Torrieri D. Principles of Spread-Spectrum Communication Systems

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CHAPTER 2. DIRECT-SEQUENCE SYSTEMS

Figure 2.3: Spectra of desired signal and interference: (a) wideband-filter out-

put and (b) demodulator input.

Whatever the precise definition of a bandwidth, W and B are proportional to

and respectively, with the same proportionality constant. Therefore,

which links the processing gain with the interference rejection illustrated in the

figure. Since its spectrum is unchanged by the despreading, white Gaussian

noise is not suppressed by a direct-sequence system.

In practical systems, the wideband filter in the transmitter is used to limit

the out-of-band radiation. This filter and the propagation channel disperse

the chip waveform so that it is no longer confined to To avoid interchip

interference in the receiver, the filter might be designed to generate a pulse that

satisfies the Nyquist criterion for no intersymbol interference. A convenient

representation of a direct-sequence signal when the chip waveform may extend

beyond is

where denotes the integer part of When the chip waveform is assumed

to be confined to then (2-6) can be expressed by (2-1) and (2-2).

2.2

Spreading Sequences and Waveforms

Random Binary Sequence

A random binary sequence is a stochastic process that consists of indepen-

dent, identically distributed symbols, each of duration T. Each symbol takes

2.2.

SPREADING SEQUENCES AND WAVEFORMS

Figure 2.4: Sample function of a random binary sequence.

the value +1 with probability or the value –1 with probability Therefore,

for all and

The process is wide-sense stationary if the location of the first symbol transition

or start of a new symbol after is a random variable uniformly distributed

over the half-open interval (0,T]. A sample function of a wide-sense-stationary

random binary sequence is illustrated in Figure 2.4.

The autocorrelation of a stochastic process is defined as

If is a wide-sense stationary process, then is a function of alone,

and the autocorrelation is denoted by From (2-7) and the definitions of

an expected value and a conditional probability, it follows that the autocorre-

lation of a random binary sequence is

where denotes the conditional probability of event A given the occur-

rence of event B. From the theorem of total probability, it follows that

Since both of the following probabilities are equal to the probability that

and differ,

Substitution of (2-10) and (2-11) into (2-9) yields

CHAPTER 2.

DIRECT-SEQUENCE SYSTEMS

If then and are independent random variables because

and are in different symbol intervals. Therefore,

and (2-6) implies that for then and

are independent only if a symbol transition occurs in the half-open interval

Consider any half-open interval of length that includes

Exactly one transition occurs in Since the first transition for is

assumed to be uniformly distributed over the probability that a transition

in occurs in is If a transition occurs in then and

are independent and differ with probability otherwise,

Consequently, if Substitution

of the preceding results into (2-12) confirms the wide-sense stationarity of

and gives the autocorrelation of the random binary sequence:

where the triangular function is defined by

Shift-Register Sequences

Ideally, one would prefer a random binary sequence as the spreading sequence.

However, practical synchronization requirements in the receiver force one to

use periodic binary sequences. A shift-register sequence is a periodic binary

sequence generated by combining the outputs of feedback shift registers. A

feedback shift register, which is diagrammed in Figure 2.5, consists of consecutive

two-state memory or storage stages and feedback logic. Binary sequences drawn

from the alphabet {0,1} are shifted through the shift register in response to clock

pulses. The contents of the stages, which are identical to their outputs, are

logically combined to produce the input to the first stage. The initial contents

of the stages and the feedback logic determine the successive contents of the

stages. If the feedback logic consists entirely of modulo-2 adders (exclusive-OR

gates), a feedback shift register and its generated sequence are called linear.

Figure 2.6(a) illustrates a linear feedback shift register with three stages and

an output sequence extracted from the final stage. The input to the first stage

is the modulo-2 sum of the contents of the second and third stages. After each

clock pulse, the contents of the first two stages are shifted to the right, and

the input to the first stage becomes its content. If the initial contents of the

shift-register stages are 0 0 1, the subsequent contents after successive shifts are

listed in Figure 2.6(b). Since the shift register returns to its initial state after 7

shifts, the periodic output sequence extracted from the final stage has a period

of 7 bits.

The state of the shift register after clock pulse is the vector

2.2.

SPREADING SEQUENCES AND WAVEFORMS

Figure 2.5: General feedback shift register with stages.

Figure 2.6: (a) Three-stage linear feedback shift register and (b) contents after

successive shifts.

CHAPTER 2.

DIRECT-SEQUENCE SYSTEMS

where denotes the content of stage after clock pulse and S(0) is the

initial state. The definition of a shift register implies that

where denotes the input to stage 1 after clock pulse If denotes the

state of bit of the output sequence, then The state of a feedback

shif

shift-register sequence. The period N

of a

periodic sequence is defined as

the smallest positive integer for which Since the number of

distinct states of an shift register is the sequence of states and the

shift-register sequence have period

The Galois field of two elements, which is denoted by GF(2), consists of

the symbols 0 and 1 and the operations of modulo-2 addition and modulo-2

multiplication. These binary operations are defined by

where denotes modulo-2 addition. From these equations, it is easy to verify

that the field is closed under both modulo-2 addition and modulo-2 multipli-

cation and that both operations are associative and commutative. Since –1 is

defined as that element which when added to 1 yields 0, we have –1 = 1, and

subtraction is the same as addition. From (2-11), it follows that the additive

identity element is 0, the multiplicative identity is 1, and the multiplicative

inverse of 1 is The substitutions of all possible symbol combinations

verify the distributive laws:

where and can each equal 0 or 1. The equality of subtraction and addition

implies that if then

The input to stage 1 of a linear feedback shift register is

where the operations are modulo-2 and the feedback coefficient

equals either

0 or 1, depending on whether the output of stage feeds a modulo-2 adder.

