Transcript 投影片 1

Chapter 2
Introduction to spread-spectrum
communications
Part II
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• As discussed in Chapter 1 Part I, a spread spectrum modulation
produces a transmitted spectrum much wider than the minimum
bandwidth required. There are many ways to generate spread
spectrum signals.
• We are going to introduce some of the most common spread
spectrum techniques such as direct sequence (DS) and frequency
hop (FH).
• Of course, one can also mix these spread spectrum techniques to
form hybrids which have the advantages of different techniques.
• Spread spectrum originates from military needs and finds most
applications in hostile communication environments.
• We will start by briefly looking at the advantage of spreading the
spectrum in the presence of a Gaussian jammer as our motivation
to study spread spectrum communications.
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1.1 Motivation—a jamming analysis
• Consider the transmission of a bit stream
through an AWGN channel.
• We employ BSPK modulation at the carrier frequency
• The channel is also corrupted by an intentional jammer.
• The received signal r(t), in complex envelope representation, is
given by
(1.1)
(1.2)
–
–
–
–
–
–
s(t) is the transmitted signal.
n(t) is the AWGN with power spectrum
j(t) is the jamming signal.
T is the symbol duration.
Tc is the symbol pulse width.
P is the average transmitted power.
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• From Section 1.7, we know that the spectrum of the transmitted
signal s(t) is given by
(2.3)
• If we model the bits as iid random variables and Δas a uniform
random variable on [0; T).
• By examining the spectrum of the transmitted signal, a
reasonable jamming strategy is to put all the jamming power PJ
into the band coincides with the main lobe of the signal spectrum,
i.e., from 2π/Tc to -2π/Tc rad/sec.
• For simplicity, we assume that j(t) is a zero-mean WSS Gaussian
random process with power spectrum
otherwise.
• Moreover, n(t) and j(t) are independent.
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• Neglecting the jamming signal, the ML receiver is the matched
filter receiver developed in Section 1.2.
• We redraw the matched filter receiver in Figure 2.1 here for
convenience.
Figure 1.1: Matched filter receiver for BPSK data with jammer
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• Let us consider the performance of the matched filter when the
jamming signal is present.
• Conditioning on Δ, the sampled output of the matched filter
corresponding to the kth symbol is
(1.4)
(1.5)
(1.6)
(1.7)
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• From the assumptions above, we know that jk and nk are
independent zero-mean Gaussian random variables.
• It remains to determine their variances.
• The variance of nk is
• For jk, we note that its variance is equal to the value of the
autocorrelation function of the matched filter output component
due to j(t) at 0.
• Using the Fourier relationship between autocorrelation function
and power spectral density, we have
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• Now, we can calculate the symbol (bit) error probability of the
communication system described above.
• By symmetry, we know that the average symbol error probability
is equal to the conditional symbol error probability given that, say,
bk = 1.
• Under the condition that bk = 1, the decision statistic Re[rk] is a
Gaussian random variable with mean
and variance
• Therefore, the symbol error probability is
where
is the symbol energy.
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• We suffer a loss in SNR by a factor of 1 + 0.9028PJTc/N0 with
respect to the case where the jammer is not present.
• There are two ways to reduce the loss in SNR.
– For a bandwidth limited channel, we can increase the
transmitted power P of the signal.
– If power is the main constraint, we can reduce the pulse width
Tc. This corresponds to spreading the spectrum of the
transmitted signal.
• In military applications, one consideration is that we do not want
our enemies to intercept or detect our transmission.
– The higher the transmission power the more susceptible is the
transmission being intercepted.
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• Therefore, we usually resort to spreading the spectrum of the
transmitted signal instead of raising the transmission power.
• This is the reason why spread spectrum is originally considered
for military communications.
• In terms of jamming immunity, the spreading method described
above is far from desirable.
• Simply reducing Tc is effective only for the continuous Gaussian
jammer assumed above.
• Since the continuous Gaussian jammer spreads its power across
the whole symbol period, for a small Tc, we only need to integrate
a small fraction of the symbol duration and, hence, pick up a
small jamming energy.
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• The discussion above brings out an important characteristic of
spread spectrum communications.
• In order for the receiver to perform properly, it has to know the
transmission times of the pulses.
• A sequence of pseudo-random transmission times is pre-assigned
to both the transmitter and the receiver.
• This sequence is generally referred to as a code.
• We will see that all spread spectrum techniques contain some
forms of pseudorandom codes.
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2.2 Direct sequence spread spectrum
• One non-trivial way of spreading the spectrum of the transmitted
signal is to modulate the data signal by a high rate pseudorandom sequence of phase-modulated pulses before mixing the
signal up to the carrier frequency for transmission.
