Introduction to

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Introduction to

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Title: Introduction to

1
Chapter 1

Introduction to
spread-spectrum communications
Part I

2
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.

4
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

5
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.

Why CDMA?
Military needs
Larger capacity for wireless cellular systems
Practical systems
GPS
IS-95
W-CDMA, CDMA2000
Numerous applications in the ISM band

8
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.

The received signal r(t) is given by

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 ( ) 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.

Based on this representation, we can rewrite
(1.1) as
where

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 matrix is .
The maximum likelihood (ML) receiver makes a
decision (select )
which maximizes the likelihood function defined
as the following conditional probability density
function

ML receiver picks
such that the squared Euclidean distance
between the signal vector smN and the receiver
vector rN,
is minimized.

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).

15
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.

The matched filter has the optimal property that
it is the linear filter that maximizes the output
signal-to-noise ratio (SNR).

Therefore, the matched filter s(T - t), among all
linear filters, maximizes the output SNR.

19
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.

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.

Consider the following signal set of the QPSK
scheme

A simple basis for this signal set is
Using this basis, the corresponding signal
vectors are

24
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

25
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 gt 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

Suppose s0(t) is being sent, then Y is a Gaussian
random variable with mean E and variance

27
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.

A diagram showing the signal points and their
corresponding decision regions is known as the
Voronoi diagram of a modulation scheme.

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.

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,

The first special case we consider is the binary
signaling case (M 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 is the variance of an
element of the noise vector nN.

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

33
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) ( ), in the signal set.

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

By symmetry, we have
which is slightly larger than the exact symbol
error probability given in (1.26).

36
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.

37
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).

Using (1.34) and (1.35), we can reconstruct the
real-valued bandpass signal s(t) back from its
complex envelope .

How to obtain the complex envelope from
the signal s(t)
The complex envelope as
The Fourier transform of s(t) is given

The complex envelope is sometimes called
the lowpass equivalent signal of s(t).

42
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 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,

From (1.37), the Fourier transform of the complex
envelope, 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.

44
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.

Let us define the complex envelope of the
random process n(t)

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

The complex envelope of n(t) is given by
Using the result above and (1.37), the power
spectral density of the complex envelope
is given by
Taking inverse Fourier transform, we get

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.

49
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 gt 0 represents the channel gain
(attenuation)
?represents the carrier phase shift due to
propagation delay, local oscillator mismatch, and
etc.
n(t) is the complex AWGN with autocorrelation
function

Suppose the receiver knows the value of ?, 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.

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 2p).
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.

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.

54
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

aks are independent identically distributed
(iid) random variables with mean zero and
variance A2.
bks are also iid random variables with mean zero
and variance B2.
The two data streams
are independent.
? 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.

Our objective is to evaluate the autocorrelation
function of s(t).
First, let us model?as 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.