An shift register is defined to have otherwise, the final state

would not contribute to the generation of the output sequence, but would only

provide a one-shift delay. For example, Figure 2.6 gives

and A general representation of a linear feedback shift

if it is open.

Since the output bit (2-16) and (2-19) imply that for

2.2.

SPREADING SEQUENCES AND WAVEFORMS

Figure 2.7: Linear feedback shift register: (a) standard representation and (b)

high-speed form.

which indicates that each output bit satisfies the linear recurrence relation:

The first output bits are determined solely by the initial state:

Figure 2.7(a) is not necessarily the best way to generate a particular shift-

higher-speed operation. From this diagram, it follows that

Repeated application of (2-22) implies that

CHAPTER 2.

DIRECT-SEQUENCE SYSTEMS

Addition of these equations yields

Substituting (2-23) and then into (2-25), we obtain

Since (2-26) is the same as (2-20). Thus, the two implementations can

produce the same output sequence indefinitely if the first output bits coincide.

However, they require different initial states and have different sequences of

states. Successive substitutions into the first equation of sequence (2-24) yields

Substituting and into (2-27) and then

using binary arithmetic, we obtain

If are specified, then (2-28) gives the corresponding initial state

of the high-speed shift register.

The sum of binary sequence and binary sequence

is defined to be the binary sequence each bit of which is the

modulo-2 sum of the corresponding bits of a and b.

Thus, if we can

write

Consider sequences a and

that are generated by the same linear feedback

shift register but may differ because the initial states may be different. For the

sequence (2-29) and the associative and distributive laws of binary

fields imply that

Since the linear recurrence relation is identical, d can be generated by the same

linear feedback logic as a and

Thus, if

and

are two output sequences of

a linear feedback shift register, then is also. If

a = b,

then is the

sequence of all 0’s, which can be generated by any linear feedback shift register.

2.2.

SPREADING SEQUENCES AND WAVEFORMS

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 feed-

back shift register has exactly nonzero states, the period of its output

sequence cannot exceed A sequence of period generated by a

linear feedback shift register is called a maximal or maximal-length sequence.

If a linear feedback shift register generates a maximal sequence, then all of its

nonzero output sequences are maximal, regardless of the initial states.

Out of possible states, the content of the last stage, which is the same

as the output bit, is a 0 in states. Among the nonzero states, the output

bit is a 0 in states. Therefore, in one period of a maximal sequence,

the number of 0’s is exactly while the number of 1’s is exactly

Given the binary sequence

, let denote a shifted binary

sequence. If a is

maximal sequence and modulo then

is not the sequence of all 0’s. Since is generated by the same shift

, it must be a maximal sequence and, hence, some cyclic shift of

We conclude that the modulo-2 sum of a maximal sequence and a cyclic shift

of itself by digits, where modulo produces another cyclic shift

of the original sequence; that is,

In contrast, a non-maximal linear sequence is not necessarily a

cyclic shift of

and may not even have the same period. As an example,

consider the linear feedback shift register depicted in Figure 2.8. The pos-

sible state transitions depend on the initial state. Thus, if the initial state

is 0 1 0, then the second state diagram indicates that there are two possible

states, and, hence, the output sequence has a period of two. The output se-

quence is a = (0,1,0,1,0,1,...), which implies that a(1) = (1,0,1,0,1,0,...)

and this result indicates that there is no value of

for which (2-31) is satisfied.

Periodic Autocorrelations

A binary sequence

with components can be mapped into a

binary antipodal sequence

with components by means of the

transformation

or, alternatively, The periodic autocorrelation of a periodic binary

sequence

with period

is defined as

Substitution of (2-32) into (2-33) yields

CHAPTER 2.

DIRECT-SEQUENCE SYSTEMS

Figure 2.8: (a) Nonmaximal linear feedback shift register and (b) state dia-

grams.

where denotes the number of agreements in the corresponding bits of

and

a(j), and denotes the number of disagreements. Equivalently, is the

number of 0’s in one period of and is the number of 1’s.

Consider a maximal sequence. From (2-31), it follows that equals the

number of 0’s in a maximal sequence if modulo N. Thus,

and, similarly, if modulo N. Therefore,

The periodic autocorrelation of a periodic function with period T is

defined as

where is the relative delay variable and is an arbitrary constant. It follows

that

has period

. We derive the periodic autocorrelation of

assum-

ing an ideal periodic spreading waveform of infinite extent and a rectangular

2.2.

SPREADING SEQUENCES AND WAVEFORMS

chip waveform. If the spreading sequence has period N, then has period

Equations (2-2) and (2-36) with c = 0 yield the autocorrelation of

If where is an integer, then (2-3), and (2-37) yield

Any delay can be expressed in the form

where is an integer

and Therefore, (2-37) and give

Using (2-38) and (2-3) in (2-39), we obtain

For a maximal sequence, the substitution of (2-35) into (2-40) yields over

one period:

where is the triangular function defined by (2-14). Since it has period

the autocorrelation can be compactly expressed as

Over one period, this autocorrelation resembles that of a random binary se-

quence, which is given by (2-13) with Both autocorrelations are shown

in Figure 2.9.

A straightforward calculation or the use of tables gives the Fourier transform

of the triangular function: