NAME
Math::PRBS - Generate Pseudorandom Binary Sequences using an
Iterator-based Linear Feedback Shift Register
SYNOPSIS
use Math::PRBS;
my $x3x2 = Math::PRBS->new( taps => [3,2] );
my $prbs7 = Math::PRBS->new( prbs => 7 );
my ($i, $value) = $x3x2t->next();
my @p7 = $prbs7->generate_all();
DESCRIPTION
This module will generate various Pseudorandom Binary Sequences (PRBS).
This module creates a iterator object, and you can use that object to
generate the sequence one value at a time, or *en masse*.
The generated sequence is a series of 0s and 1s which appears random for
a certain length, and then repeats thereafter.
It is implemented using an XOR-based Linear Feedback Shift Register
(LFSR), which is described using a feedback polynomial (or reciprocal
characteristic polynomial). The terms that appear in the polynomial are
called the 'taps', because you tap off of that bit of the shift register
for generating the feedback for the next value in the sequence.
FUNCTIONS AND METHODS
Initialization
"$seq = Math::PRBS::new( *key => value* )"
Creates the sequence iterator $seq using one of the "key => value"
pairs described below.
"prbs => *n*"
"prbs" needs an integer *n* to indicate one of the "standard"
PRBS polynomials.
# example: PRBS7 = x**7 + x**6 + 1
$seq = Math::PRBS::new( ptbs => 7 );
The "standard" PRBS polynomials implemented are
polynomial | prbs | taps | poly (string)
------------------+------------+-----------------+---------------
x**7 + x**6 + 1 | prbs => 7 | taps => [7,6] | poly => '1100000'
x**15 + x**14 + 1 | prbs => 15 | taps => [15,14] | poly => '110000000000000'
x**23 + x**18 + 1 | prbs => 23 | taps => [23,18] | poly => '10000100000000000000000'
x**31 + x**28 + 1 | prbs => 31 | taps => [31,28] | poly => '1001000000000000000000000000000'
"taps => [ *tap*, *tap*, ... ]"
"taps" needs an array reference containing the powers in the
polynomial that you tap off for creating the feedback. Do *not*
include the 0 for the "x**0 = 1" in the polynomial; that's
automatically included.
# example: x**3 + x**2 + 1
# 3 and 2 are taps, 1 is not tapped, 0 is implied feedback
$seq = Math::PRBS::new( taps => [3,2] );
"poly => '...'"
"poly" needs a string for the bits "x**k" downto "x**1", with a
1 indicating the power is included in the list, and a 0
indicating it is not.
# example: x**3 + x**2 + 1
# 3 and 2 are taps, 1 is not tapped, 0 is implied feedback
$seq = Math::PRBS::new( poly => '110' );
"$seq->reset()"
Reinitializes the sequence: resets the sequence back to the starting
state. The next call to "next()" will be the initial "$i,$value"
again.
Iteration
"$value = $seq->next()"
"($i, $value) = $seq->next()"
Computes the next value in the sequence. (Optionally, in list
context, also returns the current value of the i for the sequence.)
"$seq->rewind()"
Rewinds the sequence back to the starting state. The subsequent call
to "next()" will be the initial "$i,$value" again. (This is actually
an alias for "reset()").
"$i = $seq->tell_i()"
Return the current "i" position. The subsequent call to "next()"
will return this "i".
"$state = $seq->tell_state()"
Return the current internal state of the feedback register. Useful
for debug, or plugging into "->seek_to_state($state)" to get back to
this state at some future point in the program.
"$seq->seek_to_i( $n )"
"$seq->ith( $n )"
Moves forward in the sequence until "i" reaches $n. If "i > $n"
already, will internally "rewind()" first. If "$n > period", it will
stop at the end of the period, instead.
"$seq->seek_to_state( $lfsr )"
Moves forward in the sequence until the internal LFSR state reaches
$lfsr. It will wrap around, if necessary, but will stop once the
internal state returns to the starting point.
"$seq->seek_forward_n( $n )"
Moves forward in the sequence $n steps.
"$seq->seek_to_end()"
"$seq->seek_to_end( limit => $n )"
Moves forward until it's reached the end of the the period. (Will
start in the first period using "tell_i % period".)
