A Technique to Improve the Performance of Serial ... - IEEE Xplore
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 5, MAY 2005
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A Technique to Improve the Performance of Serial, Matched-Filter Acquisition in Direct-Sequence Spread-Spectrum Packet Radio Communications Arvind Swaminathan, Student Member, IEEE, and Daniel L. Noneaker, Senior Member, IEEE
Abstract—In this paper, we examine a simple method to improve the performance of serial, matched-filter acquisition in direct-sequence spread-spectrum packet radio communications. Each packet transmission includes an acquisition preamble, and the preamble sequence is changed at the boundaries of predefined time epochs based on a pseudorandom sequence generator. It is shown in previous work that the presence of an intermediate-frequency filter and the characteristics of the automatic gain-control subsystem lead to a probability of not acquiring that is a nonmonotonic function of the signal-to-noise ratio if the acquisition algorithm uses a threshold-crossing detector with a fixed threshold. The acquisition algorithm presented in this paper employs an estimator to adaptively select the acquisition threshold for each test statistic. It is shown that this technique reduces the severity of the nonmonotonicity and substantially improves the acquisition performance. Index Terms—Packet radio communications, pseudonoise coded communications, radio receivers, synchronization.
I. INTRODUCTION
D
IRECT-SEQUENCE (DS) spread-spectrum modulation is used widely in both commercial and military communication. It is employed in cellular code-division multiple-access (CDMA) networks, as well as tactical radio networks for the military. Packet-based transmission is the natural choice for a DS multiple-hop ad hoc network, and it has also been incorporated into the designs of third-generation and later CDMA cellular networks. The receiver of a DS packet transmission must achieve synchronization between the spreading sequence in the packet and a locally generated copy of the same sequence in order to demodulate the data in the packet. Synchronization usually occurs in two stages: acquisition and tracking. The acquisition stage provides a coarse alignment between the local sequence and the sequence of the arriving signal, and it is necessary because of the receiver’s a priori uncertainty concerning the time of arrival of a packet transmission. The acquisition performance of the
Manuscript received April 2, 2004; revised December 8, 2004. This work was supported in part by the DoD Multidisciplinary University Research Initiative (MURI) Program administered by the Office of Naval Research under Grant N00014-00-1-0565 and in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant DAAD19-00-1-0156. This paper was presented in part at the IEEE International Conference on Communications, Anchorage, AK, May 2003. The authors are with the Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2005.845406
receiver is often the limiting factor in the physical-layer performance of DS spread-spectrum packet radio communication, and thus the acquisition technique can be an important factor in the overall performance of a DS packet radio network. Each transmission in a DS packet radio system includes a preamble of predetermined duration that consists of a segment of the spreading sequence with no data modulation, and the spreading sequence (but not the time of arrival) is known a priori at the receiver. In a practical system design, security and other considerations dictate that the spreading sequence is changed frequently. This can be accomplished by employing a long-period pseudorandom sequence generator and frequently varying the sequence phase used for the start of packet transmissions. In this paper, we consider a DS packet radio system in which the preamble sequence is changed at regular intervals. The same preamble sequence is used for all packet transmissions during the time epoch between changes, and the pseudorandom generator and the sequence phase determining the preamble in any time epoch are known a priori at the receiver. The most practical techniques for acquiring a DS packet transmission utilize a filter matched to the preamble [1], [2], and that approach is considered in this paper. The focus of the paper is a technique to improve the performance of serial, matched-filter acquisition in DS spread-spectrum packet radio communication. The research is motivated by results obtained in [3] which account for the effect of the intermediate-frequency (IF) filter and the subsequent automatic gain-control (AGC) subsystem on acquisition performance in a superheterodyne receiver. The noncoherent, serial acquisition algorithm detects the arrival of the preamble at the receiver based on threshold crossing by a square-law statistic formed from the matched-filter output. The receiver considered in [3] employs a fixed acquisition threshold that is chosen by optimizing the average performance over all preamble sequences produced by the pseudorandom generator. It is shown that serial acquisition with a fixed threshold results in a probability of not acquiring that is a nonmonotonic function of the signal-to-noise ratio (SNR) of the received signal. In this paper, we present a simple method for adapting the acquisition threshold at the receiver that results in substantially better performance than acquisition with a fixed threshold. It has the additional benefit of compensating automatically for variations in the gain characteristics of the AGC subsystem. (Thus, it provides continuous calibration of the acquisition subsystem and more consistent acquisition performance over variations in the temperature, age, and manufacturing processes of the analog
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Fig. 1.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 5, MAY 2005
Block diagram of the receiver.
components of the IF stage in the receiver.) Although the acquisition algorithm is presented in the context of a superheterodyne receiver, the algorithm and the corresponding performance evaluation are applicable to any receiver in which an AGC subsystem occurs before chip-matched filtering of the DS signal. The paper is organized as follows. The communication system is described in Section II. The test statistics generated in serial, matched-filter acquisition are characterized in Section III, and the performance with a fixed acquisition threshold is examined in detail in Section IV. This examination motivates the design of the adaptive-threshold technique, which is described in Section V. The performance with the optimal fixed acquisition threshold and with the adaptive threshold is compared in Section VI. Conclusions are given in Section VII.
randomly selected from among all sequences of quaternary chips with equal probability. Each is a random variable uniformly distributed over the values in . This random-sequence model [5] provides a good approximation to the use of a long-period pseudonoise generator that is used to select a different preamble sequence for each time epoch. Without loss of generality, the transmitted signal is assumed to arrive unattenuated and undelayed at the receiver over an additive white Gaussian noise channel. Thus the received signal is (4) where (5)
II. SYSTEM DESCRIPTION The complex-valued, baseband-equivalent, white Gaussian has two-sided power spectral density . noise voltage
A. Transmitted Signal and Received Signal Each packet transmission consists of a preamble of quaternary chips followed by the packet’s data content. The spreading sequence used during any time epoch is known a priori at the receiver. The transmitted signal is given by (1) where (2) is the carrier frequency at the transmitter, is the power in the is the rectangular transmitted signal during the preamble, pulse with height 1 over , and is the packet’s data content of duration chips. The quaternary phase-shift-keyed (QPSK) spreading waveform is given by
where represents a complex-valued spreading sequence is time limited to and [4]. The chip waveform (3) The spreading sequence employed for transmission of a packet during a given time epoch is modeled as
B. Receiver Architecture Since the focus of this paper is not on the down-conversion of the received signal from the radio frequency to IF, the signal given by (4) is taken to be the received signal after down-conversion to IF. The receiver is modeled as an AWGN source, a bandpass IF filter, and an AGC subsystem followed by an acquisition stage as shown in Fig. 1. The impulse response of the IF filter is given by
where is the baseband-equivalent impulse response of the filter. A well designed AGC subsystem achieves minimal variation in its average output power over a wide dynamic range for the average input power. It responds quickly to a step change in the received signal power, yet its output power is relatively insensitive to variation in the signal envelope over the duration of a chip and to the random variations of the band-limited noise at its input. In this paper, we approximate such a design by considering an AGC subsystem that responds instantly to the arrival of the beginning of the packet transmission’s preamble. Square-law detection is employed in its gain-control feedback loop [6] (though the qualitative observations in the paper still hold if envelope-detection feedback is considered instead).
