of Microprocessors

PERCUSSIVE SOUND GENERATION 535
A modification to the shift register method will greatly increase its
efficiency as a white noise subroutine. Essentially, the register and
exclusive-ors are turned inside-out, which results in several of the register
bits changing in an iteration rather than one. Assuming that the register
length is the same as the word length, the following steps are performed for
an iteration:
1. Shift the register left bringing in a zero on the right and putting the
overflow bit into the carry flag.
2. If the carry flag is off, the iteration is complete.
3. If the carry flag is on, flip selected bits in the register. This may be
accomplished by exclusive-oring a mask word with the register contents.
The iteration is now complete.
In a 16-bit machine, these steps may require as few as three instructions to
generate a sample of quite acceptable, if not statistically perfect, white noise.
The table below lists mask words for common computer word lengths.
Word length Sequence length Mask in hexadecimal
8 265 ID
12 4095 ID9
16 65535 ID87
24 16777215 ID872B
32 4294967295 ID872B41
Using the Random Numbers
The output of the random number generators just discussed is a string
of unsigned integers. However, since they are random, one can interpret
them as standard twos-complement numbers as well or even as binary fractions.
Since twos complement is slightly asymmetrical (there is one more
possible negative value than possible positive values), the mean of the sequence
will be -0.5 of the least significant bit rather than O. This almost
never causes problems unless the sequence is integrated for long periods of
time.
Although the output of a random number generator when sent through
a DAC sounds like white noise and in fact gives a white Fourier transform, it
does not look at all like natural white noise. The difference is its probability
density function, which is uniform rather than gaussian. As was mentioned
in Chapter 10, one can easily convert uniformly distributed random numbers
into near-gaussian distributed numbers by adding up 12 of them (assuming a
range of 0 to 1.0) and subtracting 6.0 from the sum. The mean of the result
will be 0 (except for the error described above) and the standard deviation
will be 1. O. The simulation is not exact because the probability of a result
greater than 6 standard deviations from the mean is 0 when it should be