Torrieri D. Principles of Spread-Spectrum Communication Systems

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Figure 4.14: NMAT versus for double-dwell systems in presence of fast

Rayleigh fading. Step size is Values of and are

optimized.

pulsed Gaussian noise jamming is present. When a rake receiver (Chapter 5 ) is

used, each finger of the receiver must acquire the timing of a separate multipath

signal. Whether matched filtering or a serial search is used, some mechanism

is needed to ensure that each finger acquires a distinct multipath signal [9].

An alternative to acquisition tests of fixed dwell time or number of detector

samples is sequential detection, which uses only the number necessary for a

reliable decision. Thus, some sample sequences may allow a quick decision,

while others may warrant using a large number of samples in the evaluation of a

single phase of the spreading waveform. The sequential probability ratio test [2],

[3] entails the recalculation of the likelihood ratio after each new detector sample

is produced. This ratio is compared with both upper and lower thresholds to

determine if the test is terminated and no more samples need to be extracted.

If the upper threshold is exceeded, the receiver declares acquisition and the

lock mode is entered. If the likelihood ratio drops below the lower threshold,

the test fails, and another code phase is tested. As long as the likelihood ratio

lies between the two thresholds, a decision is postponed and the ratio continues

to be updated. Although the sequential detector is capable of significantly

reducing the mean acquisition time relative to detectors that use a fixed number

of samples, it has a number of practical limitations. Chief among them is

the computational complexity of the calculation of the likelihood ratio or log-

likelihood ratio.

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4.4

4.4.

CODE TRACKING

Code Tracking

Coherent code-tracking loops operate at baseband following the coherent re-

moval of the carrier of the received signal. An impediment to their use is

that the input SNR is usually too low for carrier synchronization prior to code

synchronization and the subsequent despreading of the received signal. Further-

more, coherent loops cannot easily accommodate the effects of data modulation.

Noncoherent loops operate directly on the received signals and are unaffected

by the data modulation.

To motivate the design of the noncoherent loop, one may adapt the statistic

(4-9). If the maximum-likelihood estimate is assumed to be within the interior

of its timing uncertainty region and is a differentiable function of

then the estimate that maximizes may be found by setting

A major problem with this approach is that given by (4-9) is not

differentiable if the chip waveform is rectangular. This problem is circumvented

by using a difference equation as an approximation of the derivative. Thus, for

a positive we set

This equation implies that the solution of (4-95) may be approximately obtained

by a device that finds the such that

To derive an alternative to this equation, we assume that no noise is present,

and that the correct timing offset of the received signal is

Substituting (4-2) with into (4-9) and using trigonometry, we obtain

If is modeled as the spreading waveform for a random binary sequence

and the interval [0, T] includes many chips, then the integral is reasonably

approximated by its expected value, which is proportional to the autocorrelation

where the triangular function is defined by (2-14). Substituting this result into

(4-97), we find that the maximum-likelihood estimate is approximately obtained

by a device that finds the such that

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Figure 4.15: Delay-locked loop.

The noncoherent delay-locked loop [10], which is diagrammed in Figure 4.15,

implements an approximate computation of the difference on the left-hand side

of (4-100) and then continually adjusts so that this difference remains near

zero. The estimate is used to produce the synchronized local spreading sequence

that is used for despreading the received direct-sequence signal. The code gen-

erator produces three sequences, one of which is the reference sequence used

for acquisition and demodulation. The other two sequences are advanced and

delayed, respectively, by relative to the reference sequence. The product

is usually equal to the acquisition step size, and thus usually but

other values are plausible. The advanced and delayed sequences are multiplied

by the received direct-sequence signal in separate branches.

For the received direct-sequence signal (4-2), the signal portion of the upper-

branch mixer output is

where and is the delay of the reference sequence relative to

the received sequence. Although is a function of time because of the loop

dynamics, the time dependence is suppressed for notational convenience. Since

each bandpass filter has a bandwidth on the order of where is the

duration of each symbol, is not significantly distorted by the filtering.

Nearly all spectral components except the slowly varying expected value of

are blocked by the upper-branch bandpass filter. Since

this expected value is the autocorrelation of the spreading sequence, the filter

output is

Any double-frequency component produced by the square-law device is ulti-

mately suppressed by the loop filter and thus is ignored. Since the

4.4.