• This spreading method is called direct sequence spread spectrum
(DS-SS).
• More precisely, suppose the data signal is
(1.11)
–
is the symbol sequence.
– T is the symbol duration.
• Note that all the signals here are complex envelopes unless
otherwise indicated.
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• We modulate the data signal b(t) by a spreading signal a(t) which
is given by
(1.12)
–
is called the signature sequence
–
is called the chip waveform, which is time limited to [0; Tc).
• We impose that condition that T = NTc, where N, which is usually
referred to as the processing gain or the spreading gain, is the
number of chips in a symbol and Tc is the separation between
consecutive chips.
• We normalize the energy of the chip waveforms to Tc.
• The spread spectrum signal is given by
(1.13)
where
is the largest integer which is smaller than or equal to x
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• This general model for DS-SS contains many different
modulation and spreading schemes.
• Some of the common examples are listed in Table 1.1.
• For example, a pictorial description of the BPSK modulation with
BPSK spreading scheme is given in Figure 1.2.
Table 1.1
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Figure 1.2: BPSK modulation and BPSK spreading scheme
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• To obtain the power spectrum of the spread spectrum signal, we
model the spreading elements al as iid zero-mean random
variables with
and the propagation delay as a
uniform random variable.
• Moreover, we also normalize the average symbol energy to PT,
i.e.,
• Then the power spectrum of the spread spectrum signal s(t) is
(1.14)
where
is the Fourier transform of the chip waveform
• The power spectra of the spread signals for the four schemes
shown in Table 1.1 are all given by
(1.15)
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• Comparing this to the power spectrum of the original data signal
(1.16)
• we see that the spectrum is spread N times wider by the direct
sequence technique in (1.15).
• In practice, the spreading sequences are pseudo-random.
• We will discuss, in Chapter 2, different ways to generate
sequences which have properties close to those of random
sequences.
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• In an AWGN channel, the ML receiver for the spread spectrum
signal is the matched filter receiver shown in Figure 1.3.
• We note that the matched filter is time-varying unless the
spreading sequence
is periodic with period N.
• For the kth symbol, the impulse response of the matched filter
hk(t) is given by
(1-17)
Figure 1.3: Matched filter receiver for the kth symbol of the DS-SS signal
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• For BPSK modulation, the decision device gives decision
• For QPSK modulation, the decision device gives decision
• Alternatively, we can implement the matched filter receiver as
shown in Figure 1.4.
Figure 1.4: Equivalent implementation of the matched filter receiver in Figure 1.3
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1.3 Frequency hop spread spectrum
• Another common method to spread the transmission spectrum of
a data signal is to (pseudo) randomly hop the data signal over
different carrier frequencies.
• This spreading method is called frequency hop spread spectrum
(FH-SS).
• Usually, the available band is divided into non-overlapping
frequency bins.
• The data signal occupies one and only one bin for a duration Tc
and hops to another bin afterward.
• When the hopping rate is faster than the symbol rate (i.e., T > Tc),
the FH scheme is referred to as fast hopping.
• Otherwise, it is referred to as slow hopping.
• A typical FH-SS transmitter and the corresponding receiver are
shown in Figures 1.6 and 1.7, respectively.
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Figure 1.6: Transmitter for FH-SS
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Figure 1.7: Receiver for FH-SS
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• Because it is practically difficult to build coherent frequency
synthesizers, modulation schemes, such as M-ary FSK, which
allow noncoherent detection are usually employed for the data
signal.
• For M-ary FSK, the data signal can be expressed as
(1.25)
• The frequency synthesizer outputs a hopping signal
(1.26)
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• This means that there are L frequency bins in the FH-SS system.
• T = NTc for fast hopping,
Tc = NT for slow hopping.
• For fast hopping
, the FH-SS signal is given by
(1.27)
• For slow hopping
, the FH-SS signal is given by
(1.28)
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• The orthogonality requirement for the FSK signals forces the
separation between adjacent FSK symbol frequencies be at least
2π/Tc for fast hopping, or 2π/T for slow hopping.
• Hence, the minimum separation between adjacent hopping
frequencies is 2Mπ/Tc for fast hopping, or 2Mπ/T for slow
hopping.
• For example, Figure 1.8 depicts the operation of a fast FH-SS
system with 2-FSK modulation (M = 2), 8 hopping bins (L = 8),
and 2 hops per symbol (T = 2Tc, N = 2).
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Figure 1.8 Fast FH-SS system with 2-FSK modulation, 8 hopping bins,
and 2 hops per symbol
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• To obtain the power spectrum of the M-ary FH-SS signal, we
model the phases
as iid random variables uniformly
distributed on [0; 2π).
• The hopping frequencies
are modelled as iid random
variables taking values from the set
with
equal probabilities.
