282 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 10, OCTOBER 1999
A Differentially Coherent Delay-Locked Loop
for Spread-Spectrum Tracking Receivers
Chun-Chieh Fan and Zsehong Tsai,
Member, IEEEAbstract— A novel differentially coherent delay-locked loop
(DCDLL) for accurate code tracking is proposed for direct se-quence spread spectrum systems. Due to the use of the differential decoder and exactly one correlator, the proposed scheme avoids the problems of gain imbalance. The tracking error variance is derived by linear analysis. When the proposed DCDLL scheme is applied in ranging with additive white Gaussian noise (AWGN) channel, the performance of the proposed DCDLL scheme is about 1.4 dB better than that of one-correlator tau–dither loop (TDL), and near that of noncoherent DLL.
Index Terms— Correlator, delay-locked loop, direct sequence
spread spectrum, tracking error variance.
I. SYSTEMDESCRIPTION AND SIGNAL MODEL
I
N THIS letter, we present a code tracking receiver with less complexity, by employing a differentially coherent tech-nique originally proposed for pseudonoise (PN) acquisition receiver [3]. The proposed differentially coherent delay-locked loop (DCDLL) scheme is shown in Fig. 1. The received signal is first filtered by front-end band-pass filter (BPF) and the bandwidth of BPF is . is set to be chip rate ( , where is the chip duration). Then this proposed DCDLL scheme processes the received signal using a differential decoder with a delay of -chip duration in the delay path. The decoder output is then correlated with the difference of the advanced (early) and retarded (late) versions of the local PN code to produce an error signal. After the error signal is filtered by a low-pass filter (LPF), then it drives the voltage-controlled clock (VCC) through the loop filter and corrects the code phase error of the local PN code generator. In this proposed system, the bandwidth of LPF, denoted as , is set to be the system data rate ( , where is the data bit duration). The processing gain of this direct-sequence spread-spectrum (DS/SS) system is thus given by or . Usually, if the system is applied in ranging, andif the receiver is used for wireless communication. As in other DS/SS communication systems, the output of front-end BPF can be expressed as
(1)
Manuscript received August 7, 1998. The associate editor coordinating the review of this letter and approving of for publication was Prof. C. D. Georghiades.
C. C. Fan is with the Department of Electronic Engineering, St. John’s and St. Mary’s Institute of Technology, Tam-Shiu, Taipei, Taiwan, R.O.C.
Z. Tsai is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: ztsai@cc.ee.ntu.edu.tw).
Publisher Item Identifier S 1089-7798(99)08041-2.
Fig. 1. A DCDLL for DS/SS.
where is the average signal power, is a random binary data sequence, is the transmitted PN sequence, is the unknown time delay to be tracked by the code tracking loop and is a function of , and is a slowly time-varying carrier phase function, which is assumed to be constant over several successive chips. The noise is an additive white Gaussian noise (AWGN) process with two-sided power spectral density of W/Hz, and can be expressed in the quadrature form
(2) Note that , and are zero mean, of equal variance, and mutually independent Gaussian random processes with the same power spectral density as .
Next, one may express , the -chip delay form
of , as
(3)
where is assumed to equal to and
. Again, can be written in the following quadrature form:
(4) where and are processes with stochastic property
identical to and . Hence, and have
1089–7798/99$10.00 1999 IEEE
FAN AND TSAI: DIFFERENTIALLY COHERENT DLL 283
Fig. 2. The simulation for the correlation ofn1(t) and n1(t 0 KTc).
the same property and the correlation of and can be made negligible. By the simulation shown in Fig. 2 and the fact verified in [3], or 2 is enough to generate nearly
uncorrelated and .
II. PERFORMANCE ANALYSIS
A. The Derivation of S-Curve
In Fig. 1, the output of the differential decoder is the
product of and , and the error signal
can be written as
(5) where is the Laplace transfer function of the impulse response of LPF, , and is phase detector gain. Hence, denoted the Heaviside operator operating
on the time function [1], [2]. The form is
resulted from differential decoder, i.e.,
(6) where is an integer and ( is the period of the PN sequence). Equation (6) holds due to the following shift-and-add property in -sequence: if an -sequence is added to a proper phase shift of itself, then the resulting sequence is just another shift of the original -sequence [2].