If "limit =" $n> is used, will not seek beyond "tell_i == $n".
"@all = $seq->generate( *n* )"
Generates the next *n* values in the sequence, wrapping around if it
reaches the end. In list context, returns the values as a list; in
scalar context, returns the string concatenating that list.
"@all = $seq->generate_all( )"
"@all = $seq->generate_all( *limit => $max_i* )"
Returns the whole sequence, from the beginning, up to the end of the
sequence; in list context, returns the list of values; in scalar
context, returns the string concatenating that list. If the sequence
is longer than the default limit of 65535, or the limit given by
$max_i if the optional "limit => $max_i" is provided, then it will
stop before the end of the sequence.
"@all = $seq->generate_to_end( )"
"@all = $seq->generate_to_end( *limit => $max_i* )"
Returns the remaining sequence, from whatever state the list is
currently at, up to the end of the sequence; in list context,
returns the list of values; in scalar context, returns the string
concatenating that list. The limits work just as with
"generate_all()".
Information
"$i = $seq->description"
Returns a string describing the sequence in terms of the polynomial.
$prbs7->description # "PRBS from polynomial x**7 + x**6 + 1"
"$i = $seq->taps"
Returns an array-reference containing the list of tap identifiers,
which could then be passed to "->new(taps => ...)".
my $old_prbs = ...;
my $new_prbs = Math::PRBS->new( taps => $old_prbs->taps() );
"$i = $seq->period( *force => 'estimate' | $n | 'max'* )"
Returns the period of the sequence.
Without any arguments, will return undef if the period hasn't been
determined yet (ie, haven't travelled far enough in the sequence):
$i = $seq->period(); # unknown => undef
If *force* is set to 'estimate', will return "period = 2**k - 1" if
the period hasn't been determined yet:
$i = $seq->period(force => 'estimate'); # unknown => 2**k - 1
If *force* is set to an integer $n, it will try to generate the
whole sequence (up to "tell_i <= $n"), and return the period if
found, or undef if not found.
$i = $seq->period(force => $n); # look until $n; undef if sequence period still not found
If *force* is set 'max', it will loop thru the entire sequence (up
to "i = 2**k - 1"), and return the period that was found. It will
still return undef if still not found, but all sequences should find
the period within "2**k-1". If you find a sequence that doesn't,
feel free to file a bug report, including the "Math::PRBS->new()"
command listing the taps array or poly string; if "k" is greater
than 32, please include a code that fixes the bug in the bug report,
as development resources may not allow for debug of issues when "k >
32".
$i = $seq->period(force => 'max'); # look until 2**k - 1; undef if sequence period still not found
"$i = $seq->oeis_anum"
For known polynomials, return the On-line Encyclopedia of Integer
Sequences "A" number. For example, you can go to
to look at the sequence A011686.
Not all maximum-length PRBS sequences (binary m-sequences) are in
OEIS. Of the four "standard" PRBS (7, 15, 23, 31) mentioned above,
only PRBS7 is there, as A011686 . If you
have the A-number for other m-sequences that aren't included below,
please let the module maintainer know.
