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Chapter 2
Direct-Sequence Systems
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2.1 Definitions and Concepts
• Spread-spectrum signal
– A signal that has an extra modulation that expands the signal
bandwidth beyond what is required by the underlying data
modulation.
• Spread-spectrum communication systems
– suppressing interference
– making interception difficult
– accommodating fading
– multipath channels
– providing a multiple-access capability
• The most practical and dominant methods of spread-spectrum
communications
– direct-sequence modulation
– frequency hopping
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• A direct-sequence signal
– a spread-spectrum signal generated by the direct mixing of the
data with a spreading waveform before the final carrier
modulation.
• Ideally, a direct-sequence signal with binary phase-shift keying
(PSK) or differential PSK (DPSK) data modulation can be
represented by
– A is the signal amplitude,
– d(t) is the data modulation
– p(t) is the spreading waveform
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• An amplitude
• An amplitude
if the associated data symbol is a 1.
if the associated data symbol is a 0.
• The spreading waveform has the form
– each pi equals +1 or –1 and represents one chip of the
spreading sequence.
• The chip waveform
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• Message privacy
– If a transmitted message cannot be recovered without
knowledge of the spreading sequence.
• The processing gain
–
– An integer equal to the number of chips in a symbol interval.
– If W is the bandwidth of p(t) and B is the bandwidth of d(t),
the spreading due to ensures that has a bandwidth W >> B
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•
• Therefore, if the filtered signal is given by (2-1), the multiplication
yields the despread signal s1(t) at the input of the PSK demodulator.
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• An approximate measure of the interference rejection capability
is given by the ratio W/B.
• W and B are proportional to
respectively.
• A convenient representation of a direct-sequence signal when the
chip waveform may extend beyond
is
where
denotes the integer part of x.
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2.2 Spreading Sequences and Waveforms
• Random Binary Sequence x(t)
– A stochastic process that consists of independent, identically
distributed symbols, each of duration T.
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• The autocorrelation of a stochastic process x(t) is defined as
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• Autocorrelation of the random binary sequence:
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• Shift-Register Sequences
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• The state of the shift register after clock pulse i is the vector
• The definition of a shift register implies that
where s0(i) denotes the input to stage 1 after clock pulse i.
• If denotes the ai state of bit i of the output sequence, then
• Since the number of distinct states of an m-stage shift register is
2m the sequence of states and the shift-register sequence have
period
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• The Galois field , GF(2),
– Consists of the symbols 0 and 1
– The operations of modulo-2 addition and modulo-2 multiplication.
• The input to stage 1 of a linear feedback shift register is
•
Figure 2.7: Linear feedback shift register:
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• Since the output bit
, (2-16) and (2-19) imply that for
• Each output bit satisfies the linear recurrence relation:
• Figure 2.7(a) is not necessarily the best way to generate a
particular shift register sequence.
• Figure 2.7(b) illustrates an implementation that allows higherspeed operation.
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• Since
(2-26) is the same as (2-20).
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• Successive substitutions into the first equation of sequence (2-24)
yields
• If
are specified, then (2-28) gives the
corresponding initial state of the high-speed shift register.
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• If a linear feedback shift register reached the zero state with all its
contents equal to 0 at some time, it would always remain in the
zero state, and the output sequence would subsequently be all 0’s.
• Since a linear m-stage feedback shift register has exactly
nonzero states, the period of its output sequence cannot exceed
• maximal or maximal-length sequence
– A sequence of period
generated by a linear feedback
shift register.
– If a linear feedback shift register generates a maximal
sequence, then all of its nonzero output sequences are
maximal, regardless of the initial states.
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• Given the binary sequence a, let
denote a
shifted binary sequence.
• If a is a maximal sequence and
then
– It is not the sequence of all 0’s.
– It must be a maximal sequence.
– The modulo-2 sum of a maximal sequence and a cyclic shift
of itself by j digits, produces another cyclic shift of the
original sequence; that is,
• A non-maximal linear sequence
is not necessarily a
cyclic shift of a and may not even have the same period.
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Periodic Autocorrelations
• A binary sequence a with components
can be
mapped into a binary antipodal sequence p with components by
means of the transformation
• The periodic autocorrelation of a periodic binary sequence a
with period N is defined as
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• Consider a maximal sequence.
• The periodic autocorrelation of a periodic function with period T
is defined as
• If the spreading sequence has period N, then has period
Equations (2-2) and (2-36) yield the autocorrelation of p(t)
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• If
then
, (2-3) and (2-37) yield
• If
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• Using (2-38) and (2-3) in (2-39), we obtain
• For a maximal sequence, the substitution of (2-35) into (2-40)
yields
• Since it has period NTc
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• The power spectral density of p(t) which is defined as the Fourier
transform of
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• A pseudonoise or pseudorandom sequence
– A periodic binary sequence with a nearly even balance of 0’s
and 1’s.
– An autocorrelation that roughly resembles, over one period,
the autocorrelation of a random binary sequence.
– Pseudonoise sequences, which include the maximal sequences,
provide practical spreading sequences because their
autocorrelations facilitate code synchronization in the receiver
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• Average autocorrelation of x(t)
• Average power spectral density
– It is defined as the Fourier transform of the average
autocorrelation
.
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• The autocorrelation of the direct-sequence signal s(t)
• The average power spectral density of s(t)
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Polynomials over the Binary Field
• A polynomial over the binary field GF(2) has the form
– where the coefficients
are elements of GF(2)
• Ex:
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• The characteristic polynomial associated with a linear feedback
shift register of m stages is defined as
• The generating function associated with the output sequence is
defined as
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• Substitution of (2-20) into this equation yields
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• Combining this equation with (2-56), and defining c0=1, we
obtain
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• The generating function of the output sequence generated by a
linear feedback shift register with characteristic polynomial f(x)
may be expressed in the form
– where the degree ψ(x) of is less than the degree of f(x).
