comm.GoldSequence
Generate Gold sequence from set of sequences
Description
The comm.GoldSequence
System object™ generates a binary sequence with small periodic crosscorrelation
properties from a bounded set of sequences. For more information, see Gold Sequences.
To generate a Gold sequence from set of sequences:
Create the
comm.GoldSequence
object and set its properties.Call the object with arguments, as if it were a function.
To learn more about how System objects work, see What Are System Objects?
Creation
Description
creates
a Gold sequence generator System object. This object generates a pseudorandom Gold sequence of binary
numbers.goldseq
= comm.GoldSequence
sets properties using one or more namevalue arguments. Enclose each property
name in quotes. For example, goldseq
= comm.GoldSequence(Name
=Value
)goldseq =
comm.GoldSequence(Shift=1)
specifies a one sample offset of the
output sequence from the starting point.
Properties
Unless otherwise indicated, properties are nontunable, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
release
function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
FirstPolynomial
— Generator polynomial for first preferred PN sequence
'z^6 + z + 1'
(default)  character vector  string scalar  binaryvalued row vector  integervalued row vector
Generator polynomial for the first preferred PN sequence, specified as one of these options:
Character vector or string scalar of a polynomial whose constant term is
1
. For more information, see Representation of Polynomials in Communications Toolbox.Binaryvalued row vector that represents the coefficients of the polynomial in order of descending powers. The length of this vector must be N + 1, where N is the degree of the polynomial. The first and last entries must be
1
, indicating the leading term with degree N and a constant term of 1.Integervalued row vector of elements that represent the exponents for the nonzero terms of the polynomial in order of descending powers. The last entry must be
0
, indicating a constant term of 1.
This property determines the feedback connections for the shift register
of the first preferred PN sequence generator. The degree of the first
generator polynomial must equal the degree of the second generator
polynomial specified by the SecondPolynomial
property. For more information, see Preferred Pairs of Sequences.
Example: 'z^8 + z^2 + 1'
, [1 0 0 0 0 0 1 0
1]
, and [8 2 0]
represent the same
polynomial, p(z) =
z^{8} +
z^{2} +
1.
Data Types: double
 char
FirstInitialConditions
— Initial conditions used for shift register of first preferred PN sequence generator
[0 0 0 0 0 1]
(default)  binaryvalued scalar  binaryvalued vector
Initial conditions used for the shift register of the first preferred PN sequence generator when the simulation starts, specified as a binaryvalued scalar or binaryvalued vector.
If you set this property to a scalar, the initial value of all cells in the shift register are the specified scalar value.
If you set this property to a vector, each element of the vector corresponds to the initial value of the corresponding cell in the shift register. The length of the vector must equal the degree of the generator polynomial specified by the
FirstPolynomial
property.
Note
For the object to generate a nonzero sequence, at least one element of the initial conditions for the first or second preferred PN sequence generator must be nonzero. Specifically, the initial state of at least one of the shift registers must be nonzero.
Data Types: double
SecondPolynomial
— Generator polynomial for second preferred PN sequence
'z^6 + z^5 + z^2 + z + 1'
(default)  character vector  string scalar  binaryvalued row vector  integervalued row vector
Generator polynomial for the first preferred PN sequence, specified as one of these options:
Character vector or string scalar of a polynomial whose constant term is
1
. For more information, see Representation of Polynomials in Communications Toolbox.Binaryvalued row vector that represents the coefficients of the polynomial in order of descending powers. The length of this vector must be N + 1, where N is the degree of the polynomial. The first and last entries must be
1
, indicating the leading term with degree N and a constant term of 1.Integervalued row vector of elements that represent the exponents for the nonzero terms of the polynomial in order of descending powers. The last entry must be
0
, indicating a constant term of 1.
This property determines the feedback connections for the shift register
of the second preferred PN sequence generator. The degree of the second
generator polynomial must equal the degree of the first generator polynomial
specified by the FirstPolynomial
property. For more information, see Preferred Pairs of Sequences.
Data Types: double
 char
SecondInitialConditionsSource
— Source of initial conditions used for shift register of second preferred PN sequence generator
'Property'
(default)  'Input port'
Source of the initial conditions used for the shift register of the second preferred PN sequence generator, specified as one of these values:
'Property'
— Specify PN sequence generator initial conditions by using theSecondInitialConditions
property.'Input port'
— Specify PN sequence generator initial conditions by using thesecondinitcond
input argument.
Data Types: string
 char
SecondInitialConditions
— Initial conditions used for shift register of second preferred PN sequence generator
[0 0 0 0 0 1]
(default)  binaryvalued scalar  binaryvalued vector
Initial conditions used for the shift register of the second preferred PN sequence generator when the simulation starts, specified as a binaryvalued scalar or binaryvalued vector.