Polynomial | Taps | OEIS
----------------------------------------------+-----------------------+---------
x**2 + x**1 + 1 | [ 2, 1 ] | A011655
x**3 + x**2 + 1 | [ 3, 2 ] | A011656
x**3 + x**1 + 1 | [ 3, 1 ] | A011657
x**4 + x**3 + x**2 + x**1 + 1 | [ 4, 3, 2, 1 ] | A011658
x**4 + x**1 + 1 | [ 4, 1 ] | A011659
x**5 + x**4 + x**2 + x**1 + 1 | [ 5, 4, 2, 1 ] | A011660
x**5 + x**3 + x**2 + x**1 + 1 | [ 5, 3, 2, 1 ] | A011661
x**5 + x**2 + 1 | [ 5, 2 ] | A011662
x**5 + x**4 + x**3 + x**1 + 1 | [ 5, 4, 3, 1 ] | A011663
x**5 + x**3 + 1 | [ 5, 3 ] | A011664
x**5 + x**4 + x**3 + x**2 + 1 | [ 5, 4, 3, 2 ] | A011665
x**6 + x**5 + x**4 + x**1 + 1 | [ 6, 5, 4, 1 ] | A011666
x**6 + x**5 + x**3 + x**2 + 1 | [ 6, 5, 3, 2 ] | A011667
x**6 + x**5 + x**2 + x**1 + 1 | [ 6, 5, 2, 1 ] | A011668
x**6 + x**1 + 1 | [ 6, 1 ] | A011669
x**6 + x**4 + x**3 + x**1 + 1 | [ 6, 4, 3, 1 ] | A011670
x**6 + x**5 + x**4 + x**2 + 1 | [ 6, 5, 4, 2 ] | A011671
x**6 + x**3 + 1 | [ 6, 3 ] | A011672
x**6 + x**5 + 1 | [ 6, 5 ] | A011673
x**7 + x**6 + x**5 + x**4 + x**3 + x**2 + 1 | [ 7, 6, 5, 4, 3, 2 ] | A011674
x**7 + x**4 + 1 | [ 7, 4 ] | A011675
x**7 + x**6 + x**4 + x**2 + 1 | [ 7, 6, 4, 2 ] | A011676
x**7 + x**5 + x**2 + x**1 + 1 | [ 7, 5, 2, 1 ] | A011677
x**7 + x**5 + x**3 + x**1 + 1 | [ 7, 5, 3, 1 ] | A011678
x**7 + x**6 + x**4 + x**1 + 1 | [ 7, 6, 4, 1 ] | A011679
x**7 + x**6 + x**5 + x**4 + x**2 + x**1 + 1 | [ 7, 6, 5, 4, 2, 1 ] | A011680
x**7 + x**6 + x**5 + x**3 + x**2 + x**1 + 1 | [ 7, 6, 5, 3, 2, 1 ] | A011681
x**7 + x**1 + 1 | [ 7, 1 ] | A011682
x**7 + x**5 + x**4 + x**3 + x**2 + x**1 + 1 | [ 7, 5, 4, 3, 2, 1 ] | A011683
x**7 + x**4 + x**3 + x**2 + 1 | [ 7, 4, 3, 2 ] | A011684
x**7 + x**6 + x**3 + x**1 + 1 | [ 7, 6, 3, 1 ] | A011685
x**7 + x**6 + 1 | [ 7, 6 ] | A011686
x**7 + x**6 + x**5 + x**4 + 1 | [ 7, 6, 5, 4 ] | A011687
x**7 + x**5 + x**4 + x**3 + 1 | [ 7, 5, 4, 3 ] | A011688
x**7 + x**3 + x**2 + x**1 + 1 | [ 7, 3, 2, 1 ] | A011689
x**7 + x**3 + 1 | [ 7, 3 ] | A011690
x**7 + x**6 + x**5 + x**2 + 1 | [ 7, 6, 5, 2 ] | A011691
x**8 + x**6 + x**4 + x**3 + x**2 + x**1 + 1 | [ 8, 6, 4, 3, 2, 1 ] | A011692
x**8 + x**5 + x**4 + x**3 + 1 | [ 8, 5, 4, 3 ] | A011693
x**8 + x**7 + x**5 + x**3 + 1 | [ 8, 7, 5, 3 ] | A011694
x**8 + x**7 + x**6 + x**5 + x**4 + x**2 + 1 | [ 8, 7, 6, 5, 4, 2 ] | A011695
x**8 + x**7 + x**6 + x**5 + x**4 + x**3 + 1 | [ 8, 7, 6, 5, 4, 3 ] | A011696
x**8 + x**4 + x**3 + x**2 + 1 | [ 8, 4, 3, 2 ] | A011697
x**8 + x**6 + x**5 + x**4 + x**2 + x**1 + 1 | [ 8, 6, 5, 4, 2, 1 ] | A011698
x**8 + x**7 + x**5 + x**1 + 1 | [ 8, 7, 5, 1 ] | A011699
x**8 + x**7 + x**3 + x**1 + 1 | [ 8, 7, 3, 1 ] | A011700
x**8 + x**5 + x**4 + x**3 + x**2 + x**1 + 1 | [ 8, 5, 4, 3, 2, 1 ] | A011701
x**8 + x**7 + x**5 + x**4 + x**3 + x**2 + 1 | [ 8, 7, 5, 4, 3, 2 ] | A011702
x**8 + x**7 + x**6 + x**4 + x**3 + x**2 + 1 | [ 8, 7, 6, 4, 3, 2 ] | A011703