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
We approximate the gain of the AGC subsystem prior to the arrival of the transmission as a constant that is inversely proportional to the expected (noise) power at its input. Its gain after the arrival of the transmission is constant and inversely proportional to the expected power at its input after the IF filter transients have settled, where the expectation is with respect to the random noise process, the random preamble sequence, and a random chip offset. An expression is developed in [3] that characterizes the gain under this approximation. It is shown that the is given by AGC output
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(9)
is verified, a locally generated copy of the spreading waveform for data demodulation is synchronized to the delay and the receiver enters the data-detection mode. If verification fails, however, the receiver returns to the acquisition mode. Packet acquisition thus occurs if the receiver is in the acquisiis generated. tion mode and a hit occurs when the statistic is genBut if the receiver is in the acquisition mode when erated and the test does not result in a hit, a miss occurs and the packet is not acquired. On the other hand, if the detector declares , a false alarm occurs. If the a hit for a test statistic other than receiver is in verification mode due to a false alarm at the time is generated, then acquisition fails due to the false alarm. The verification interval is the amount of time required for the receiver to determine that a false alarm has occurred and return to the acquisition mode, and in this paper, it is taken to be a con. Hence, a packet is not acquired if either a miss occurs stant . In the or a false alarm occurs for , remainder of this paper, the term “probability of miss” refers to the probability that a miss occurs if the receiver is in acquisition mode at the time the end of the preamble signal is received. The term “probability of false alarm” refers to the probability that a false alarm results in failed acquisition of the packet. The filters for the acquisition stage that are considered in the analysis are those having the form
(10)
(13)
where is the Fourier transform of the chip waveform and is the baseband equivalent of the IF filter’s frequency response. The parameter serves as a key measure of the ratio of the IF filter’s bandwidth to the bandwidth of the DS signal.
is the spreading sequence for the acquiwhere sition preamble. Furthermore, the impulse response of the baseand the function that are conband-equivalent IF filter sidered satisfy the joint constraint
(6) where (7) The parameters in (7) are given by (8)
C. Acquisition Algorithm
for all
(14)
The acquisition stage employs noncoherent square-law combining of the baseband filter outputs to form the test statistics. The inphase and quadrature branches of the receiver each conand , as shown in Fig. 1. tain two filters, denoted by The outputs in each branch at time are summed to form
with the IF filter’s Under this constraint, the convolution of baseband-equivalent impulse response is matched to the acquisition preamble.
(11) (12)
In this section, the test statistics are characterized for both a and for a random given preamble sequence preamble sequence. It follows from (4)–(12) that conditioned on the preamble sequence
where and . The for receiver samples the outputs of the filters at time . (The each integer , and it forms the test statistic assumption of chip-level synchronism at the receiver results in an accurate approximation to the acquisition performance with over-sampling and an arbitrary sample-timing error [7].) Thus, , the statistics if a packet’s preamble is received starting at correspond to the period before the packet’s arrival, correspond to the period during reception of corresponds to the reception of the preamble signal, and the full preamble signal. The test statistics are utilized sequentially as they are formed is if the receiver is in the acquisition mode, and the statistic , and compared with a threshold . A hit is declared if the receiver then enters the verification mode in which it determines if synchronization has been achieved. If synchronization
III. CHARACTERIZATION OF THE TEST STATISTICS
(15)
where is the aperiodic autocorrelation function of the sequence given by (16) and SNR is the preamble signal-to-noise ratio given by
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(The preamble signal-to-noise ratio is referred to simply as the signal-to-noise ratio in the remainder of the paper. Note that it is much greater than the data-channel-symbol SNR for a given received signal power, since the number of chips in the preamble is much greater than the number of chips per data channel symbol in any practical packet radio system.) Also ,
and lated. The
(17) are mutually uncorrecollection
of
uncorrelated random variables are jointly Gaussian and, therefore, independent. Hence, the test statistics are independent chi-square random variables with two degrees of freedom. The test statistics are central chi-square random variables, are noncentral chi-square and the test statistics random variables in general. Suppose instead that the preamble sequence is selected randomly from among all possible sequences. It is shown in [3] that for all are mutually uncorrelated the set of statistics random variables with (18) where the expectation is with respect to a random preamble sequence. It is also shown that
Fig. 2. Fixed-threshold acquisition performance with ideal preamble sequence of length 26.
IV. PERFORMANCE OF FIXED-THRESHOLD ACQUISITION In this section, we examine in detail the performance of serial, matched-filter packet acquisition with a fixed acquisition threshold. A key characteristic of acquisition performance is revealed that is undesirable with respect to performance robustness in channels of a priori unknown signal quality. Consideration of the underlying cause leads naturally to the adaptivethreshold technique introduced in the next section. Consider a DS packet radio system employing a fixed preamble sequence. Suppose that a fixed acquisition threshold is for all ). It follows from the results employed (so that of the previous section that the probability of not acquiring can be expressed as
. (19) The joint distribution function of the collection of uncorrelated random variables can be approximated for a random preamble sequence by treating the random variables as jointly Gaussian, and therefore independent, with first and second moments given by (18) and (19), respectively. Under this approximation, the test statisare central chi-square random tics is a noncentral chi-square variables and the test statistic random variable, each with two degrees of freedom [3]. It is shown in [3, eqs. (32)–(34)] that the Gaussian approximation results in a simple, closed-form expression for the probability of not acquiring (averaged over all preamble sequences) with a fixed acquisition threshold. We have shown through simulation that the Gaussian approximation leads to extremely accurate results for the probability of not acquiring with a fixed threshold for the preamble sequence lengths and verification intervals of practical interest. In this paper, however, we employ the Gaussian approximation only to obtain some qualitative insights in the next section.
(20) is given by (15), where from (17), and is the Marcum Q-function [8]. [In pracfor this system by Monte Carlo tice, we choose to evaluate simulation rather than by using (20).] Suppose the fixed preamble sequence is an (unrealizable) ideal quaternary sequence, that is, a preamble sequence with a discrete aperiodic autocorrelation function for which each sidelode is equal to zero. Suppose, for example, that the system chips, the verification has an ideal preamble of length chips, the chip waveform and interval has a duration of , and the the response of the receiver’s IF filter result in . The acquisition threshold is given by acquisition performance of the system is shown in Fig. 2. As the
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Fig. 3.
Fixed-threshold acquisition performance with preamble sequence A.
SNR increases, the statistic corresponding to correct synchroincreases stochastically so that the probability nization of miss decreases monotonically and approaches zero asymptotically. Furthermore, each of the statistics decreases stochastically toward a limiting distribution as the SNR increases, due to the change in the AGC attenuation upon the arrival of the beginning of the packet transmission. Thus the probability of false alarm decreases toward a nonzero limiting value as the SNR increases, though the dependence on the SNR is slight. Consequently, acquisition performance with the ideal preamble sequence closely approximates the constant-false-alarm-rate performance predicted in well-known acquisition models that don’t account for the effect of the AGC subsystem. Similar behavior is observed with fixed-threshold acquisition and an ideal preamble sequence of any length. Suppose instead that the fixed preamble sequence is a realizable quaternary sequence with very good autocorrelation properties. This is illustrated by a system that uses the preamble sequence of length 26 in the Appendix, denoted sequence A, which has a maximum aperiodic autocorrelation sidelobe magnitude of only 3.16. The parameters of the system are the same as those in the previous example, except that the fixed . The acquisition threshold is given by acquisition performance of the system is shown in Fig. 3. The probability of miss depends on the SNR in the same manner as in the previous example. But each of the quadrature statistics exhibits an increasing mean that determine and decreasing variance as the SNR increases. The resulting probability of false alarm is a nonmonotonic function of the SNR in this example, though the region in which it is increasing is indiscernible from the figure. Consequently, the probability of not acquiring is also a nonmonotonic function of the SNR. Since the probability of false alarm exhibits only minimal dependence on the SNR, however, the acquisition performance once again approximates that of a constant false-alarm-rate system if preamble sequence A is used. The nonmonotonicity can be much more pronounced if the fixed preamble sequence
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Fig. 4. Average fixed-threshold acquisition performance over all preamble sequences of length 26.
