wireless communication and networks part3
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190
CHAPTER
7 / SPREAD
SPECTRUM
If
m,
a,
C,
and
X
o
are integers, then this technique will produce a sequence of integers
with each integer in the range 0
:5
X
n
<
m.
An
essential characteristic of a pseudoran-
dom number generator
is
that the generated sequence should appear random.
Although the sequence
is
not random, because it
is
generated deterministically, there
is
a variety of statistical tests that can be used to assess the degree to which a sequence
exhibits randomness. Another desirable characteristic
is
that the function should be a
full-period generating function. That
is,
the function should generate all the numbers
between 0 and
m
before repeating.
With the linear congruential algorithm, a choice of parameters
that
provides a
full period does
not
necessarily provide a good randomization. For example, consider
the two generators:
X
n
+
1
=
(6X
n
)mod
13
X
n
+
1
=
(7X
n
)mod
13
Write out the two sequences to show that both are full period. Which one appears
more random to you?
7.10 We would like
m
to be very large so
that
there
is
the potential for producing a long
series of distinct random numbers. A common criterion is
that
m
be
nearly equal
to
the
maximum representable nonnegative integer for a given computer. Thus, a value
of
m
near to or equal to 2
31
is typically chosen. Many experts recommend a value
of
2
31
-
1.
You may wonder why one should not simply use 2
3
\
because this latter num-
ber
can
be
represented with no additional bits, and the mod operation should
be
eas-
ier
to
perform.
In
general, the modulus
2
k
-
1
is
preferable to
2
k
.
Why
is
this so?
7.11
In
any use of pseudorandom numbers, whether for encryption, simulation,
or
statisti-
cal design, it
is
dangerous to trust blindly the random number generator that happens
to
be
available in your computer's system library. [PARK88] found
that
many con-
temporary textbooks and programming packages make use
of
flawed algorithms for
pseudorandom
number
generation. This exercise will enable you to test your system.
The test
is
based on a theorem attributed to Ernesto Cesaro (see [KNUT98] for
a proof), which states that the probability
is
equal
to
-;
that
the greatest common
7T
divisor of two randomly chosen integers
is
1.
Use this theorem in a program to deter-
mine statistically the value of
7T.
The main program should call three subprograms: the
random number generator from the system library to generate the random integers; a
subprogram to calculate the greatest common divisor of two integers using Euclid's
algorithm; and a subprogram that calculates square roots.
If
these latter two programs
are not available, you will have to write them
as
welL
The main program should loop
through a large number of random numbers to give an estimate of the aforementioned
probability. From this, it
is
a simple matter to solve for your estimate of
7T.
If
the result is close to 3.14, congratulations!
If
not, then the result
is
probably
low,
usually a value of around 2.7. Why would such an inferior result
be
obtained?
7.12 This problem demonstrates that different LFSRs can be used to generate an m-sequence.
a. Assume an initial state of 10000 in the
LFSR
of Figure 7.19a. In a manner similar
to Figure 7.13b, show the generation of
an
m-sequence.
b. Now assume the configuration
of
Figure 7.19b, with the same initial state, and
repeat part
a.
Show
that
this configuration also produces an m-sequence,
but
that
it
is
a different sequence from
that
produced by
the
first LFSR.
7.13 Demonstrate
that
the codes in an 8
X
8Walsh matrix are orthogonal to each other by
showing that multiplying any code by any other code produces a result of zero.
7.14 Consider a
CDMA
system in which users A and B have the Walsh codes
(
-1
1
-11
-1
1
-11)
and (
-1
-111
-1
-111),
respectively.
a. Show the
output
at the receiver if A transmits a data bit 1 and B does not transmit.
b. Show the
output
at the receiver ifA transmits a data bit 0 and B does not transmit.
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米饭蚊子2015-02-01老师上课参考教材之一，很清晰，要是合在一起就最好了
