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Chapter 1
Introduction to
spread-spectrum communications
Part I
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1.1 What is spread spectrum?
• Spread spectrum:
– A modulation technique that produces a spectrum for
the transmitted signal much wider than the usual
bandwidth needed to convey a particular stream of
information.
• Narrowband modulation:
– A modulation technique that produces a transmitted
signal with the usual bandwidth as opposed to a
spread spectrum modulation.
– BPSK, QPSK, QAM, and MSK are common
examples of narrowband modulation techniques.
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1.2 Why spread spectrum?
•
•
•
•
•
Resistant to jamming and interference
Difficult to intercept
Better multipath resolution, i.e., resistant to fading
Time and range measurement
Code division multiple access
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1.3 What is code division multiple access (CDMA)?
• CDMA:
– A multiple access scheme which allows multiple users to
communicate simultaneously using the same frequency band
by assigning different ‘codes’ to different users. Usually,
CDMA is achieved by spread spectrum techniques.
• FDMA:
– A multiple access scheme which allows multiple users to
communicate simultaneously by assigning non-overlapping
frequency bands to different users.
• TDMA:
– A multiple access scheme which allows multiple users to
communicate using the same frequency band by restricting
different users to transmit in non-overlapping time slots.
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• Why CDMA?
– Military needs
– Larger capacity for wireless cellular systems
• Practical systems
– GPS
– IS-95
– W-CDMA, CDMA2000
– Numerous applications in the ISM band
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Review of Digital Communication Theory
• Maximum likelihood receiver
– Assume that the communication channel is corrupted
by an additive white Gaussian noise (AWGN) with
two-sided power spectral density N0/2 W/Hz.
– The transmitter sends a signal chosen from the set of
M signals
– Further assume that all the M signals are time-limited
to [0; T], where T is called the symbol duration.
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– The received signal r(t) is given by
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• Our goal is to develop a receiver which observes the received
signal r(t) and determines which one of the M signals is being
sent based on maximizing the likelihood function.
• Define what the likelihood function is.
– By employing the Gram-Schmidt procedure, we can construct
a set of N (N  M) orthonormal functions
(all are time limited to [0; T]) which spans the signal space
formed by
– We augment this set of functions by another set of
orthonormal functions
so that the augmented set
forms an orthonormal basis for the space of
square-integrable functions.
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• Based on this representation, we can rewrite (1.1) as
where
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•
is a sufficient statistic for determining which signal is being sent,
i.e., determining the value of m.
• Rewriting (1.2) with these finite dimensional vectors, we have
– nN is a zero mean Gaussian random vector whose covariance
N
matrix is 20 I .
• The maximum likelihood (ML) receiver makes a decision
(select
) which maximizes the likelihood
function defined as the following conditional probability density
function:
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• ML receiver picks
such that the squared
Euclidean distance between the signal vector smN and the receiver
vector rN,
is minimized.
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•

is called the correlation metric between the received
signal r(t) and the transmitted signal sm(t).
Em is the energy of the transmitted signal sm(t).
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1.2 Matched filter receiver
• The correlators in Figure 1.2 can be replaced by the linear filters
and samplers as shown in Figure 1.3.
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• The matched filter has the optimal property that it is the
linear filter that maximizes the output signal-to-noise
ratio (SNR).
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• Therefore, the matched filter s(T - t), among all linear filters, maximizes the
output SNR.
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1.3 Signal space representation
• Suppose
is an orthonormal basis for the
signal space spanned by a set of square integrable signal
waveforms
• We represent the signal waveforms by a set of M
N-dimensional vectors with respect to the basis
• More precisely, for
sm(t) is represented
by the N-dimensional vector
• Given the basis, we can uniquely determine the signal
sm(t) from the vector sm or vice versa.
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• The notation denotes the Euclidean norm of a vector.
• The first identity states that the inner products in the function
space and the vector space are equivalent.
• The second identity states that the squared distance in the
function space is the same as the squared Euclidean distance in
the vector space.
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• Consider the following signal set of the QPSK scheme:
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• A simple basis for this signal set is
－
• Using this basis, the corresponding signal vectors are
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1.4 ML receiver error analysis
• A symbol error event occurs when the decision made by the
receiver is different from the transmitted symbol.
• Let
denotes the conditional symbol error probability given
that sm(t) is being transmitted.
• Pm denotes that probability that the transmitter sends sm(t).
• The average symbol error probability, Ps, is given by
• For simplicity, we assume all the signals are equally likely to be
transmitted, i.e., Pm = 1/M.