has larger aperiodic autocorrelation sidelodes, however, in which case the probability of not acquiring is not accurately approximated by a constant false-alarm rate. In tactical packet radio communication, security concerns will often dictate that the preamble sequence is changed at regular intervals among a large set of candidate sequences. Thus, an important measure of acquisition performance is the average performance over a large set of preamble sequences. If the number of sequences is sufficiently large, the average performance can be closely approximated by evaluating the expected performance with a randomly generated preamble sequence. The expected acquisition performance for a random preamble for any sequence is evaluated in [3]. It is shown that combination of the DS chip waveform and the IF filter. (Indeed, except in the unrealizable case of a Nyquist-pulse chip waveform and an Nyquist-bandwidth ideal bandpass IF filter.) Thus, it follows from (18) and (19) that under the Gaussian inapproximation discussed in Section III, crease stochastically with the SNR. Consequently, the average probability of false alarm is a strictly increasing function of the SNR under the Gaussian approximation if the acquisition threshold is fixed, and the average probability of not acquiring is a nonmonotonic function of the SNR. These conclusions based on the Gaussian approximation are borne out by Monte Carlo simulations that do not employ the approximation. For example, consider the acquisition performance and a expected for a system with a preamble of random preamble sequence, which is illustrated in Fig. 4. The parameters of the system are the same as those in the previous example, except that the fixed acquisition threshold is given by . The example clearly illustrates the nonmonotonicity of the acquisition performance as a function of the SNR. The nonmonotonicity is even more pronounced if the system employs a longer preamble sequence. This is illustrated in Fig. 5
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this undesirable phenomenon. It is shown in the next section that the modification leads to improved acquisition performance. The modified serial acquisition algorithm employs an acquisition threshold that is adapted for each test statistic. The acquiis compared sition threshold against which the test statistic is given by (21) where is a constant and is the scaling factor employed at time instant to scale the threshold . (Normalization by the is used for convenience in the analysis and constant does not entail any loss of generality in the result.) The scaling is the average of the previous factor employed at time test statistics given by (22) Fig. 5. Average fixed-threshold acquisition performance over all preamble sequences of length 400.
in which the average acquisition performance is shown for a system with a preamble of 400 chips. The system’s verification chips, the chip waveform interval has a duration of , and the fixed acquisition and the IF filter result in . threshold is given by Thus, DS packet acquisition exhibits a surprising average performance characteristic if a fixed acquisition threshold is used with a randomly selected preamble sequence. It is shown in Section VI that this characteristic limits the ability to achieve robust acquisition performance in a channel with a SNR that is unknown a priori. The severity of the nonmonotonicity in acquisition performance is greater for larger values of [3] so that the phenomenon is most pronounced if the receiver uses an (inexpensive) IF filter with a bandwidth several-fold greater than the bandwidth of the desired signal. The severity of the nonmonotonicity is mitigated to some extent in any actual receiver by the saturation effects in the radio-frequency and AGC subsystems. For any practical radio design, however, the dynamic range of the receiver is large enough for the phenomenon to occur. Indeed, a nonmonotonic average probability of not acquiring with increasing SNR has been observed in tests of a prototype DS packet radio system [9]. V. ACQUISITION USING AN ADAPTIVE THRESHOLD It is shown in the previous section that the average probability of not acquiring is a nonmonotonic function of the SNR if the serial acquisition algorithm employs a fixed acquisition threshold with a randomly selected preamble sequence. In particular, the off-peak test statistics generated during reception of the pre) increase amble signal (that is, the statistics stochastically as the SNR increases, and consequently, the probability that one of them results in a false alarm increases with an increase in the SNR. In this section, we introduce a simple modification to the serial acquisition algorithm that mitigates
where is the window size. No knowledge of the distribution function of the test statistic is required by the receiver in order to determine . (Note also that the form of the scaling factor does not depend on the preamble sequence used in any given time epoch.) The scaling factor is intended to compensate for the dependence on the SNR for each off-peak test statistic’s distribution (denoted function. In particular, the mean of the test statistic ) depends on the SNR, and the scaling factor can be viewed as an estimator of . From (18) and (19), it follows that
(23)
where and expectation is with respect to the noise process and the random preamble sequence. Since for , is an unbiased estimator it follows from (19) and (23) that if (i.e., for noise-only test statistics). But it has a of negative bias if . are identically disThe independent random variables so that for . By the strong law tributed for with probability one as the window of large numbers, size is increased for , and a large window size results in . For , however, a highly accurate estimate of for increases with increasing window size (and inthe bias in creasing SNR). Thus, for any given set of operating conditions, the optimal window size with respect to estimator accuracy is some finite value. VI. COMPARISON OF THE ACQUISITION TECHNIQUES In this section, we compare the performance that results with a fixed acquisition threshold and the performance that results with the adaptive threshold described in the previous section. Two performance criteria and the corresponding measures of performance are introduced and used in the comparison.
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Both the performance with a fixed preamble sequence and the expected performance with a randomly selected preamble sequence are considered. All performance results are based on Monte Carlo simulations using the exact characterization of the test statistics. (The Gaussian approximation discussed in Section III is not employed.) One natural criterion for acquisition performance is that the acquisition algorithm achieves the best possible worst-case performance over a specified range of AWGN channels, where each channel is characterized by the SNR at the receiver. This criterion is referred to as the min-max criterion. A performance measure that reflects the min-max criterion is , the maximum probability of not acquiring over all AWGN channels for which , where is a predetermined value that characterizes the poorest quality channel for which acquisition performance is of interest. For a given acquisition technique, optimal performance with respect to the min-max criterion is achievable for defined as the smallest possible value of . the specified value of Another natural criterion for acquisition performance is that the acquisition algorithm achieves or exceeds a desired level of performance over the widest possible range of AWGN channels. This criterion is referred to as the max-range criterion. A performance measure that reflects the max-range criterion is , the smallest SNR such that the maximum probability over all AWGN chanof not acquiring does not exceed , where is a predetermined nels for which value that characterizes the poorest acceptable acquisition performance. For a given acquisition technique, optimal performance with respect to the max-range criterion is defined as the achievable for the specified smallest possible value of . value of The length of the preamble is 400 chips in all of the examples, and the chip waveform and response of the receiver’s IF filter . The verification interval is chips. result in chips, which The size of the estimator window is provides optimal performance under these circumstances if the adaptive threshold is used. With the exception of the results illustrated in Fig. 10, the value of used with each acquisition technique is the one that results in min-max optimal perfordB. (Recall that this corresponds to mance for a much lower channel-symbol SNR for the data payload in any packet format of interest.) For this range of optimization, the , while min-max optimal fixed threshold is the min-max optimal value for in (21) used with the adap. For both acquisition techtive threshold is niques, optimization is with respect to the expected performance with a randomly selected preamble sequence. A. Comparison of Performance With a Fixed Preamble Sequence We can gain some insight into the performance of the two acquisition techniques by considering the performance that results for examples of specific preamble sequences. In this subsection, we consider the performance with three fixed preamble sequences, denoted sequences B, C, and D, each of which is
Fig. 6.
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Acquisition performance with preamble sequence B.
given in the Appendix. The three sequences illustrate the range of acquisition performance that may be observed among the preamble sequences generated by a long-period pseudonoise generator. The probability of not acquiring is shown as a function of the SNR in Fig. 6, for both acquisition techniques with preamble sequence B. The probability of not acquiring with the fixed threshold is low if the SNR is high, and the performance over the . (Recall that the range of interest is poorest if threshold is chosen to optimize the average performance over all preamble sequences rather than the performance for this particular preamble sequence.) In contrast, the worst-case probability and as SNR of not acquiring occurs at both approaches infinity if the adaptive threshold is used. Moreover, the worst-case probability of not acquiring for is more than an order of magnitude lower with the adaptive threshold than with the fixed threshold. The performance for both acquisition techniques is shown in Fig. 7 for preamble sequence C. The worst-case probability of not acquiring over the range of interest occurs at an intermediate value of SNR if the fixed threshold is used. Similarly, the results for preamble sequence D are shown in Fig. 8. If the fixed threshold is used with this preamble sequence, the worstcase probability of not acquiring occurs as SNR approaches infinity. For both preamble sequence C and preamble sequence D, the worst-case probability of not acquiring occurs at if the adaptive threshold is used. With either preamble sequence, the worst-case probability of not acquiring is more than an order of magnitude lower with the adaptive threshold than with the fixed threshold. Not only does the adaptive threshold result in much better worst-case performance than the optimal fixed threshold for each preamble sequence, it results in much more consistent performance for different preamble sequences and over all values of the SNR. If the optimal fixed threshold is used, the
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Fig. 7. Acquisition performance with preamble sequence C.
Fig. 9. Acquisition performance averaged over all sequences with SNR
=
Fig. 10.
=
16 dB.
Fig. 8. Acquisition performance with preamble sequence D. Acquisition performance averaged over all sequences with P .
