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Chapter 2 Direct-Sequence Systems 1 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 2 • 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 3 • 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 4 5 • 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 6 7 • • 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. 8 • 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. 9 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. 10 • The autocorrelation of a stochastic process x(t) is defined as 11 • Autocorrelation of the random binary sequence: 12 • Shift-Register Sequences 13 14 • 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 15 • 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: 16 17 • 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. 18 19 • Since (2-26) is the same as (2-20). 20 • 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. 21 • 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. 22 • 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. 23 24 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 25 • 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) 26 • If then , (2-3) and (2-37) yield • If 27 • 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 28 29 • The power spectral density of p(t) which is defined as the Fourier transform of 30 31 • 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 32 • Average autocorrelation of x(t) • Average power spectral density – It is defined as the Fourier transform of the average autocorrelation . 33 • The autocorrelation of the direct-sequence signal s(t) • The average power spectral density of s(t) 34 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: 35 • 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 36 • Substitution of (2-20) into this equation yields 37 • Combining this equation with (2-56), and defining c0=1, we obtain 38 • 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 39 Output sequence: 40 41 • 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). 42 • 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 43 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. 44 • octal numbers in increasing order (e.g. ) 45 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. 46 47 • 48 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 49 – – – – 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. 50 • (2-75) to (2-79) indicate that the demodulated sequence corresponding to this data symbol is 51 • The input to the decision device is • The decision device produces the symbol 1 if V > 0 and the symbol 0 if V < 0. 52 • The white Gaussian noise has autocorrelation N0 Rn ( ) 2 • The mean value of the decision variable is 53 • Since pi and pj are independent for 54 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 55 • 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 56 • 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 57 • Using the Gaussian density to evaluate • Assuming ψ that is uniformly distributed over the symbol error probability , we obtain 58 General Tone Interference 59 • 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 60 • 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. 61 • 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 62 GS/I (G = 50) 63 64 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. 65 66 • 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 67 • The lower decision variable at the end of a channel-symbol interval where • Since the energy per channel symbol is 68 • 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, 69 • The quaternary system provides a slight advantage relative to the binary system against tone interference. • Both systems provide the same and nearly the same . 70 71 2.5 References [1] D. Torrieri, Principles of spread spectrum communications theory, Springer 2005. 72