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AM, FM, and Digital Modulated Systems Chap. 5
where g m (t) = A c m(t). Thus, for DS, where g c (t) = c(t) is used in Eq. (5–120b), the complex
envelope for the SS signal becomes
g(t) = A c m(t)c(t)
(5–121)
The resulting s(t) = Re{g(t)e jvct } is called a binary phase-shift keyed data, direct
sequence spreading, spread spectrum signal (BPSK-DS-SS), and c(t) is a polar spreading
signal. Furthermore, let this spreading waveform be generated by using a pseudonoise (PN)
code generator, as shown in Fig. 5–39b, where the values of c(t) are ;1. The pulse width of
c(t) is denoted by T c and is called a chip interval (as contrasted with a bit interval). The code
generator uses a modulo-2 adder and r shift register stages that are clocked every T c sec. It can
be shown that c(t) is periodic. Furthermore, feedback taps from the stage of the shift registers
and modulo-2 adders are arranged so that the c(t) waveform has a maximum period of N
chips, where N = 2 r - 1. This type of PN code generator is said to generate a maximum-length
sequence, or m-sequence, waveform.
Properties of Maximum-Length Sequences.
m-sequences [Peterson, Ziemer, and Borth, 1995]:
The following are some properties of
Property 1. In one period, the number of l’s is always one more than the number of 0’s.
Property 2. The modulo-2 sum of any m-sequence, when summed chip by chip with a
shifted version of the same sequence, produces another shifted version of the same sequence.
Property 3. If a window of width r (where r is the number of stages in the shift register)
is slid along the sequence for N shifts, then all possible r-bit words will appear
exactly once, except for the all 0 r-bit word.
Property 4. If the 0’s and 1’s are represented by -1 and +1 V, the autocorrelation of
the sequence is
1, k = N
R c (k) = c
- 1 N , k Z N
(5–122)
where R c 1k2 ! 11N2 c n c n k and c n =;1.
The autocorrelation of the waveform c(t) is
R c (t) = a1 - t e
bR
T c (k) + t e
R
c T c (k + 1)
c
where R c ( t) = 8c(t) c (t + t) 9 and tP is defined by
Equation (5–123) reduces to
t = kT c + t e ,
with 0 … t e 6 T c
/=q
R c (t) = c a a1 + 1
/=-q N b a t -/NT c
bd - 1 T c N
(5–123)
(5–124)
(5–125)