2:15 2 10
maximum probability of not acquiring is 0.028, 0.1, and 0.75 for sequences B, C, and D, respectively, whereas the corresponding value with the adaptive threshold is 0.0018 for each sequence. Furthermore, the variation in the probability of not acquiring over the range of SNRs of interest is several orders of magnitude for a given sequence if the optimal fixed threshold is used, but the variation is no more than fourfold over the same range for any of the three sequences if the adaptive threshold is used. B. Comparison of Average Performance Over All Preamble Sequences The average acquisition performance over all preamble sequences is compared for the two acquisition techniques using both the min-max criterion and the max-range criterion. The
comparison using the min-max criterion is shown in Fig. 9 for dB. The worst-case probability of not acquiring dB is 0.0018 if the adaptive threshold is used, over while the worst-case probability of not acquiring with the optimal fixed threshold is 0.021. Thus, the performance is more than a order of magnitude better with the adaptive threshold. The comparison using the max-range criterion is shown in Fig. 10 in which the performance requirement is that the prob10 ability of not acquiring is no greater than for . The constant is chosen for each acquisition for the technique to achieve the smallest possible value of . With this performance requirement, the specified value of optimal value of in (21) for the adaptive threshold technique is
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Fig. 11. Acquisition performance with an optimum fixed threshold with errors in the AGC gain.
. If the optimal fixed threshold is used, dB, the performance requirement can be satisfied for but if the adaptive threshold is used, the performance requiredB. ment can be satisfied over the wider range of Thus, the adaptive threshold yields a performance improvement of 1.65 dB over the optimal fixed threshold with respect to this performance criterion. The difference in the performance of the two acquisition techniques is even greater if the value of is larger (e.g., if the IF filter bandwidth is larger). C. Performance With Variation in IF Component Characteristics The performance of any communication device is susceptible to variation in the characteristics of its analog components that occurs as the components age and the operating temperature of the device changes. Furthermore, differences in analog components due to variations in manufacturing processes can result in differing performance among devices based on the same design. The performance of serial, matched-filter acquisition can be affected significantly by variations in the components of the IF portion of the superheterodyne receiver. The performance is sensitive to variations in the nominal steady-state output voltage of the AGC subsystem or the response of the IF filter if a fixed threshold is used. But the use of the adaptive threshold effectively eliminates performance sensitivity to first of these impairments, and its superior performance is preserved in the presence of the second impairment. The effect of the steady-state AGC output voltage on the acquisition performance is illustrated in Fig. 11. The performance averaged over all preamble sequences is shown for a receiver using a fixed threshold that has been optimized for the nominal steady-state output voltage of the AGC subsystem dB using the min-max criterion. The and
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average probability of not acquiring is shown as a function of the SNR for three circumstances: the steady-state AGC output voltage is equal to its nominal value, the voltage is 15% greater than its nominal value, and the voltage is 15% less than its nominal value. If the AGC output voltage is at the nominal level for which the threshold is chosen, the worst-case dB is 0.021. But probability of not acquiring for if the AGC output voltage is 15% higher than the nominal value, the maximum probability of not acquiring increases to 0.068, and if it is 15% lower than the nominal value, the worst-case probability of not acquiring increases to 0.079. In contrast, if the adaptive threshold is employed, variations in the steady-state output voltage of the AGC subsystem are compensated for automatically, and the acquisition performance is not affected by deviations of the steady-state voltage from its nominal value. Similarly, the acquisition performance is highly sensitive to the bandwidth of the IF filter if a fixed threshold is used, and moderate deviations from the nominal filter bandwidth can result in substantial degradation in acquisition performance. (Differences in the bandwidth of the IF filter are reflected in differences in the value of the parameter introduced earlier.) Acquisition performance with the adaptive threshold is also affected by deviations from the nominal filter bandwidth. For any given deviation from the nominal bandwidth, however, acquisition with the adaptive threshold results in much better performance than fixed-threshold acquisition. VII. CONCLUSION In this paper, we present an algorithm to adapt the threshold used with serial, matched-filter acquisition in DS spread-spectrum packet radio communications. The algorithm adapts the threshold for each test statistic based on a windowed average of the recent past test statistics. It is shown that the acquisition performance that results with the adaptive threshold is better than the performance that results if the optimal fixed threshold is used. The adaptive threshold results in better performance than a fixed threshold if a fixed preamble sequence is employed, it results in performance that is more consistent across different preamble sequences and over a range of values of the SNR, and it results in better average performance over the set of all possible preamble sequences. In addition, the adaptive threshold is shown to result in more robust acquisition performance in the presence of analog components with parameters that differ from their nominal values. APPENDIX The individual quaternary preamble sequences considered in the paper are given below. The binary representation of each in-phase and quadrature sequence is padded with trailing zeros to obtain its unique hexadecimal representation shown below. Sequence A has a length of 26 quaternary symbols. Sequences B, C, and D have a length of 400 quaternary symbols, as shown on the next page.
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Real part of sequence A
Imaginary part of sequence A
Real part of sequence B
Imaginary part of sequence B
Real part of sequence C
Imaginary part of sequence C
Real part of sequence D
Imaginary part of sequence D
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
REFERENCES [1] C. R. Cahn, “Spread spectrum applications and state-of-the-art equipments,” in Proc. AGARD-NATO Lecture Series, Spread-Spectrum Commun., Bolkesjø, Norway, May 1973, p. 5. [2] A. Polydoros and C. L. Weber, “A unified approach to serial search spread-spectrum code acquisition—Part II: A matched filter receiver,” IEEE Trans. Commun., vol. COM-32, no. 5, pp. 550–560, May 1984. [3] D. L. Noneaker, A. R. Raghavan, and C. W. Baum, “The effect of automatic gain control on serial, matched-filter acquisition in direct-sequence packet radio communications,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 1140–1150, Jul. 2001. [4] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, no. 5, pp. 593–619, May 1980. [5] K. K. Chawla and D. V. Sarwate, “Acquisition of PN sequences in chip synchronous DS/SS systems using a random sequence model and the SPRT,” IEEE Trans. Commun., vol. 42, no. 6, pp. 2325–2334, Jun. 1994. [6] J. R. Smith, Modern Communication Circuits, 2nd ed. New York: McGraw-Hill, 1998. [7] D. L. Noneaker, “The performance of serial, matched-filter acquisition in direct-sequence packet radio communications,” in Proc. IEEE Military Commun. Conf., McLean, VA, Oct. 2001, pp. 1045–1049. [8] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [9] J. R. McChesney, ITT Aerospace/Communications Division, private correspondence.
Arvind Swaminathan (S’99) was born in Chennai, India, on February 4, 1978. He received the B.Tech. degree in electronics and communications engineering from the Regional Engineering College, Calicut, India, in 1999 and the M.S. degree in electrical engineering from Clemson University, Clemson, SC, in 2002. For his M.S. thesis, he worked on techniques to improve the performance of serial acquisition in direct-sequence spread-spectrum packet radio networks. He is currently working towards the Ph.D. degree in electrical engineering at Clemson University, where he is a Research Assistant in the Wireless Communications Laboratory. He is working on protocol design for ad hoc networks with directional antennas for his Ph.D. dissertation. Mr. Swaminathan is a member of Alpha Epsilon Lambda.
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Daniel L. Noneaker (SM’93) was born in Montgomery, AL, on December 10, 1957. He received the B.S. degree (high honors) in mathematics from Auburn University, Auburn, AL, in 1977, the M.S. degree in mathematics from Emory University, Atlanta, GA, in 1979, the M.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1984, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Urbana, in 1993. He has industrial experience in both hardware and software design for communication systems. From 1979 to 1982, he was with Sperry-Univac, Salt Lake City, UT, and from 1984 to 1988, he was with the Motorola Government Electronics Group, Scottsdale, AZ. He was a Research Assistant in the Coordinated Science Laboratory, University of Illinois at Urbana–Champaign from 1988 to 1993. Since August 1993, he has been with the Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, SC, where he currently holds the position of Associate Professor. He has published numerous papers on the design and analysis of multiple-access systems for both cellular communication and ad hoc packet radio networks. He is engaged in research on wireless communication for both military and commercial applications with emphases on spread-spectrum communications, error-control coding for fading channels, and protocols for mobile radio networks.