x**8 + x**6 + x**3 + x**2 + 1 | [ 8, 6, 3, 2 ] | A011704
x**8 + x**7 + x**3 + x**2 + 1 | [ 8, 7, 3, 2 ] | A011705
x**8 + x**6 + x**5 + x**2 + 1 | [ 8, 6, 5, 2 ] | A011706
x**8 + x**7 + x**6 + x**4 + x**2 + x**1 + 1 | [ 8, 7, 6, 4, 2, 1 ] | A011707
x**8 + x**7 + x**6 + x**3 + x**2 + x**1 + 1 | [ 8, 7, 6, 3, 2, 1 ] | A011708
x**8 + x**7 + x**2 + x**1 + 1 | [ 8, 7, 2, 1 ] | A011709
x**8 + x**7 + x**6 + x**1 + 1 | [ 8, 7, 6, 1 ] | A011710
x**8 + x**7 + x**6 + x**5 + x**2 + x**1 + 1 | [ 8, 7, 6, 5, 2, 1 ] | A011711
x**8 + x**7 + x**5 + x**4 + 1 | [ 8, 7, 5, 4 ] | A011712
x**8 + x**6 + x**5 + x**1 + 1 | [ 8, 6, 5, 1 ] | A011713
x**8 + x**4 + x**3 + x**1 + 1 | [ 8, 4, 3, 1 ] | A011714
x**8 + x**6 + x**5 + x**4 + 1 | [ 8, 6, 5, 4 ] | A011715
x**8 + x**7 + x**6 + x**5 + x**4 + x**1 + 1 | [ 8, 7, 6, 5, 4, 1 ] | A011716
x**8 + x**5 + x**3 + x**2 + 1 | [ 8, 5, 3, 2 ] | A011717
x**8 + x**6 + x**5 + x**4 + x**3 + x**1 + 1 | [ 8, 6, 5, 4, 3, 1 ] | A011718
x**8 + x**5 + x**3 + x**1 + 1 | [ 8, 5, 3, 1 ] | A011719
x**8 + x**7 + x**4 + x**3 + x**2 + x**1 + 1 | [ 8, 7, 4, 3, 2, 1 ] | A011720
x**8 + x**6 + x**5 + x**3 + 1 | [ 8, 6, 5, 3 ] | A011721
x**9 + x**4 + 1 | [ 9, 4 ] | A011722
x**10 + x**3 + 1 | [ 10, 3 ] | A011723
x**11 + x**2 + 1 | [ 11, 2 ] | A011724
x**12 + x**7 + x**4 + x**3 + 1 | [ 12, 7, 4, 3 ] | A011725
x**13 + x**4 + x**3 + x**1 + 1 | [ 13, 4, 3, 1 ] | A011726
x**14 + x**12 + x**11 + x**1 + 1 | [ 14, 12, 11, 1 ] | A011727
x**15 + x**1 + 1 | [ 15, 1 ] | A011728
x**16 + x**5 + x**3 + x**2 + 1 | [ 16, 5, 3, 2 ] | A011729
x**17 + x**3 + 1 | [ 17, 3 ] | A011730
x**18 + x**7 + 1 | [ 18, 7 ] | A011731
x**19 + x**6 + x**5 + x**1 + 1 | [ 19, 6, 5, 1 ] | A011732
x**20 + x**3 + 1 | [ 20, 3 ] | A011733
x**21 + x**2 + 1 | [ 21, 2 ] | A011734
x**22 + x**1 + 1 | [ 22, 1 ] | A011735
x**23 + x**5 + 1 | [ 23, 5 ] | A011736
x**24 + x**4 + x**3 + x**1 + 1 | [ 24, 4, 3, 1 ] | A011737
x**25 + x**3 + 1 | [ 25, 3 ] | A011738
x**26 + x**8 + x**7 + x**1 + 1 | [ 26, 8, 7, 1 ] | A011739
x**27 + x**8 + x**7 + x**1 + 1 | [ 27, 8, 7, 1 ] | A011740
x**28 + x**3 + 1 | [ 28, 3 ] | A011741
x**29 + x**2 + 1 | [ 29, 2 ] | A011742
x**30 + x**16 + x**15 + x**1 + 1 | [ 30, 16, 15, 1 ] | A011743
x**31 + x**3 + 1 | [ 31, 3 ] | A011744
x**32 + x**28 + x**27 + x**1 + 1 | [ 32, 28, 27, 1 ] | A011745
THEORY
A pseudorandom binary sequence (PRBS) is the sequence of N unique bits,
in this case generated from an LFSR. Once it generates the N bits, it
loops around and repeats that seqence. While still within the unique N
bits, the sequence of N bits shares some properties with a truly random
sequence of the same length. The benefit of this sequence is that, while
it shares statistical properites with a random sequence, it is actually
deterministic, so is often used to deterministically test hardware or
software that requires a data stream that needs pseudorandom properties.