If you set this property to a scalar, the initial value of all cells in the shift register are the specified scalar value.
If you set this property to a vector, each element of the vector corresponds to the initial value of the corresponding cell in the shift register. The length of the vector must equal the degree of the generator polynomial specified by the
SecondPolynomial
property.
Note
For the object to generate a nonzero sequence, at least one element of the initial conditions for the first or second preferred PN sequence generator must be nonzero. Specifically, the initial state of at least one of the shift registers must be nonzero.
Dependencies
To enable this property set the SecondInitialConditionsSource
property to
'Property'
.
Data Types: double
Index
— Sequence index
0
(default)  integer
Sequence index of the output sequence from the set of Gold sequences, specified as an integer in the range [–2, 2^{n}–2]. n is the degree of the preferred polynomials. For more information, see Sequence Index and Gold Sequences.
Data Types: double
Shift
— Offset of output sequence from starting point
0
(default)  integer
Offset of the output sequence from the starting point, specified as an integer. Units are in samples. The object wraps shift values that are negative or greater than the length of the Gold sequence. For an example using shift, see Generate Gold Sequences with Various Shift Values.
Note
Calculate the mask vector by using the shift2mask
function.
Data Types: double
VariableSizeOutput
— Enable variablesize outputs
false
(default)  true
Enable variablesize outputs, specified as a numeric or logical
0
(false
) or 1
(true
). To enable variablesize outputs by using the
outputsize
input argument, set this property to
true
. The enabled input specifies the output size of
the Gold sequence. The input value must be less than or equal to the value
of the MaximumOutputSize
property.
When you set this property to false
, the SamplesPerFrame
property specifies the number of output samples.
Data Types: logical
 double
MaximumOutputSize
— Maximum output frame size
[10 1]
(default)  vector of the form [m 1]
Maximum output frame size, specified as a vector of the form
[m 1], where m is a positive
integer. The first element of the vector indicates the maximum length of the
output frame and the second element of the vector must be
1
.
Example: [20 1]
specifies a maximum frame output size of
20
by1
.
Dependencies
To enable this property, set the VariableSizeOutput
property to
true
.
Data Types: double
SamplesPerFrame
— Number of samples output per frame
1
(default)  positive integer
Number of samples output per frame, specified as a positive integer. If
you set this property to a value of M, the object outputs
M samples of a Gold sequence that has a period of N =
2^{n} –
1, where n is the degree of the generator
polynomials specified by the FirstPolynomial
and SecondPolynomial
properties.
Dependencies
To enable this property, set the VariableSizeOutput
property to
false
.
Data Types: double
ResetInputPort
— Enable generator reset input
false
(default)  true
Enable the generator reset input, specified as a numeric or logical
0
(false
) or 1
(true
). To enable the ability to reset the sequence
generator using the resetseq
input argument, set this property to
true
. This input resets the states of the two shift
registers of the Gold sequence generator to the initial conditions specified
by the FirstInitialConditions
and SecondInitialConditions
properties. For an example using the
resetseq
input, see Generate Gold Sequences with Various Reset Values.
Dependencies
To enable this property, set the SecondInitialConditionsSource
property to
'Property'
.
Data Types: logical
 double
OutputDataType
— Data type of output
'double'
(default)  'logical'
 'Smallest unsigned integer'
 'Smallest integer'
Data type of the output, specified as 'double'
,
'logical'
, 'Smallest unsigned
integer'
, or 'Smallest integer'
.
To use the Smallest unsigned integer
option, you must
have the FixedPoint Designer™ product.
Data Types: string
 char
Usage
Syntax
Description
outputs a
frame of the Gold sequence in a column vector based on the configured
object.outSequence
= goldseq()
uses outSequence
= goldseq(secondinitcond
)secondinitcond
as the initial conditions for the
second PN sequence.
To enable this syntax, set the SecondInitialConditionsSource
property to 'Input
port'
and the ResetInputPort
property to false
.
uses outSequence
= goldseq(outputsize
)outputsize
as the output size.
To enable this syntax, set the VariableSizeOutput
property to true
.
uses outSequence
= goldseq(resetseq
)resetseq
as the reset signal.
To enable this syntax, set the ResetInputPort
property to true
and the SecondInitialConditionsSource
property to
'Property'
.
specifies inputs for the second PN sequence initial conditions and the output
size. outSequence
= goldseq(secondinitcond
,outputsize
)
To enable this syntax, set the SecondInitialConditionsSource
property to 'Input
port'
, the ResetInputPort
property to false
, and set the VariableSizeOutput
property to true
.
specifies inputs for the output size and a reset signal.outSequence
= goldseq(outputsize
,resetseq
)
To enable this syntax, set the VariableSizeOutput
property to true
, the
SecondInitialConditionsSource
property to
'Property'
, and the ResetInputPort
property to true
.