• Then the problem reduces to evaluating
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1.4.2 BPSK
• For the case of BPSK (binary antipodal signaling), the
matched filter receiver in Section 1.2 is the ML receiver.
• The receiver compares the sampled output Y of the
matched filter to the threshold zero.
• If Y > 0, the receiver decides that s(t) = s0(t) is sent.
Otherwise, it decides that s(t) = s1(t) is sent.
• From (1.14) and (1.15), we know that the noise sample
Yn is a zero mean Gaussian random variable with
variance
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• Suppose s0(t) is being sent, then Y is a Gaussian random
variable with mean E and variance
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1.4.2 General case (a geometric approach)
• Now assume that we employ M-ary signaling, i.e., the transmitter
sends a signal out from the set
• Using the vector representation in Section 1.1, we know that the
ML receiver decides that the m-th signal is sent when the
Euclidean distance
is the smallest among all the M
signal vectors.
• If we draw the signal vectors as points in the constellation
diagram as shown in Figure 1.6, the geometric meaning of the
ML decision rule is that the signal smN closest to the receiver
vector rN is selected.
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• A diagram showing the signal points and their corresponding
decision regions is known as the Voronoi diagram of a
modulation scheme.
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• Equivalently, we can construct a decision region (based on the
minimum distance principle) for each of the signal point in the
constellation diagram.
• Decide a specific signal point is sent if the received vector rN falls
into the corresponding decision region.
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• Suppose sm(t), for some
is being sent, and let
Rm denotes the decision region for sm(t).
• We make an error if the received vector rN falls outside Rm.
• Therefore, the conditional symbol error probability given that sm(t)
is sent,
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• The first special case we consider is the binary signaling case
(M = 2, N  2 ).
• It is intuitive that the decision regions for the signal points s0
and s1 are separated by the hyperplane half-way between the
signal points and perpendicular to the line joining the two signal
points.
• The next step is to evaluate the integral in (1.24).
• Suppose s0(t) is being sent, we know that
where
nN.
is the variance of an element of the noise vector
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• The next special case we consider is the QPSK example given in
Section 1.3 (M = 4, N = 2).
• It is again obvious that the decision region for a signal point is the
quadrant in which the signal point is located.
• Suppose s0 (t) is being sent, then
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1.4.3 Union bound
• When the exact symbol error probability is too difficult to
evaluate, we resort to bounds and approximations.
• One of such methods is the union bound.
• Suppose s0 (t) is being transmitted,
• The event
in (1.27) is exactly the same
as the error event as if there were only two signals, s0 (t) and sm (t)
( m  1), in the signal set.
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• The union bound of the conditional symbol error probability as
• By averaging over all the signals, we obtain the union bound for
the average symbol error probability as
• The union bound for the symbol error probability for the QPSK
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• By symmetry, we have
which is slightly larger than the exact symbol error
probability given in (1.26).
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1.5 Complex envelope
• Very often in a communication system, we do not
transmit the lowpass baseband signal directly.
• Instead, we mix the baseband signal with a carrier up to
a certain frequency, which matches the electromagnetic
propagation characteristic of the channel.
• As a result, the actual transmitted signal is a bandpass
signal.
• In this section, we introduce the concept of complex
envelope which provides a convenient way to represent
bandpass signals.
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1.5.1 Narrowband signal
• Suppose s(t) is a (real-valued) bandpass signal with most
of its frequency content concentrated in a narrow band
in the vicinity of a center frequency fc.
• A sufficient condition is that the Fourier transform of s(t)
satisfies
We refer to this condition as
the narrowband assumption.
• For a bandpass signal s(t) satisfying the narrowband
assumption stated above, it can be shown that s(t) can be
represented by an in-phase component x(t) and a
quadrature component y(t).
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•
• Using (1.34) and (1.35), we can reconstruct the real-valued
bandpass signal s(t) back from its complex envelope s (t ) .
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• How to obtain the complex envelope s (t ) from the signal s(t)
– The complex envelope as
– The Fourier transform S ( ) of s(t) is given
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• The complex envelope s (t ) is sometimes called the lowpass
equivalent signal of s(t).
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1.5.2 Bandpass filter
• We can use the complex envelope in the previous section to
represent the impulse response h(t) of a bandpass filter given that
h(t) satisfies the narrowband assumption stated before.
• Hence, if h (t ) is the complex envelope of h(t), then
• If a bandpass signal (satisfying the narrowband assumption) si(t)
is the input to the bandpass filter h(t), then the output from the
filter so(t) also satisfies the narrowband assumption and
• In the frequency domain,
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• From (1.37), the Fourier transform of the complex envelope, s0 (t )
of so(t) is given by
• By taking inverse Fourier transform on both sides of (1.42), we
obtain
• Hence, we can convolute the complex envelopes of h(t) and si(t)
and then convert the result back to obtain the output bandpass
signal.