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A Technique to Improve the Performance of Serial, Matched-Filter Acquisition in Direct-Sequence Spread-Spectrum Packet Radio Communications Arvind Swaminathan, Student Member, IEEE, and Daniel L. Noneaker, Senior Member, IEEE
Abstract—In this paper, we examine a simple method to improve the performance of serial, matched-filter acquisition in direct-sequence spread-spectrum packet radio communications. Each packet transmission includes an acquisition preamble, and the preamble sequence is changed at the boundaries of predefined time epochs based on a pseudorandom sequence generator. It is shown in previous work that the presence of an intermediate-frequency filter and the characteristics of the automatic gain-control subsystem lead to a probability of not acquiring that is a nonmonotonic function of the signal-to-noise ratio if the acquisition algorithm uses a threshold-crossing detector with a fixed threshold. The acquisition algorithm presented in this paper employs an estimator to adaptively select the acquisition threshold for each test statistic. It is shown that this technique reduces the severity of the nonmonotonicity and substantially improves the acquisition performance. Index Terms—Packet radio communications, pseudonoise coded communications, radio receivers, synchronization.
I. INTRODUCTION
D
IRECT-SEQUENCE (DS) spread-spectrum modulation is used widely in both commercial and military communication. It is employed in cellular code-division multiple-access (CDMA) networks, as well as tactical radio networks for the military. Packet-based transmission is the natural choice for a DS multiple-hop ad hoc network, and it has also been incorporated into the designs of third-generation and later CDMA cellular networks. The receiver of a DS packet transmission must achieve synchronization between the spreading sequence in the packet and a locally generated copy of the same sequence in order to demodulate the data in the packet. Synchronization usually occurs in two stages: acquisition and tracking. The acquisition stage provides a coarse alignment between the local sequence and the sequence of the arriving signal, and it is necessary because of the receiver’s a priori uncertainty concerning the time of arrival of a packet transmission. The acquisition performance of the
Manuscript received April 2, 2004; revised December 8, 2004. This work was supported in part by the DoD Multidisciplinary University Research Initiative (MURI) Program administered by the Office of Naval Research under Grant N00014-00-1-0565 and in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant DAAD19-00-1-0156. This paper was presented in part at the IEEE International Conference on Communications, Anchorage, AK, May 2003. The authors are with the Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/JSAC.2005.845406
receiver is often the limiting factor in the physical-layer performance of DS spread-spectrum packet radio communication, and thus the acquisition technique can be an important factor in the overall performance of a DS packet radio network. Each transmission in a DS packet radio system includes a preamble of predetermined duration that consists of a segment of the spreading sequence with no data modulation, and the spreading sequence (but not the time of arrival) is known a priori at the receiver. In a practical system design, security and other considerations dictate that the spreading sequence is changed frequently. This can be accomplished by employing a long-period pseudorandom sequence generator and frequently varying the sequence phase used for the start of packet transmissions. In this paper, we consider a DS packet radio system in which the preamble sequence is changed at regular intervals. The same preamble sequence is used for all packet transmissions during the time epoch between changes, and the pseudorandom generator and the sequence phase determining the preamble in any time epoch are known a priori at the receiver. The most practical techniques for acquiring a DS packet transmission utilize a filter matched to the preamble [1], [2], and that approach is considered in this paper. The focus of the paper is a technique to improve the performance of serial, matched-filter acquisition in DS spread-spectrum packet radio communication. The research is motivated by results obtained in [3] which account for the effect of the intermediate-frequency (IF) filter and the subsequent automatic gain-control (AGC) subsystem on acquisition performance in a superheterodyne receiver. The noncoherent, serial acquisition algorithm detects the arrival of the preamble at the receiver based on threshold crossing by a square-law statistic formed from the matched-filter output. The receiver considered in [3] employs a fixed acquisition threshold that is chosen by optimizing the average performance over all preamble sequences produced by the pseudorandom generator. It is shown that serial acquisition with a fixed threshold results in a probability of not acquiring that is a nonmonotonic function of the signal-to-noise ratio (SNR) of the received signal. In this paper, we present a simple method for adapting the acquisition threshold at the receiver that results in substantially better performance than acquisition with a fixed threshold. It has the additional benefit of compensating automatically for variations in the gain characteristics of the AGC subsystem. (Thus, it provides continuous calibration of the acquisition subsystem and more consistent acquisition performance over variations in the temperature, age, and manufacturing processes of the analog
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Fig. 1.
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Block diagram of the receiver.
components of the IF stage in the receiver.) Although the acquisition algorithm is presented in the context of a superheterodyne receiver, the algorithm and the corresponding performance evaluation are applicable to any receiver in which an AGC subsystem occurs before chip-matched filtering of the DS signal. The paper is organized as follows. The communication system is described in Section II. The test statistics generated in serial, matched-filter acquisition are characterized in Section III, and the performance with a fixed acquisition threshold is examined in detail in Section IV. This examination motivates the design of the adaptive-threshold technique, which is described in Section V. The performance with the optimal fixed acquisition threshold and with the adaptive threshold is compared in Section VI. Conclusions are given in Section VII.
randomly selected from among all sequences of quaternary chips with equal probability. Each is a random variable uniformly distributed over the values in . This random-sequence model [5] provides a good approximation to the use of a long-period pseudonoise generator that is used to select a different preamble sequence for each time epoch. Without loss of generality, the transmitted signal is assumed to arrive unattenuated and undelayed at the receiver over an additive white Gaussian noise channel. Thus the received signal is (4) where (5)
II. SYSTEM DESCRIPTION The complex-valued, baseband-equivalent, white Gaussian has two-sided power spectral density . noise voltage
A. Transmitted Signal and Received Signal Each packet transmission consists of a preamble of quaternary chips followed by the packet’s data content. The spreading sequence used during any time epoch is known a priori at the receiver. The transmitted signal is given by (1) where (2) is the carrier frequency at the transmitter, is the power in the is the rectangular transmitted signal during the preamble, pulse with height 1 over , and is the packet’s data content of duration chips. The quaternary phase-shift-keyed (QPSK) spreading waveform is given by
where represents a complex-valued spreading sequence is time limited to and [4]. The chip waveform (3) The spreading sequence employed for transmission of a packet during a given time epoch is modeled as
B. Receiver Architecture Since the focus of this paper is not on the down-conversion of the received signal from the radio frequency to IF, the signal given by (4) is taken to be the received signal after down-conversion to IF. The receiver is modeled as an AWGN source, a bandpass IF filter, and an AGC subsystem followed by an acquisition stage as shown in Fig. 1. The impulse response of the IF filter is given by
where is the baseband-equivalent impulse response of the filter. A well designed AGC subsystem achieves minimal variation in its average output power over a wide dynamic range for the average input power. It responds quickly to a step change in the received signal power, yet its output power is relatively insensitive to variation in the signal envelope over the duration of a chip and to the random variations of the band-limited noise at its input. In this paper, we approximate such a design by considering an AGC subsystem that responds instantly to the arrival of the beginning of the packet transmission’s preamble. Square-law detection is employed in its gain-control feedback loop [6] (though the qualitative observations in the paper still hold if envelope-detection feedback is considered instead).
