Torrieri D. Principles of Spread-Spectrum Communication Systems
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CHAPTER 2.
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Figure 2.23: Performance against pulsed interference for convolutional code
with white-noise metric, K = 7, and
metric, calculations similar to the preceding ones yield
This equation and (2-157) give the upper bound on for the white-noise met-
ric. Figure 2.23 illustrates the upper bound on versus for K = 7,
and several values of The figure demonstrates
the vulnerability of soft-decision decoding with the white-noise metric to short
high-power pulses if interference power is conserved. The high values of
for are due to the domination of the metric by a few degraded symbol
metrics.
Consider a coherent PSK demodulator that erases its output and, hence,
a received symbol whenever an interference pulse occurs. The presence of the
pulse might be detected by examining a sequence of the demodulator outputs
and determining which ones have inordinately large magnitudes compared to the
others. Alternatively, the demodulator might decide that a pulse has occurred
if an output has a magnitude that exceeds a known upper bound for the desired
signal. Consider an ideal demodulator that unerringly detects the pulses and
erases the corresponding received symbols. Following the deinterleaving of the
demodulated symbols, the decoder processes symbols that have a probability
of being erased equal to The unerased symbols are decoded by using the
2.5.
PULSED INTERFERENCE
99
Figure 2.24: Performance against pulsed interference for convolutional code
with erasures, K = 7,
and
white-noise metric.
The
erasing
of
symbols
causes
two
sequences
that
differ
in symbols to be compared on the basis of symbols where As
a result,
The substitution of this equation into (2-157) give the upper bound on for
errors-and-erasures decoding.
The upper bound on is illustrated in Figure 2.24 for K = 7,
20 dB, and several values of In this example, erasures provide no ad-
vantage over the white-noise metric in reducing the required for
if but are increasingly useful as decreases. Consider an ideal de-
modulator that activates erasures only when is small enough that the erasures
are more effective than the white-noise metric. When this condition does not
occur, the white-noise metric is used. The upper bound on for this ideal
erasure decoding, worst-case pulsed interference, dB, and several
binary convolutional codes is illustrated in Figure 2.25. The required
at is roughly 2 dB less than for worst-case hard-decision decoding.
However, a practical demodulator will sometimes erroneously make erasures or
fail to erase, and its performance advantage may be much more modest.
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CHAPTER 2.
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Figure 2.25: Worst-case performance against pulsed interference for convo-
lutional codes with ideal erasure decoding, constraint length K, rate and
2.6
Despreading with Matched Filters
Despreading short spreading sequences with matched filters provides inherent
code synchronization. The spreading waveform for a short sequence may be
expressed as
where is one period of the spreading waveform and T is its period. If the
short spreading sequence has length N, then
where and
Consider a signal that is zero outside the interval [0, T]. A filter is said to
be matched to this signal if the impulse response of the filter is
2.6.
DESPREADING WITH MATCHED FILTERS
101
When
is applied to a filter matched to it, the filter output is
The aperiodic autocorrelation of a deterministic signal with finite energy is de-
fined as
Therefore, the response of a matched filter to the matched signal is
If this output is sampled at then the signal energy.
Consider a bandpass matched filter that is matched to
where is one period of a spreading waveform and is the desired carrier
frequency. We evaluate the filter response to the received signal corresponding
to a single data symbol:
where is a measure of the unknown arrival time, the polarity of A is de-
termined by the data symbol, and
is the received carrier frequency, which
differs from because of oscillator instabilities and the Doppler shift. The
matched-filter output is
If then substituting (2-173) into (2-174) yields
where
is the phase mismatch and
If
the carrier-frequency error is inconsequential, and
where
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CHAPTER 2.
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In the absence of noise, the matched-filter output is a sinusoidal spike with
a polarity determined by A. Assuming that (2-77) is applicable, the peak
magnitude, which occurs at equals However, if
then (2-175) is not well-approximated by (2-176), and the matched-filter output
is significantly degraded.
The response of the matched filter to the interference plus noise, denoted
by may be expressed as
where
These equations exhibit the spreading of the interference spectrum.
The envelope of the matched-filter output is
Define such that is an integer times If
is su
fficiently
large that then (2-176) and (2-178) imply that if is sampled
at
where If
then (2-181) implies that
A comparison of this equation with (2-182) indicates that there is relatively
little degradation in using an envelope detector after the matched filter rather
than directly detecting the peak magnitude of the matched-filter output, which
is much more difficult.
Figure 2.26 illustrates the basic form of a surface-acoustic-wave (SAW)
transversal filter, which is a passive matched filter that essentially stores a
replica of the underlying spreading sequence and waits for the received se-
quence to align itself with the replica. The SAW delay line consists primarily
2.6.
DESPREADING WITH MATCHED FILTERS
103
Figure 2.26: Matched filter that uses a SAW transversal filter.
of a piezoelectric substrate, which serves as the acoustic propagation medium,
and interdigital transducers, which serve as the taps and the input transducer.
The transversal filter is matched to one period of the spreading waveform, the
propagation delay between taps is and is an integer. The chip matched
filter following the summer is matched to It is easily veri-
fied that the impulse response of the transversal filter is that of a filter matched
to
A convolver is an active matched filter that produces the convolution of
the received signal with a local reference [8]. When used as a direct-sequence
matched filter, a convolver uses a recirculating, time-reversed replica of the
spreading waveform as a reference waveform. In a SAW elastic convolver, which
is depicted in Figure 2.27, the received signal and the reference are applied to
interdigital transducers that generate acoustic waves at opposite ends of the
substrate. The acoustic waves travel in opposite directions with speed and
Figure 2.27: SAW elastic convolver.