Input Arguments
secondinitcond
— Initial conditions used for shift register of second sequence
polynomial
binary scalar  binary vector
Initial conditions used for the shift register of the second preferred PN sequence generator when the simulation starts, specified as a binaryvalued scalar or binaryvalued vector.
If you set this input to a scalar, the initial value of all cells in the shift register are the specified scalar value.
If you set this input to a vector, each element of the vector corresponds to the initial value of the corresponding cell in the shift register. The length of the vector must equal the degree of the generator polynomial specified by the
SecondPolynomial
property.
Note
For the object to generate a nonzero sequence, at least one element of the initial conditions for the first or second preferred PN sequence generator must be nonzero. Specifically, the initial state of at least one of the shift registers must be nonzero.
This object accepts variablesize inputs. After the object is locked, you can change the size of each input channel, but you cannot change the number of channels. For more information, see VariableSize Signal Support with System Objects.
Dependencies
To enable this input argument, set the SecondInitialConditionsSource
property to
'Input port'
.
Data Types: double
outputsize
— Length of output sequence
nonnegative integer  vector of the form [n 1]
Length of the output sequence, specified as a nonnegative integer,
n, or a vector of the form [n
1], where n is a positive integer. The first element
of the vector indicates the length of the output frame and the second
element of the vector must be 1
.
The scalar or the first element of the row vector must be less than or
equal to the first element of the MaximumOutputSize
property value.
Dependencies
To enable this input argument, set the VariableSizeOutput
property to
true
.
Data Types: double
resetseq
— Reset sequence generator
scalar  column vector
Reset sequence generator, specified as a scalar or a column vector
with length equal to the number of samples per frame specified by the
SamplesPerFrame
property.
When you specify this input as a nonzero scalar, the object resets to the specified initial conditions and then generates a new output frame.
When you specify this input as a column vector, the object resets to the specified initial conditions at each sample in the output frame that aligns with a nonzero value in the reset vector.
For an example using reset, see Generate Gold Sequences with Various Reset Values.
Dependencies
To enable this input argument, set the ResetInputPort
property to
true
.
Data Types: double
 logical
Output Arguments
outSequence
— Gold sequence
column vector
Gold sequence, returned as a column vector. For more information, see Gold Sequences.
Object Functions
To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named obj
, use
this syntax:
release(obj)
Examples
Generate Gold Sequence Samples
Generate 10 samples of a Gold sequence with a period of $${2}^{5}1$$.
goldseq = comm.GoldSequence('FirstPolynomial','x^5+x^2+1', ... 'SecondPolynomial','x^5+x^4+x^3+x^2+1', ... 'FirstInitialConditions',[0 0 0 0 1], ... 'SecondInitialConditions',[0 0 0 0 1], ... 'Index',4,'SamplesPerFrame',10); x = goldseq()
x = 10×1
1
1
1
0
0
0
0
0
0
1
Generate Gold Sequences with Various Shift Values
Generate Gold sequences using different values for the shift input to demonstrate that the object wraps shift values that are negative or greater than the sequence length.
Create a Gold sequence System object™, specifying 15 samples per frame and no shift. Generate the 7sample Gold sequence. The output frame begins at the start of the Gold sequence.
goldseq = comm.GoldSequence( ... "FirstPolynomial","x^3+x^2+1", ... "SecondPolynomial","x^3+x+1", ... "FirstInitialConditions",[0 0 1], ... "SecondInitialConditions",[0 1 1], ... "Index",3, ... "SamplesPerFrame",15)
goldseq = comm.GoldSequence with properties: FirstPolynomial: "x^3+x^2+1" FirstInitialConditions: [0 0 1] SecondPolynomial: "x^3+x+1" SecondInitialConditionsSource: 'Property' SecondInitialConditions: [0 1 1] Index: 3 Shift: 0 VariableSizeOutput: false SamplesPerFrame: 15 ResetInputPort: false OutputDataType: 'double'
outseq = goldseq()'
outseq = 1×15
1 1 0 0 0 0 0 1 1 0 0 0 0 0 1
Release the object, adjust the shift to 1, and generate the shifted output. Repeat this process for shift values of 6 and 13. For each of these shift settings, the output frame begins at the 6th sample of this 7sample long Gold sequence.
release(goldseq); goldseq.Shift = 1; outseq = goldseq()'
outseq = 1×15
0 1 1 0 0 0 0 0 1 1 0 0 0 0 0
release(goldseq); goldseq.Shift = 6; outseq = goldseq()'
outseq = 1×15
0 1 1 0 0 0 0 0 1 1 0 0 0 0 0
release(goldseq); goldseq.Shift = 13; outseq = goldseq()'
outseq = 1×15
0 1 1 0 0 0 0 0 1 1 0 0 0 0 0
Generate Gold Sequences with Various Reset Values
Generate Gold sequences using different values for the reset input.