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
We approximate the gain of the AGC subsystem prior to the arrival of the transmission as a constant that is inversely proportional to the expected (noise) power at its input. Its gain after the arrival of the transmission is constant and inversely proportional to the expected power at its input after the IF filter transients have settled, where the expectation is with respect to the random noise process, the random preamble sequence, and a random chip offset. An expression is developed in [3] that characterizes the gain under this approximation. It is shown that the is given by AGC output
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(9)
is verified, a locally generated copy of the spreading waveform for data demodulation is synchronized to the delay and the receiver enters the data-detection mode. If verification fails, however, the receiver returns to the acquisition mode. Packet acquisition thus occurs if the receiver is in the acquisiis generated. tion mode and a hit occurs when the statistic is genBut if the receiver is in the acquisition mode when erated and the test does not result in a hit, a miss occurs and the packet is not acquired. On the other hand, if the detector declares , a false alarm occurs. If the a hit for a test statistic other than receiver is in verification mode due to a false alarm at the time is generated, then acquisition fails due to the false alarm. The verification interval is the amount of time required for the receiver to determine that a false alarm has occurred and return to the acquisition mode, and in this paper, it is taken to be a con. Hence, a packet is not acquired if either a miss occurs stant . In the or a false alarm occurs for , remainder of this paper, the term “probability of miss” refers to the probability that a miss occurs if the receiver is in acquisition mode at the time the end of the preamble signal is received. The term “probability of false alarm” refers to the probability that a false alarm results in failed acquisition of the packet. The filters for the acquisition stage that are considered in the analysis are those having the form
(10)
(13)
where is the Fourier transform of the chip waveform and is the baseband equivalent of the IF filter’s frequency response. The parameter serves as a key measure of the ratio of the IF filter’s bandwidth to the bandwidth of the DS signal.
is the spreading sequence for the acquiwhere sition preamble. Furthermore, the impulse response of the baseand the function that are conband-equivalent IF filter sidered satisfy the joint constraint
(6) where (7) The parameters in (7) are given by (8)
C. Acquisition Algorithm
for all
(14)
The acquisition stage employs noncoherent square-law combining of the baseband filter outputs to form the test statistics. The inphase and quadrature branches of the receiver each conand , as shown in Fig. 1. tain two filters, denoted by The outputs in each branch at time are summed to form
with the IF filter’s Under this constraint, the convolution of baseband-equivalent impulse response is matched to the acquisition preamble.
(11) (12)
In this section, the test statistics are characterized for both a and for a random given preamble sequence preamble sequence. It follows from (4)–(12) that conditioned on the preamble sequence
where and . The for receiver samples the outputs of the filters at time . (The each integer , and it forms the test statistic assumption of chip-level synchronism at the receiver results in an accurate approximation to the acquisition performance with over-sampling and an arbitrary sample-timing error [7].) Thus, , the statistics if a packet’s preamble is received starting at correspond to the period before the packet’s arrival, correspond to the period during reception of corresponds to the reception of the preamble signal, and the full preamble signal. The test statistics are utilized sequentially as they are formed is if the receiver is in the acquisition mode, and the statistic , and compared with a threshold . A hit is declared if the receiver then enters the verification mode in which it determines if synchronization has been achieved. If synchronization
III. CHARACTERIZATION OF THE TEST STATISTICS
(15)
where is the aperiodic autocorrelation function of the sequence given by (16) and SNR is the preamble signal-to-noise ratio given by
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(The preamble signal-to-noise ratio is referred to simply as the signal-to-noise ratio in the remainder of the paper. Note that it is much greater than the data-channel-symbol SNR for a given received signal power, since the number of chips in the preamble is much greater than the number of chips per data channel symbol in any practical packet radio system.) Also ,
and lated. The
(17) are mutually uncorrecollection
of
uncorrelated random variables are jointly Gaussian and, therefore, independent. Hence, the test statistics are independent chi-square random variables with two degrees of freedom. The test statistics are central chi-square random variables, are noncentral chi-square and the test statistics random variables in general. Suppose instead that the preamble sequence is selected randomly from among all possible sequences. It is shown in [3] that for all are mutually uncorrelated the set of statistics random variables with (18) where the expectation is with respect to a random preamble sequence. It is also shown that
Fig. 2. Fixed-threshold acquisition performance with ideal preamble sequence of length 26.
IV. PERFORMANCE OF FIXED-THRESHOLD ACQUISITION In this section, we examine in detail the performance of serial, matched-filter packet acquisition with a fixed acquisition threshold. A key characteristic of acquisition performance is revealed that is undesirable with respect to performance robustness in channels of a priori unknown signal quality. Consideration of the underlying cause leads naturally to the adaptivethreshold technique introduced in the next section. Consider a DS packet radio system employing a fixed preamble sequence. Suppose that a fixed acquisition threshold is for all ). It follows from the results employed (so that of the previous section that the probability of not acquiring can be expressed as
. (19) The joint distribution function of the collection of uncorrelated random variables can be approximated for a random preamble sequence by treating the random variables as jointly Gaussian, and therefore independent, with first and second moments given by (18) and (19), respectively. Under this approximation, the test statisare central chi-square random tics is a noncentral chi-square variables and the test statistic random variable, each with two degrees of freedom [3]. It is shown in [3, eqs. (32)–(34)] that the Gaussian approximation results in a simple, closed-form expression for the probability of not acquiring (averaged over all preamble sequences) with a fixed acquisition threshold. We have shown through simulation that the Gaussian approximation leads to extremely accurate results for the probability of not acquiring with a fixed threshold for the preamble sequence lengths and verification intervals of practical interest. In this paper, however, we employ the Gaussian approximation only to obtain some qualitative insights in the next section.
(20) is given by (15), where from (17), and is the Marcum Q-function [8]. [In pracfor this system by Monte Carlo tice, we choose to evaluate simulation rather than by using (20).] Suppose the fixed preamble sequence is an (unrealizable) ideal quaternary sequence, that is, a preamble sequence with a discrete aperiodic autocorrelation function for which each sidelode is equal to zero. Suppose, for example, that the system chips, the verification has an ideal preamble of length chips, the chip waveform and interval has a duration of , and the the response of the receiver’s IF filter result in . The acquisition threshold is given by acquisition performance of the system is shown in Fig. 2. As the
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Fig. 3.
Fixed-threshold acquisition performance with preamble sequence A.
SNR increases, the statistic corresponding to correct synchroincreases stochastically so that the probability nization of miss decreases monotonically and approaches zero asymptotically. Furthermore, each of the statistics decreases stochastically toward a limiting distribution as the SNR increases, due to the change in the AGC attenuation upon the arrival of the beginning of the packet transmission. Thus the probability of false alarm decreases toward a nonzero limiting value as the SNR increases, though the dependence on the SNR is slight. Consequently, acquisition performance with the ideal preamble sequence closely approximates the constant-false-alarm-rate performance predicted in well-known acquisition models that don’t account for the effect of the AGC subsystem. Similar behavior is observed with fixed-threshold acquisition and an ideal preamble sequence of any length. Suppose instead that the fixed preamble sequence is a realizable quaternary sequence with very good autocorrelation properties. This is illustrated by a system that uses the preamble sequence of length 26 in the Appendix, denoted sequence A, which has a maximum aperiodic autocorrelation sidelobe magnitude of only 3.16. The parameters of the system are the same as those in the previous example, except that the fixed . The acquisition threshold is given by acquisition performance of the system is shown in Fig. 3. The probability of miss depends on the SNR in the same manner as in the previous example. But each of the quadrature statistics exhibits an increasing mean that determine and decreasing variance as the SNR increases. The resulting probability of false alarm is a nonmonotonic function of the SNR in this example, though the region in which it is increasing is indiscernible from the figure. Consequently, the probability of not acquiring is also a nonmonotonic function of the SNR. Since the probability of false alarm exhibits only minimal dependence on the SNR, however, the acquisition performance once again approximates that of a constant false-alarm-rate system if preamble sequence A is used. The nonmonotonicity can be much more pronounced if the fixed preamble sequence
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Fig. 4. Average fixed-threshold acquisition performance over all preamble sequences of length 26.
has larger aperiodic autocorrelation sidelodes, however, in which case the probability of not acquiring is not accurately approximated by a constant false-alarm rate. In tactical packet radio communication, security concerns will often dictate that the preamble sequence is changed at regular intervals among a large set of candidate sequences. Thus, an important measure of acquisition performance is the average performance over a large set of preamble sequences. If the number of sequences is sufficiently large, the average performance can be closely approximated by evaluating the expected performance with a randomly generated preamble sequence. The expected acquisition performance for a random preamble for any sequence is evaluated in [3]. It is shown that combination of the DS chip waveform and the IF filter. (Indeed, except in the unrealizable case of a Nyquist-pulse chip waveform and an Nyquist-bandwidth ideal bandpass IF filter.) Thus, it follows from (18) and (19) that under the Gaussian inapproximation discussed in Section III, crease stochastically with the SNR. Consequently, the average probability of false alarm is a strictly increasing function of the SNR under the Gaussian approximation if the acquisition threshold is fixed, and the average probability of not acquiring is a nonmonotonic function of the SNR. These conclusions based on the Gaussian approximation are borne out by Monte Carlo simulations that do not employ the approximation. For example, consider the acquisition performance and a expected for a system with a preamble of random preamble sequence, which is illustrated in Fig. 4. The parameters of the system are the same as those in the previous example, except that the fixed acquisition threshold is given by . The example clearly illustrates the nonmonotonicity of the acquisition performance as a function of the SNR. The nonmonotonicity is even more pronounced if the system employs a longer preamble sequence. This is illustrated in Fig. 5
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this undesirable phenomenon. It is shown in the next section that the modification leads to improved acquisition performance. The modified serial acquisition algorithm employs an acquisition threshold that is adapted for each test statistic. The acquiis compared sition threshold against which the test statistic is given by (21) where is a constant and is the scaling factor employed at time instant to scale the threshold . (Normalization by the is used for convenience in the analysis and constant does not entail any loss of generality in the result.) The scaling is the average of the previous factor employed at time test statistics given by (22) Fig. 5. Average fixed-threshold acquisition performance over all preamble sequences of length 400.
