1 Chapter 2 Direct-Sequence Systems

2 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

3 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

4 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 p i equals +1 or –1 and represents one chip of the spreading sequence. The chip waveform

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6 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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8 Therefore, if the filtered signal is given by (2-1), the multiplication yields the despread signal s 1 (t) at the input of the PSK demodulator.

9 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.

Spreading Sequences and Waveforms Random Binary Sequence x(t) –A stochastic process that consists of independent, identically distributed symbols, each of duration T.

11 The autocorrelation of a stochastic process x(t) is defined as

12 Autocorrelation of the random binary sequence:

13 Shift-Register Sequences

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15 The state of the shift register after clock pulse i is the vector The definition of a shift register implies that where s 0 (i) denotes the input to stage 1 after clock pulse i. If denotes the a i state of bit i of the output sequence, then Since the number of distinct states of an m-stage shift register is 2 m the sequence of states and the shift-register sequence have period

16 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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18 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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20 Since (2-26) is the same as (2-20).

21 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.

22 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.

23 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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25 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

26 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)

27 If then, (2-3) and (2-37) yield If

28 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 NT c

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30 The power spectral density of p(t) which is defined as the Fourier transform of

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32 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

33 Average autocorrelation of x(t) Average power spectral density –It is defined as the Fourier transform of the average autocorrelation.

34 The autocorrelation of the direct-sequence signal s(t) The average power spectral density of s(t)

35 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:

36 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

37 Substitution of (2-20) into this equation yields

38 Combining this equation with (2-56), and defining c 0 =1, we obtain

39 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

40 Output sequence:

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42 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).

43 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.

44 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.

45 octal numbers in increasing order (e.g. )

46 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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Systems with PSK Modulation Assuming that the chip waveform is well approximated by a waveform of duration T c, the received signal is where p i is equal to +1 or –1 The processing gain, defined as

50 –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.

51 (2-75) to (2-79) indicate that the demodulated sequence corresponding to this data symbol is

52 The input to the decision device is The decision device produces the symbol 1 if V > 0 and the symbol 0 if V < 0.

53 The white Gaussian noise has autocorrelation The mean value of the decision variable is

54 Since p i and p j are independent for

55 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

56 Let k 1 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

57 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

58 Using the Gaussian density to evaluate Assuming ψ that is uniformly distributed over, we obtain the symbol error probability

59 General Tone Interference

60 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

61 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 f d = 0 but this advantage decreases as increases and ultimately disappears.

62 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

63 GS/I (G = 50)

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Quaternary Systems A received quaternary direct-sequence signal with ideal carrier synchronization and a chip waveform of duration T c can be represented by –t 0 is the relative delay between the in-phase and quadrature components of the signal. –For QPSK, t 0 =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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67 Let T s 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 J i and N i are given by (2-82) and (2-83), respectively. The term representing crosstalk, is negligible if

68 The lower decision variable at the end of a channel-symbol interval where Since the energy per channel symbol is

69 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,

70 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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References [1] D. Torrieri, Principles of spread spectrum communications theory, Springer 2005.