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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 if the associated data symbol is a 1.

• An amplitude 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

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– 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 higher-speed 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 )

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• 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 initial

states.

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• 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 t

he 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 tha

n the data-symbol duration. • Short sequences are susceptible to interception and linear sequen

ces are inherently susceptible to mathematical cryptanalysis.• Long nonlinear pseudonoise sequences are needed for communic

ations with a high level of security. • However, if a modest level of security is acceptable, short or mod

erate-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

• The mean value of the decision variable is

0 ( )2n

NR

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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 for which

• The number for which is

• Equations (2-93), (2-3), and (2-94) yield

• 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 , we obtain the symbol error probability

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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.