in which the average acquisition performance is shown for a system with a preamble of 400 chips. The system’s verification chips, the chip waveform interval has a duration of , and the fixed acquisition and the IF filter result in . threshold is given by Thus, DS packet acquisition exhibits a surprising average performance characteristic if a fixed acquisition threshold is used with a randomly selected preamble sequence. It is shown in Section VI that this characteristic limits the ability to achieve robust acquisition performance in a channel with a SNR that is unknown a priori. The severity of the nonmonotonicity in acquisition performance is greater for larger values of [3] so that the phenomenon is most pronounced if the receiver uses an (inexpensive) IF filter with a bandwidth several-fold greater than the bandwidth of the desired signal. The severity of the nonmonotonicity is mitigated to some extent in any actual receiver by the saturation effects in the radio-frequency and AGC subsystems. For any practical radio design, however, the dynamic range of the receiver is large enough for the phenomenon to occur. Indeed, a nonmonotonic average probability of not acquiring with increasing SNR has been observed in tests of a prototype DS packet radio system [9]. V. ACQUISITION USING AN ADAPTIVE THRESHOLD It is shown in the previous section that the average probability of not acquiring is a nonmonotonic function of the SNR if the serial acquisition algorithm employs a fixed acquisition threshold with a randomly selected preamble sequence. In particular, the off-peak test statistics generated during reception of the pre) increase amble signal (that is, the statistics stochastically as the SNR increases, and consequently, the probability that one of them results in a false alarm increases with an increase in the SNR. In this section, we introduce a simple modification to the serial acquisition algorithm that mitigates
where is the window size. No knowledge of the distribution function of the test statistic is required by the receiver in order to determine . (Note also that the form of the scaling factor does not depend on the preamble sequence used in any given time epoch.) The scaling factor is intended to compensate for the dependence on the SNR for each off-peak test statistic’s distribution (denoted function. In particular, the mean of the test statistic ) depends on the SNR, and the scaling factor can be viewed as an estimator of . From (18) and (19), it follows that
(23)
where and expectation is with respect to the noise process and the random preamble sequence. Since for , is an unbiased estimator it follows from (19) and (23) that if (i.e., for noise-only test statistics). But it has a of negative bias if . are identically disThe independent random variables so that for . By the strong law tributed for with probability one as the window of large numbers, size is increased for , and a large window size results in . For , however, a highly accurate estimate of for increases with increasing window size (and inthe bias in creasing SNR). Thus, for any given set of operating conditions, the optimal window size with respect to estimator accuracy is some finite value. VI. COMPARISON OF THE ACQUISITION TECHNIQUES In this section, we compare the performance that results with a fixed acquisition threshold and the performance that results with the adaptive threshold described in the previous section. Two performance criteria and the corresponding measures of performance are introduced and used in the comparison.
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Both the performance with a fixed preamble sequence and the expected performance with a randomly selected preamble sequence are considered. All performance results are based on Monte Carlo simulations using the exact characterization of the test statistics. (The Gaussian approximation discussed in Section III is not employed.) One natural criterion for acquisition performance is that the acquisition algorithm achieves the best possible worst-case performance over a specified range of AWGN channels, where each channel is characterized by the SNR at the receiver. This criterion is referred to as the min-max criterion. A performance measure that reflects the min-max criterion is , the maximum probability of not acquiring over all AWGN channels for which , where is a predetermined value that characterizes the poorest quality channel for which acquisition performance is of interest. For a given acquisition technique, optimal performance with respect to the min-max criterion is achievable for defined as the smallest possible value of . the specified value of Another natural criterion for acquisition performance is that the acquisition algorithm achieves or exceeds a desired level of performance over the widest possible range of AWGN channels. This criterion is referred to as the max-range criterion. A performance measure that reflects the max-range criterion is , the smallest SNR such that the maximum probability over all AWGN chanof not acquiring does not exceed , where is a predetermined nels for which value that characterizes the poorest acceptable acquisition performance. For a given acquisition technique, optimal performance with respect to the max-range criterion is defined as the achievable for the specified smallest possible value of . value of The length of the preamble is 400 chips in all of the examples, and the chip waveform and response of the receiver’s IF filter . The verification interval is chips. result in chips, which The size of the estimator window is provides optimal performance under these circumstances if the adaptive threshold is used. With the exception of the results illustrated in Fig. 10, the value of used with each acquisition technique is the one that results in min-max optimal perfordB. (Recall that this corresponds to mance for a much lower channel-symbol SNR for the data payload in any packet format of interest.) For this range of optimization, the , while min-max optimal fixed threshold is the min-max optimal value for in (21) used with the adap. For both acquisition techtive threshold is niques, optimization is with respect to the expected performance with a randomly selected preamble sequence. A. Comparison of Performance With a Fixed Preamble Sequence We can gain some insight into the performance of the two acquisition techniques by considering the performance that results for examples of specific preamble sequences. In this subsection, we consider the performance with three fixed preamble sequences, denoted sequences B, C, and D, each of which is
Fig. 6.
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Acquisition performance with preamble sequence B.
given in the Appendix. The three sequences illustrate the range of acquisition performance that may be observed among the preamble sequences generated by a long-period pseudonoise generator. The probability of not acquiring is shown as a function of the SNR in Fig. 6, for both acquisition techniques with preamble sequence B. The probability of not acquiring with the fixed threshold is low if the SNR is high, and the performance over the . (Recall that the range of interest is poorest if threshold is chosen to optimize the average performance over all preamble sequences rather than the performance for this particular preamble sequence.) In contrast, the worst-case probability and as SNR of not acquiring occurs at both approaches infinity if the adaptive threshold is used. Moreover, the worst-case probability of not acquiring for is more than an order of magnitude lower with the adaptive threshold than with the fixed threshold. The performance for both acquisition techniques is shown in Fig. 7 for preamble sequence C. The worst-case probability of not acquiring over the range of interest occurs at an intermediate value of SNR if the fixed threshold is used. Similarly, the results for preamble sequence D are shown in Fig. 8. If the fixed threshold is used with this preamble sequence, the worstcase probability of not acquiring occurs as SNR approaches infinity. For both preamble sequence C and preamble sequence D, the worst-case probability of not acquiring occurs at if the adaptive threshold is used. With either preamble sequence, the worst-case probability of not acquiring is more than an order of magnitude lower with the adaptive threshold than with the fixed threshold. Not only does the adaptive threshold result in much better worst-case performance than the optimal fixed threshold for each preamble sequence, it results in much more consistent performance for different preamble sequences and over all values of the SNR. If the optimal fixed threshold is used, the
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Fig. 7. Acquisition performance with preamble sequence C.
Fig. 9. Acquisition performance averaged over all sequences with SNR
=
Fig. 10.
=
16 dB.
Fig. 8. Acquisition performance with preamble sequence D. Acquisition performance averaged over all sequences with P .