• The output sequence is said to be generated by f(x).
• Equation (2-60) explicitly shows that the output sequence is
completely determined by the feedback coefficients
and the initial state
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Output sequence:
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• The polynomial p(x) is said to divide the polynomial b(x) if there
is a polynomial h(x) such that
• A polynomial p(x) over GF(2) of degree m is called irreducible
– If p(x) is not divisible by any polynomial over GF(2) of
degree less than m but greater than zero. (m < degree <0 )
–
• An irreducible polynomial over GF(2) must have an odd number
of terms, but this condition is not sufficient for irreducibility.
– If has an even number of terms, then
and the
fundamental theorem of algebra implies that
divides
p(x).
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• If a shift-register sequence is periodic with period n then its
generating function
may be expressed as
•
,
– which has the form of (2-62).
• Thus, f (x) generates a sequence of period n for all and, hence, all
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initial states.
• A polynomial over GF(2) of degree m is called primitive.
– If the smallest positive integer n for which the polynomial
divides
• A primitive characteristic polynomial of degree m can generate a
sequence of period
which is the period of a maximal
sequence generated by a characteristic polynomial of degree m.
• A primitive characteristic polynomial must be irreducible.
• A characteristic polynomial of degree m generates a maximal
sequence of period
if and only if it is a primitive
polynomial.
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• octal numbers in increasing order (e.g.
)
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Long Nonlinear Sequences
• Long sequence or long code
– A spreading sequence with a period that is much longer than
the data-symbol duration and may even exceed the message
duration.
• A short sequence or short code
– A spreading sequence with a period that is equal to or less
than the data-symbol duration.
• Short sequences are susceptible to interception and linear
sequences are inherently susceptible to mathematical
cryptanalysis.
• Long nonlinear pseudonoise sequences are needed for
communications with a high level of security.
• However, if a modest level of security is acceptable, short or
moderate-length pseudonoise sequences are preferable for rapid
acquisition, burst communications, and multiuser detection.
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•
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2.3 Systems with PSK Modulation
• Assuming that the chip waveform is well approximated by a
waveform
of duration Tc, the received signal is
where pi is equal to +1 or –1
• The processing gain, defined as
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–
–
–
–
i(t) the interference.
n(t) denotes the zero-mean white Gaussian noise.
The chip matched filter has impulse response
Its output is sampled at the chip rate to provide G samples per data symbol.
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• (2-75) to (2-79) indicate that the demodulated sequence
corresponding to this data symbol is
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• The input to the decision device is
• The decision device produces the symbol 1 if V > 0 and the
symbol 0 if V < 0.
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• The white Gaussian noise has autocorrelation
N0
Rn 
 ( )
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• The mean value of the decision variable is
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• Since pi and pj are independent for
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Tone Interference at Carrier Frequency
• The tone interference has the form
• (2-82), (2-85), (2-92) and a change of variables give
• For rectangular chip waveform has
• For sinusoidal chips in the spreading waveform
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• Let k1 denote the number of chips in
• The number for which is
• Equations (2-93), (2-3), and (2-94) yield
for which
• These equations indicate that the use of sinusoidal chip
waveforms instead of rectangular ones effectively reduces the
interference power by a factor
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• Equation (2-95) indicates that tone interference at the
carrier frequency would be completely rejected if
in every symbol interval.
• The conditional symbol error probability given the value of ψ is
–
is the conditional symbol error probability given
the values of
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• Using the Gaussian density to evaluate
• Assuming ψ that is uniformly distributed over
the symbol error probability
, we obtain
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General Tone Interference
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• The conditional symbol error probability is well approximated by
–
: equivalent two-sided power spectral density of the
interference plus noise, given the value of φ
• For sinusoidal chip waveforms, a similar derivation yields (2-110)
with
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• To explicitly exhibit the reduction of the interference power by the
factor G, we may substitute
in (2-111) or (2-112).
• A comparison of these two equations (2-111) and (2-112) confirms
that sinusoidal chip waveforms provide a
dB
advantage when fd = 0 but this advantage decreases as increases
and ultimately disappears.
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• If in (2-109) is modeled as a random variable that is uniformly
distributed over
then the character of in (2-111) implies
that its distribution is the same as it would be if were
uniformly distributed over
• The symbol error probability, which is obtained by averaging
over the range of ψis
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GS/I
(G = 50)
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2.4 Quaternary Systems
• A received quaternary direct-sequence signal with ideal carrier
synchronization and a chip waveform of duration Tc can be
represented by
– t0 is the relative delay between the in-phase and quadrature
components of the signal.
– For QPSK, t0=0
– For offset QPSK (OQPSK) or minimum-shift keying (MSK),
– For OQPSK, the chip waveforms are rectangular.
– For MSK, the chip waveforms are sinusoidal.
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• Let Ts denote the duration of the data symbols before the
generation of (2-123).
• Let
denote the duration of the channel symbols,
which are transmitted in pairs.
– where Ji and Ni are given by (2-82) and (2-83), respectively.
• The term representing crosstalk,
is negligible if
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• The lower decision variable at the end of a channel-symbol
interval
where
• Since
the energy per channel symbol is
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• Using the tone-interference model of Section 2.3, and averaging
the error probabilities for the two parallel symbol streams, we
obtain the conditional symbol error probability:
– For rectangular chip waveforms (QPSK and OQPSK signals)
– For sinusoidal chip waveforms,
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• The quaternary system provides a slight advantage relative to the
binary system against tone interference.
• Both systems provide the same
and nearly the
same
.
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2.5 References
[1] D. Torrieri, Principles of spread spectrum communications
theory, Springer 2005.
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