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CHAPTER 2.
DIRECT-SEQUENCE SYSTEMS
the acoustic terminations suppress reflections. The signal wave is launched at
position and the reference wave at The signal wave travels to the
right in the substrate and has the form
where is the modulation at position The reference wave travels to
the left and has the form
where is the modulation at position Both and are assumed
to have bandwidths much smaller than The beam compressors, which con-
sist of thin metallic strips, focus the acoustic energy to increase the convolver’s
efficiency. When the acoustic waves overlap beneath the central electrode, a
nonlinear piezoelectric effect causes a surface charge distribution that is spa-
tially integrated by the electrode. The primary component of the convolver
output is proportional to
Substituting (2-185) and (2-186) into (2-187) and using trigonometry, we find
that is the sum of a number of terms, some of which are negligible if
Others are slowly varying and are easily blocked by a filter. The
most useful component of the convolver output is
where Changing variables, we find that the amplitude
of the output is
where the factor results from the counterpropagation of the two acoustic
waves.
Suppose that an acquisition pulse is a single period of the spreading wave-
form. Then and where is the uncertainty
in the arrival time of an acquisition pulse relative to the launching of the ref-
erence signal at The periodicity of allows the time origin to be
selected so that Equations (2-189) and (2-167) and a change of
variables yield
2.6.
DESPREADING WITH MATCHED FILTERS
105
Since unless unless For
every positive integer let
Only one term in (2-190) can be nonzero when and
The maximum possible magnitude of is produced if and
that is, if
Since (2-191) indicates that there is some that satisfies
(2-193) if
Thus, if L is large enough, then there is some such that and
the envelope of the convolver output at has the maximum possible
magnitude If and only one peak value occurs in
response to the single received pulse.
As an example, let and The chips propagating
in the convolver for three separate time instants and are
illustrated in Figure 2.28. The top diagrams refer to the counterpropagating
periodic reference signal, whereas the bottom diagrams refer to the single re-
ceived pulse of four chips. The chips are numbered consecutively. The received
pulse is completely contained within the convolver during The
maximum magnitude of the output occurs at time which is the instant
of perfect alignment of the reference signal and the received chips.
Figure 2.28: Chip configurations within convolver at time instants
and when and
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CHAPTER 2
.
DIRECT-SEQUENCE SYSTEMS
Figure 2.29: Direct-sequence system with binary code-shift keying: (a) trans-
mitter and (b) receiver.
Noncoherent Systems
In a noncoherent direct-sequence system with binary code-shift keying (CSK),
one of two orthogonal spreading sequences is transmitted, as shown in Figure
2.29(a). One sequence represents the symbol 1, and the other represents the
symbol 0. The receiver uses two matched filters, each matched to a different
sequence and followed by an envelope detector, as shown in Figure 2.29(b). In
the absence of noise and interference, each sequence causes only one envelope
detector to produce a significant output. The data is recovered by comparing
the two detector outputs every symbol period.
Since each of the two orthogonal sequences has a period equal to the symbol
duration, symbol or bit synchronization is identical to code synchronization.
The symbol synchronizer, which provides timing pulses to the comparator or
decision device, must lock onto the autocorrelation spikes appearing in the
envelope-detector outputs. Ideally, these spikes have a triangular shape. The
symbol synchronizer must be impervious to the autocorrelation sidelobe peaks
and any cross-correlation peaks. A simple implementation with a single thresh-
2.6.
DESPREADING WITH MATCHED FILTERS
107
old detector would result in an unacceptable number of false alarms, premature
detections, or missed detections when the received signal amplitude is unknown
and has a wide dynamic range. Limiting or automatic gain control only exacer-
bates the problem when the signal power level is below that of the interference
plus noise. More than one threshold detector with precedence given to the
highest threshold crossed will improve the accuracy of the decision timing or
sampling instants produced by the symbol synchronizer [9]. Another approach
is to use peak detection based on a differentiator and a zero-crossing detector.
Finally, a phase-locked or feedback loop of some type could be used in the
symbol synchronizer. A preamble may be transmitted to initiate accurate syn-
chronization so that symbols are not incorrectly detected while synchronization
is being established.
Consider the detection of a symbol represented by (2-173), where is
the CSK waveform to which filter 1 is matched. Assuming perfect symbol
synchronization, the channel symbol is received during the interval
From (2-176) to (2-181) with and we find that the output of
envelope detector 1 at is
where
Similarly, if filter 2 is matched to sequence then the output of envelope
detector 2 at is
where
and the response to the transmitted symbol at is zero because of the
orthogonality of the sequences.
Suppose that the interference plus noise is modeled as zero-mean,
Gaussian interference, and the spreading sequences are modeled as determin-
istic and orthogonal. Then and If
is assumed to be wideband enough that its autocorrelation is approximated by
(2-87), then straightforward calculations using and the orthogonal-
ity of and indicate that and are all uncorrelated with
each other. The jointly Gaussian character of the random variables then implies
that they are statistically independent of each other, and hence and are
independent. Analogous results can be obtained when the transmitted symbol