In an LFSR, the polynomial description (like "x**3 + x**2 + 1")
indicates which bits are "tapped" to create the feedback bit: the taps
are the powers of x in the polynomial (3 and 2). The 1 is really the
"x**0" term, and isn't a "tap", in the sense that it isn't used for
generating the feedback; instead, that is the location where the new
feedback bit comes back into the shift register; the 1 is in all
characteristic polynomials, and is implied when creating a new instance
of Math::PRBS.
If the largest power of the polynomial is "k", there are "k+1" bits in
the register (one for each of the powers "k..1" and one for the "x**0 =
1"'s feedback bit). For any given "k", the largest sequence that can be
produced is "N = 2^k - 1", and that sequence is called a maximum length
sequence or m-sequence; there can be more than one m-sequence for a
given "k". One useful feature of an m-sequence is that if you divide it
into every possible partial sequence that's "k" bits long (wraping from
N-1 to 0 to make the last few partial sequences also "k" bits), you will
generate every possible combination of "k" bits (*), except for "k"
zeroes in a row. For example,
# x**3 + x**2 + 1 = "1011100"
"_101_1100 " -> 101
"1_011_100 " -> 011
"10_111_00 " -> 111
"101_110_0 " -> 110
"1011_100_ " -> 100
"1_0111_00 " -> 001 (requires wrap to get three digits: 00 from the end, and 1 from the beginning)
"10_1110_0 " -> 010 (requires wrap to get three digits: 0 from the end, and 10 from the beginning)
The Wikipedia:LFSR article (see "REFERENCES") lists some polynomials
that create m-sequence for various register sizes, and links to Philip
Koopman's complete list up to "k=64".
If you want to create try own polynonial to find a long m-sequence, here
are some things to consider: 1) the number of taps for the feedback
(remembering not to count the feedback bit as a tap) must be even; 2)
the entire set of taps must be relatively prime; 3) those two conditions
are necesssary, but not sufficient, so you may have to try multiple
polynomials to find an m-sequence; 4) keep in mind that the time to
compute the period (and thus determine if it's an m-sequence) doubles
every time "k" increases by 1; as the time increases, it makes more
sense to look at the complete list up to "k=64"), and pure-perl is
probably tpp wrong language for searching "k>64".
(*) Since a maximum length sequence contains every k-bit combination
(except all zeroes), it can be used for verifying that software or
hardware behaves properly for every possible sequence of k-bits.
REFERENCES
* Wikipedia:Linear-feedback Shift Register (LFSR) at
* Article includes a list of some maximum length polynomials
* Article links to Philip Koopman's complete list of maximum
length polynomials, up to "k = 64" at
* Wikipedia:Pseudorandom Binary Sequence (PRBS) at
* The underlying algorithm in Math::PRBS is based on the C code in
this article's "Practical Implementation"
* Wikipedia:Maximum Length Sequence (m-sequence) at
* Article describes some of the properties of m-sequences
AUTHOR
Peter C. Jones ""
Please report any bugs or feature requests thru the web interface at
COPYRIGHT
Copyright (C) 2016 Peter C. Jones
LICENCE
This program is free software; you can redistribute it and/or modify it
under the terms of either: the GNU General Public License as published
by the Free Software Foundation; or the Artistic License.
See for more information.