For high processing gain and appropriately selected , the
term . This indicates
that there are almost no degradation effects due to data modulation. Hence, one need not consider the phenomenon of modulation self-noise. By neglecting the code self-noise, (5) can be reduced to
(7)
where is the code phase error normalized w.r.t. . In general, is a function of and can be defined by
. is the local estimate of the incoming code delay and is early–late discriminator offset. is the total noise. The autocorrelation of PN sequence, , is defined by
(8) After some algebra, it can be also shown the so-called S-curve
as
(9) From (9), we can find the S-curve of the proposed DCDLL scheme is the same as that of the coherent baseband delay-locked loop (BDLL) [2].
B. The Linear Analysis
For high SNR, the normalized tracking phase error can be kept small most of time. Hence, in the dynamic range there exists a linear region around . Within the
lin-ear region, one can assume , where
is the slope of the loop S-curve at (the origin). The closed-loop transfer function of the linear model can be shown as [1], [2]
(10) where indicates the Laplace transform.
In linear analysis, assuming that the loop bandwidth is narrow compared to the spectrum of total noise , the normalized (by ) tracking error variance, , can be ap-proximated as [2]
(11) where is the single-sided bandwidth of , and is defined by
(12) By definition, satisfies [2]
(13)
where is the autocorrelation
function of the total noise . After some algebra [2], it can be shown that
PN
(14) where
(15)
284 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 10, OCTOBER 1999
Finally, the tracking error variance for the proposed DCDLL, , can be derived from (11), (13), and (14). Namely, the closed form for is
(16) The is equal to 2 for the proposed DCDLL. For the popular choice of , the tracking error variance of the proposed scheme is given by
(17) The tracking error formulas of other schemes are available in the literature. For example, the tracking error variance of tau–dither loop (TDL) for is given by [2]
(18) and that of conventional noncoherent delay-locked loop
(NDLL) for is given by [1], [2]
(19) where is the equivalent bandwidth of LPF, and , the effect of data modulation, satisfies . For a system employing nonreturn-to-zero (NRZ) data modulation and ideal LPF’s, the value of is 0.902 [1].
III. NUMERICAL RESULTS AND DISCUSSION
In order to present more meaningful numerical results, we
define , and let be the ratio
of loop-SNR( ) to data-SNR( ), or equivalently the ratio of data bandwidth to loop bandwidth. Then it is straightforward to see
(20) and the previous (16) can be rewritten as
(21) and is set to be 1, i.e., . Data modulation is not considered here. This situation indicates that we only consider the application in ranging in AWGN channel. This can be shown in Fig. 3. From Fig. 3, we find the standard deviation
Fig. 3. Deviationas a function of ddB for DCDLL, TDL, and NDLL in linear analysis, with parameterso= 102; 103, and1 = 1=2.
of DCDLL is about 1.4 dB better than that of TDL and near that of NDLL with the linear analysis for fixed , and different values of . Fig. 3 also shows that the standard deviation decreases with the increasing data SNR .
IV. CONCLUSIONS
In this paper, we provide a DCDLL code tracking system like TDL with fairly less complexity in structure.
Compared with the conventional NDLL, the proposed DCDLL only employs one-correlator. Hence, it has less complexity than NDLL for implementation, and it also has evolved no difficulties in gain imbalance. When the proposed tracking receiver is applied in ranging, numerical results confirm that the performance of the proposed DCDLL is about 1.4 dB better than that of one-correlator TDL in AWGN channel environment.
REFERENCES
[1] A. Polydoros and C. L. Weber, “Analysis and optimization of cor-relative code-tracking loops in spread-spectrum systems,” IEEE Trans.
Commun., vol. COM-33, pp. 30–43, Jan. 1985.
[2] J. K. Holmes, Coherent Spread Spectrum Systems. New York: Wiley 1982.
[3] C. D. Chung, “Differentially coherent detection technique for direct-sequence code acquisition in a Rayleigh fading mobile channel,” IEEE
Trans. Commun., vol. 43, pp. 1116–1126, Feb./Mar./Apr. 1995.