2:15 2 10
maximum probability of not acquiring is 0.028, 0.1, and 0.75 for sequences B, C, and D, respectively, whereas the corresponding value with the adaptive threshold is 0.0018 for each sequence. Furthermore, the variation in the probability of not acquiring over the range of SNRs of interest is several orders of magnitude for a given sequence if the optimal fixed threshold is used, but the variation is no more than fourfold over the same range for any of the three sequences if the adaptive threshold is used. B. Comparison of Average Performance Over All Preamble Sequences The average acquisition performance over all preamble sequences is compared for the two acquisition techniques using both the min-max criterion and the max-range criterion. The
comparison using the min-max criterion is shown in Fig. 9 for dB. The worst-case probability of not acquiring dB is 0.0018 if the adaptive threshold is used, over while the worst-case probability of not acquiring with the optimal fixed threshold is 0.021. Thus, the performance is more than a order of magnitude better with the adaptive threshold. The comparison using the max-range criterion is shown in Fig. 10 in which the performance requirement is that the prob10 ability of not acquiring is no greater than for . The constant is chosen for each acquisition for the technique to achieve the smallest possible value of . With this performance requirement, the specified value of optimal value of in (21) for the adaptive threshold technique is
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
Fig. 11. Acquisition performance with an optimum fixed threshold with errors in the AGC gain.
. If the optimal fixed threshold is used, dB, the performance requirement can be satisfied for but if the adaptive threshold is used, the performance requiredB. ment can be satisfied over the wider range of Thus, the adaptive threshold yields a performance improvement of 1.65 dB over the optimal fixed threshold with respect to this performance criterion. The difference in the performance of the two acquisition techniques is even greater if the value of is larger (e.g., if the IF filter bandwidth is larger). C. Performance With Variation in IF Component Characteristics The performance of any communication device is susceptible to variation in the characteristics of its analog components that occurs as the components age and the operating temperature of the device changes. Furthermore, differences in analog components due to variations in manufacturing processes can result in differing performance among devices based on the same design. The performance of serial, matched-filter acquisition can be affected significantly by variations in the components of the IF portion of the superheterodyne receiver. The performance is sensitive to variations in the nominal steady-state output voltage of the AGC subsystem or the response of the IF filter if a fixed threshold is used. But the use of the adaptive threshold effectively eliminates performance sensitivity to first of these impairments, and its superior performance is preserved in the presence of the second impairment. The effect of the steady-state AGC output voltage on the acquisition performance is illustrated in Fig. 11. The performance averaged over all preamble sequences is shown for a receiver using a fixed threshold that has been optimized for the nominal steady-state output voltage of the AGC subsystem dB using the min-max criterion. The and
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average probability of not acquiring is shown as a function of the SNR for three circumstances: the steady-state AGC output voltage is equal to its nominal value, the voltage is 15% greater than its nominal value, and the voltage is 15% less than its nominal value. If the AGC output voltage is at the nominal level for which the threshold is chosen, the worst-case dB is 0.021. But probability of not acquiring for if the AGC output voltage is 15% higher than the nominal value, the maximum probability of not acquiring increases to 0.068, and if it is 15% lower than the nominal value, the worst-case probability of not acquiring increases to 0.079. In contrast, if the adaptive threshold is employed, variations in the steady-state output voltage of the AGC subsystem are compensated for automatically, and the acquisition performance is not affected by deviations of the steady-state voltage from its nominal value. Similarly, the acquisition performance is highly sensitive to the bandwidth of the IF filter if a fixed threshold is used, and moderate deviations from the nominal filter bandwidth can result in substantial degradation in acquisition performance. (Differences in the bandwidth of the IF filter are reflected in differences in the value of the parameter introduced earlier.) Acquisition performance with the adaptive threshold is also affected by deviations from the nominal filter bandwidth. For any given deviation from the nominal bandwidth, however, acquisition with the adaptive threshold results in much better performance than fixed-threshold acquisition. VII. CONCLUSION In this paper, we present an algorithm to adapt the threshold used with serial, matched-filter acquisition in DS spread-spectrum packet radio communications. The algorithm adapts the threshold for each test statistic based on a windowed average of the recent past test statistics. It is shown that the acquisition performance that results with the adaptive threshold is better than the performance that results if the optimal fixed threshold is used. The adaptive threshold results in better performance than a fixed threshold if a fixed preamble sequence is employed, it results in performance that is more consistent across different preamble sequences and over a range of values of the SNR, and it results in better average performance over the set of all possible preamble sequences. In addition, the adaptive threshold is shown to result in more robust acquisition performance in the presence of analog components with parameters that differ from their nominal values. APPENDIX The individual quaternary preamble sequences considered in the paper are given below. The binary representation of each in-phase and quadrature sequence is padded with trailing zeros to obtain its unique hexadecimal representation shown below. Sequence A has a length of 26 quaternary symbols. Sequences B, C, and D have a length of 400 quaternary symbols, as shown on the next page.
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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 5, MAY 2005
Real part of sequence A
Imaginary part of sequence A
Real part of sequence B
Imaginary part of sequence B
Real part of sequence C
Imaginary part of sequence C
Real part of sequence D
Imaginary part of sequence D
SWAMINATHAN AND NONEAKER: TECHNIQUE TO IMPROVE THE PERFORMANCE OF SERIAL, MATCHED-FILTER ACQUISITION
REFERENCES [1] C. R. Cahn, “Spread spectrum applications and state-of-the-art equipments,” in Proc. AGARD-NATO Lecture Series, Spread-Spectrum Commun., Bolkesjø, Norway, May 1973, p. 5. [2] A. Polydoros and C. L. Weber, “A unified approach to serial search spread-spectrum code acquisition—Part II: A matched filter receiver,” IEEE Trans. Commun., vol. COM-32, no. 5, pp. 550–560, May 1984. [3] D. L. Noneaker, A. R. Raghavan, and C. W. Baum, “The effect of automatic gain control on serial, matched-filter acquisition in direct-sequence packet radio communications,” IEEE Trans. Veh. Technol., vol. 50, no. 4, pp. 1140–1150, Jul. 2001. [4] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, no. 5, pp. 593–619, May 1980. [5] K. K. Chawla and D. V. Sarwate, “Acquisition of PN sequences in chip synchronous DS/SS systems using a random sequence model and the SPRT,” IEEE Trans. Commun., vol. 42, no. 6, pp. 2325–2334, Jun. 1994. [6] J. R. Smith, Modern Communication Circuits, 2nd ed. New York: McGraw-Hill, 1998. [7] D. L. Noneaker, “The performance of serial, matched-filter acquisition in direct-sequence packet radio communications,” in Proc. IEEE Military Commun. Conf., McLean, VA, Oct. 2001, pp. 1045–1049. [8] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [9] J. R. McChesney, ITT Aerospace/Communications Division, private correspondence.
Arvind Swaminathan (S’99) was born in Chennai, India, on February 4, 1978. He received the B.Tech. degree in electronics and communications engineering from the Regional Engineering College, Calicut, India, in 1999 and the M.S. degree in electrical engineering from Clemson University, Clemson, SC, in 2002. For his M.S. thesis, he worked on techniques to improve the performance of serial acquisition in direct-sequence spread-spectrum packet radio networks. He is currently working towards the Ph.D. degree in electrical engineering at Clemson University, where he is a Research Assistant in the Wireless Communications Laboratory. He is working on protocol design for ad hoc networks with directional antennas for his Ph.D. dissertation. Mr. Swaminathan is a member of Alpha Epsilon Lambda.
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Daniel L. Noneaker (SM’93) was born in Montgomery, AL, on December 10, 1957. He received the B.S. degree (high honors) in mathematics from Auburn University, Auburn, AL, in 1977, the M.S. degree in mathematics from Emory University, Atlanta, GA, in 1979, the M.S. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1984, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Urbana, in 1993. He has industrial experience in both hardware and software design for communication systems. From 1979 to 1982, he was with Sperry-Univac, Salt Lake City, UT, and from 1984 to 1988, he was with the Motorola Government Electronics Group, Scottsdale, AZ. He was a Research Assistant in the Coordinated Science Laboratory, University of Illinois at Urbana–Champaign from 1988 to 1993. Since August 1993, he has been with the Holcombe Department of Electrical and Computer Engineering, Clemson University, Clemson, SC, where he currently holds the position of Associate Professor. He has published numerous papers on the design and analysis of multiple-access systems for both cellular communication and ad hoc packet radio networks. He is engaged in research on wireless communication for both military and commercial applications with emphases on spread-spectrum communications, error-control coding for fading channels, and protocols for mobile radio networks.