• The FSK symbol frequencies
are iid random variables taking
values from the set
with equal probabilities.
• The delay is assumed to be uniformly distributed on [0, Tc) for
fast hopping, or [0; T) for slow hopping.
• We also assume that all the random variables mentioned above
are independent.
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• With these assumptions, we can show that the power spectrum of
the FH-SS signal s(t) is given by
– for fast hopping
(1.29)
– for slow hopping
(1.30)
• Therefore, the spectrum of the original data signal b(t) is
approximately spread by a factor of LN for fast hopping, or by a
factor of L for slow hopping.
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1.4 Time hop spread spectrum
• In time hop spread spectrum (TH-SS), we spread the spectrum by
modulating the data signal by a pseudo-random pulse-positionmodulated spreading signal.
• Suppose the data signal b(t) is
(1-31)
• We modulate the data signal by the spreading signal
(1-32)
where
• The resulting spread spectrum signal s(t) is given by
(1-33)
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• To obtain the power spectrum of the TH-SS signal s(t), we model
the data symbols bk as iid zero mean random variables with
• The pulse location indices ak are assumed to be iid random
variables taking values from {0, 1, …, N} with equal
probabilities.
• The propagation delay is modelled as a uniform random variable
on [0; T) as usual.
• It can be shown that the power spectral density of the spread
spectrum signal s(t) is
(1.34)
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• The matched filter receiver for this spreading method is shown in
Figure 1.9.
• The sampler is controlled by a timing circuit which is in turn
driven by the pseudo-random pulse-location code.
• We note that there are other types of TH-SS techniques.
– For example, one can use pulse-position modulation for the
data signal.
– As a result, the spread spectrum signal will be purely pulseposition modulated.
• The hopping scheme is similar to the M-ary FSK FH-SS system
with frequency bins replaced by time bins.
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Figure 1.9: Matched filter receiver for TH-SS
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1.5 Multicarrier spread spectrum
• In FH-SS, only one of many possible frequencies is transmitted at
a time.
• The other extreme is that we transmit all the possible frequencies
simultaneously.
• The resulting spreading method is called multicarrier spread
spectrum (MC-SS).
• More precisely, suppose the data signal b(t) is given by
(1-35)
• We modulate the data signal by the spreading signal
(1-36)
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• The resulting spread spectrum signal is
•
•
•
•
(1.37)
The data and spreading sequences are phase-modulated as in the
case of DS-SS.
The carrier frequencies should be chosen so that signals at
different frequencies do not interfere each other.
The minimum frequency separation is 2π/T.
To obtain the power spectrum of the spread spectrum signal, we
model the spreading elements an,k as iid zero-mean random
variables with
and the propagation delay as a
uniform random variable on [0; T).
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• We also normalize the average symbol energy to PT by setting
• Then the power spectrum of the spread spectrum signal s(t) is
(1.38)
• The matched filter receiver for MC-SS is shown in Figure 3.10.
• When
the operation of the correlator branches in
Figure 1.10 can be approximately performed by a single FFT.
• Hence the matched filter receiver can be implemented very
efficiently.
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Figure 1.10: Matched filter receiver for MC-SS
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• To see this, consider the output of the nth correlator for the kth
symbol in Figure 1.10 and denote it by zn,k.
• Then,
(1.39)
• In above, we divide the interval
into
N sub-intervals of length Tc.
• In the lth sub-interval, for l = 0, 1,…, N-1, we approximate
(1.40)
(1.41)
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• We note that (1.41) says that for each k, the sequence
is approximately the DFT (scaled) of the sequence
• Therefore, we can implement the correlator branches
approximately by sampling r(t) at the chip rate and passing the
samples through an FFT circuit.
• Similar approximations can also be applied on the transmitter
side.
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3.5 References
[1] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread
Spectrum Communications, Prentice Hall, Inc., 1995.
[2] M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum
multiple-access communication — Part I: System analysis,” IEEE Trans.
Commun., vol. 25, no. 8, pp. 795–799, Aug. 1977.
[3] R. A. Scholtz, “Multiple access with time-hopping impulse modulation,” Proc.
MILCOM ’93, pp. 11-14, Boston, MA, Oct. 1993.
[4] N. Yee, J. M. G. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor
wireless radio networks,” IEICE Trans. Commun., vol. E77-B, no. 7, pp. 900–
904, Jul. 1994.
[5] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA
systems,” IEEE Trans. Commun., vol. 44, no. 2, pp. 238–246, Feb. 1996.
[6] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, “Spread spectrum for
mobile communications,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 313–
321, May 1991.
[7] D. Torrieri, “Principles of spread spectrum communications theory,” Springer
2005.
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