CODE TRACKING

211

data modulation is removed, and the upper-branch output is

Similarly, the output of the lower branch is

The difference between the outputs of the two branches is the error signal:

Since is an even function, the error signal is proportional to the left-hand

side of (4-100).

The substitution of (4-99) and (2-14) into (4-105) yields

where is the discriminator characteristic or S-curve of the tracking loop.

For

In both cases,

Figure 4.16 illustrates the discriminator characteristic for The

filtered error signal is applied to the voltage-controlled clock. Changes in the

clock frequency cause the reference sequence to converge toward alignment with

the received spreading sequence. When the reference sequence

is delayed relative to the received sequence. As shown in Figure 4.16, is

positive, so the clock rate is increased, and decreases. The figure indicates

that as Similarly, when we find that as

Thus, the delay-locked loop tracks the received code timing once the

acquisition system has finished the coarse alignment.

The discriminator characteristic of code-tracking loops differs from that of

phase-locked loops in that it is nonzero only within a finite range of Outside

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Figure 4.16: Discriminator characteristic of delay-locked loop for

that range, code tracking cannot be sustained, the synchronization system loses

lock, and a reacquisition search is initiated by the lock detector. Tracking

resumes once the acquisition system reduces to within the range for which the

discriminator characteristic is nonzero.

When short spreading sequences are used in a synchronous direct-sequence

network, the reduced randomness in the multiple-access interference (Chapter

6) may cause increased tracking jitter or even an offset in the discriminator

characteristic [11]. For orthogonal sequences, the interference is zero when

synchronization exists, but becomes large when there is a code-phase error in

the local spreading sequence. In the presence of a tracking error, the delay-

locked-loop arm with the larger offset relative to the correct code phase re-

ceives relatively more noise power than the other arm. This disparity reduces

the slope of the discriminator characteristic and, hence, degrades the tracking

performance. Moreover, because of the nonsymmetric character of the cross-

correlations among the spreading sequences, the discriminator characteristic

may be biased in one direction, which will cause a tracking offset.

The noncoherent tau-dither loop, which is depicted in Figure 4.17, is a lower-

complexity alternative to the noncoherent delay-locked loop. The dither gener-

ator produces the dither signal

a square wave that alternates between +1

and –1. This signal controls a switch that alternately passes an advanced or

delayed version of the spreading sequence. In the absence of noise, the output

of the switch can be represented by

where the two factors within brackets are orthogonal functions of time and

alternate between +1 and 0. Only one of the factors is nonzero at any instant.

The received direct-sequence signal is multiplied by filtered, and then

applied to a square-law device. If the bandpass filter has a sufficiently narrow

4.4.

CODE TRACKING

213

Figure 4.17: Tau-dither loop.

bandwidth, then a derivation similar to that of (4-103) indicates that the device

output is

Since and the input to

the loop filter is

which is a rectangular wave if the time variation of is ignored. Since the loop

filter has a narrow bandwidth relative to that of its output is approx-

imately the direct-current component of which is the average value of

Averaging the two terms of (4-112), we obtain the filter output:

The substitution of (4-99) yields the clock input:

where the discriminator characteristic is given by (4-107) to (4-109). Thus, the

tau-dither loop can track the code timing in a manner similar to that of the

delay-locked loop. A detailed analysis indicates that the tau-dither loop pro-

vides less accurate code tracking [2], [3]. However, the tau-dither loop requires

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less hardware than the delay-locked loop and avoids the need to balance the

gains and delays in the two branches of the delay-locked loop.

In the presence of frequency-selective fading, the discriminator characteris-

tics of tracking loops are severely distorted. Much better performance is poten-

tially available from a noncoherent tracking loop with diversity and multipath-

interference cancellation [12], but a large increase in implementation complexity

is required.

4.5

Frequency-Hopping Patterns

The synchronization of the reference frequency-hopping pattern produced by

the receiver synthesizer with the received pattern may be facilitated by pre-

cision clocks in both the transmitter and the receiver, feedback signals from

the receiver to the transmitter, or transmitted pilot signals. However, in most

applications, it is necessary or desirable for the receiver to be capable of obtain-

ing synchronization by processing the received signal. During acquisition, the

reference pattern is synchronized with the received pattern to within a fraction

of a hop duration. The tracking system further reduces the synchronization er-

ror, or at least maintains it within certain bounds. For communication systems

that require a strong capability to reject interference, matched-filter acquisition

and serial-search acquisition are the most effective techniques. The matched

filter provides rapid acquisition of short frequency-hopping patterns, but re-

quires the simultaneous synthesis of multiple frequencies. The matched filter

may also be used in the configuration of Figure 4.2 to detect short patterns

embedded in much longer frequency-hopping patterns. Such a detection can be

used to initialize or supplement serial-search acquisition, which is more reliable

and accommodates long patterns.

