðMpþ1Þ

m

L 2   JL;min. (A.38) References

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