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1.5.3 Narrowband process
• Suppose n(t) is a wide-sense stationary (WSS) process with zero
mean and power spectral density
• If
satisfies the narrowband assumption, then n(t) is called a
narrowband process.
• n(t) can also be written as
where nx(t) and ny(t) are zero-mean jointly WSS processes.
• If n(t) is Gaussian, nx(t) and ny(t) are jointly Gaussian.
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• Let us define the complex envelope n(t ) of the random process n(t)
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• If we treat the autocorrelation function
as a bandpass signal,
then
is its complex envelope.
• Hence, we can use the results in Section 1.5.1 to convert between
and
.
• A common example of narrowband process is the bandpass
additive Gaussian noise n(t) with zero mean and power spectral
density
• n(t) can be written as
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• The complex envelope of n(t) is given by
• Using the result above and (1.37), the power spectral density n(t )
of the complex envelope
is given by
• Taking inverse Fourier transform, we get
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• For the case where bandpass transmitted signals are sent through
a channel corrupted by n(t) and the bandwidths of the transmitted
signals are much smaller than the carrier frequency
, we
approximate
in (1.53) by
• This means that the lowpass equivalent of the additive bandpass
Gaussian noise looks white to the lowpass equivalents of the
transmitted signals.
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1.6 Noncoherent receiver
• As we mentioned before, since most communication systems
transmit bandpass signals instead of baseband ones, we focus on
this kind of signals and use the complex envelopes to represent
them here.
• Again, we consider the simple case of a non-dispersive channel,
for which we can model the received signal as
– Where A > 0 represents the channel gain (attenuation)
– θrepresents the carrier phase shift due to propagation delay,
local oscillator mismatch, and etc.
– n(t) is the complex AWGN with autocorrelation function
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• Suppose the receiver knows the value of θ, the problem reduces
to the one in Section 1.1.
• Hence we can use the correlation receiver in Figure 1.2 to detect
the received signal r(t).
• Generally, receivers that make use of the phase information are
referred to as coherent receivers.
• Therefore, the correlation receiver in Figure 1.2 and the matched
filter receiver in Figure 1.4 are coherent receivers.
• For coherent reception, we need to estimate the carrier phase .
– This estimation can sometimes be hard to perform, and
inaccurate estimation of the carrier phase will significantly
degrade the performance of the coherent ML receiver.
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• One alternative to coherent reception is to avoid using the phase
information.
• To do so, we model the carrier phase as a random variable
uniformly distributed on [0; 2π).
• Following steps similar to those in Section 1.2, we can develop
the ML receiver for this case.
– The resulting receiver is known as the noncoherent ML
receiver.
– For the case where the transmitted signals
have
equal energies.
– The ML receiver assumes the simple form shown in Figure
1.8.
– This receiver is usually referred to as the envelope receiver or
the square-law receiver.
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• It is difficult to evaluate the symbol error probability for a general
M-ary signal set received by the noncoherent ML receiver.
• For the special case of equal-energy binary orthogonal signals,
we state that the average symbol error probability (assuming
equal a priori probabilities) is given by
where E is the signal energy.
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1.7 Power spectrum
• In this section, we consider a more realistic model in which a
train of pulses are transmitted.
• For simplicity, we ignore the white noise and assume that the
(complex envelope of the) received signal is given by
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– ak’s are independent identically distributed (iid) random
variables with mean zero and variance A2.
– bk’s are also iid random variables with mean zero and variance
B2.
– The two data streams
are independent.
– Δ can be interpreted as the propagation delay
–
are the pulses for the in-phase and quadrature
channels, respectively.
– s(t) is a zero-mean random process.
• This model almost covers all practical quadrature modulation
schemes.
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• Our objective is to evaluate the autocorrelation function of s(t).
• First, let us modelΔas a random variable which is uniformly
distributed on [0; Ts), and is independent to both
• Then the autocorrelation function of s(t) is given by
• The last equality in (1.59) follows from the fact that the two data
streams consist of zero-mean independent random variables.
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• Similarly, we have
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• Therefore, the process s(t) is WSS and
• The power spectral density (power spectrum) of s(t) is given by
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• We consider the BPSK scheme where
• Let A2 = 2.
• We consider two cases:
– Ts = T, the power spectrum is
– Ts = T/10, the power spectrum is
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1.8 References
[1] J. G. Proakis, Digital Communications, 3rd Ed.,
McGraw-Hill, Inc., 1995.
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