Matched-Filter Acquisition

Figure 4.18 shows a programmable matched-filter acquisition system that pro-

vides substantial protection against interference [13]. It is assumed that a single

frequency channel is used during each hop interval that occurs during acqui-

sition. One or more programmable frequency synthesizers produce tones at

frequencies which are offset by a constant frequency from the

consecutive frequencies of the hopping pattern for code acquisition. Each tone

multiplies the received frequency-hopping signal and the result is filtered so that

most of the received energy is blocked, except the energy in a frequency-hopping

pulse at a specific frequency. The threshold detector of branch produces

if its threshold is exceeded, which ideally occurs only if the received sig-

nal hops to a specific frequency. Otherwise, the threshold detector produces

The use of binary detector outputs prevents the system from being

overwhelmed by a few strong interference signals. Input of the compara-

tor is the number of frequencies in the hopping pattern that were received in

4.5.

FREQUENCY-HOPPING PATTERNS

215

Figure 4.18: Matched-filter acquisition system with protection against interfer-

ence.

succession. This discrete-valued, continuous-time function is

where is the hop duration. These waveforms are illustrated in Figure 4.19(a)

for N = 8. The input to the threshold generator is

Acquisition is declared when where is an adaptive threshold

that is a function of An effective choice is

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Figure 4.19: Ideal acquisition system waveforms: (a) formation of when

N = 8, and (b) comparison of and

where is a positive integer.

In the absence of noise and interference, and during

the hop interval in which as illustrated in Figure 4.19(b). If

of the N frequency channels monitored by the matched filter receive strong,

continuous interference, then and during this hop interval

if and During other intervals,

but Therefore, and the matched filter does not declare

acquisition. False alarms are prevented because provides an estimate of

the number of frequency channels with continuous interference.

When acquisition tone is received, the signal in branch of the matched

filter

where is the intermediate frequency, the first term is the desired signal with

average power S, the second term represents tone interference with average

power I, is zero-mean, stationary Gaussian noise and interference, and

is the phase shift of the tone interference relative to the desired signal. The

power in is

where is power of the thermal noise and is the power of the statistically

independent noise interference.

Bandpass filters are used instead of filters matched to the acquisition tones

because the appropriate sampling times are unknown. The passbands of the

bandpass filters in the branches are assumed to be spectrally disjoint so that

tone interference entering one branch has negligible effect on the other branches,

and the filter outputs are statistically independent of each other. To prove the

statistical independence of the noise, let and denote the autocorre-

lation and power spectral density, respectively, of the stationary Gaussian noise

4.5.

FREQUENCY-HOPPING PATTERNS

217

in the received signal. Let and denote the impulse responses and

and the transfer functions of two bandpass filters. Since the same

Gaussian noise process enters both filters, their outputs are jointly Gaussian.

The cross-covariance of the jointly Gaussian, zero-mean filter outputs is

where all the integrals extend over Thus, C = 0, if and

are spectrally disjoint. If the noise is white and, hence, is a constant, then

C = 0 if and are orthogonal. When C = 0 for all pairs of band-

pass filters, the threshold-detector outputs in the N branches are statistically

independent.

Suppose that noise interference is present in a branch, but that tone in-

terference is absent so that I = 0. The stationary Gaussian noise has the

representation (Appendix C.2)

where and are zero-mean Gaussian processes with noise powers equal

to In practice, the matched filter of Figure 4.18 would operate in contin-

uous time so that acquisition might be declared at any moment. However, for

analytical simplicity, the detection and false-alarm probabilities are calculated

under the assumption that there is one sample taken per hop dwell time. From

(4-119) with I = 0 and (4-121), it follows that

where

Since and are statistically independent (Appendix C.2), the joint

probability density function of and at any specific time is

Let R and be implicitly defined by and The joint

density of R and is