Create a Gold sequence System object™, specifying 15 samples per frame and enabling the reset input. Generate the 7sample Gold sequence. To observe the starting point of frames output by the object relative to the generated Gold sequence, the samples per frame value includes one sample more than two times the sequence length.
goldseq = comm.GoldSequence( ... "FirstPolynomial","x^3+x^2+1", ... "SecondPolynomial","x^3+x+1", ... "FirstInitialConditions",[0 0 1], ... "SecondInitialConditions",[0 1 1], ... "Index",3, ... "ResetInputPort",true, ... "SamplesPerFrame",15);
So that the object does not reset the sequence to the initial conditions, specify a scalar reset input value of 0
. Display consecutive 15sample frames of the sequence. The second frame continues the sequence from where the first frame left off, with the second element of the Gold sequence.
resetseq = 0; x1 = goldseq(resetseq)'
x1 = 1×15
1 1 0 0 0 0 0 1 1 0 0 0 0 0 1
x2 = goldseq(resetseq)'
x2 = 1×15
1 0 0 0 0 0 1 1 0 0 0 0 0 1 1
To reset the sequence to the initial conditions, specify a scalar reset input value of 1. Display consecutive 15sample frames of the sequence. The second frame now begins with the first element of the Gold sequence.
resetseq = 1; x3 = goldseq(resetseq)'
x3 = 1×15
1 1 0 0 0 0 0 1 1 0 0 0 0 0 1
x4 = goldseq(resetseq)'
x4 = 1×15
1 1 0 0 0 0 0 1 1 0 0 0 0 0 1
To reconfigure the reset input of the object to accept a vector, release the object. Specify a reset input vector filled with 0
s. Display consecutive 15sample frames of the sequence. Since the object does not reset the sequence to the initial conditions, the second frame begins with the second element of the Gold sequence.
release(goldseq) resetseq = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]'; x5 = goldseq(resetseq)'
x5 = 1×15
1 1 0 0 0 0 0 1 1 0 0 0 0 0 1
x6 = goldseq(resetseq)'
x6 = 1×15
1 0 0 0 0 0 1 1 0 0 0 0 0 1 1
Specify a reset input vector that has two elements with nonzero values. Display consecutive 15sample frames of the sequence. The second and third frames continue the sequence from the prior frame but reset the sequence at the third and ninth samples of the frame due to the position of the nonzero entries in the vector specified for the reset input.
resetseq = [0 0 1 0 0 0 0 0 1 0 0 0 0 0 0]'; x7 = goldseq(resetseq)'
x7 = 1×15
0 0 1 1 0 0 0 0 1 1 0 0 0 0 0
x8 = goldseq(resetseq)'
x8 = 1×15
1 1 1 1 0 0 0 0 1 1 0 0 0 0 0
x9 = goldseq(resetseq)'
x9 = 1×15
1 1 1 1 0 0 0 0 1 1 0 0 0 0 0
More About
Gold Sequences
The characteristic crosscorrelation properties of Gold sequences
make them useful when multiple devices are broadcasting in the same frequency range.
The Gold sequences are defined using a specified pair of sequences,
u and v, called a preferred pair, as
defined in Preferred Pairs of Sequences. The
u and v pair of sequences has a period
N = 2^{n} – 1,
where n is the degree of the generator polynomials specified by
the FirstPolynomial
and
SecondPolynomial
properties. The set G(u, v)
of Gold sequences is defined by
$$G(u,v)=\{u,v,u\oplus v,u\oplus Tv,u\oplus {T}^{2}v,\mathrm{...},u\oplus {T}^{N1}v\}$$
T represents the operator that shifts vectors cyclically to the left by one place, and $$\oplus $$ represents addition modulo 2. G(u,v) contains N + 2 sequences of period N.
Gold sequences have the property that the crosscorrelation between any two sequences or between shifted versions of the sequences takes on one of three values: –t(n), –1, or t(n) – 2, where
$$t(n)=\{\begin{array}{ll}1+{2}^{(n+1)/2}\hfill & n\text{even}\hfill \\ 1+{2}^{(n+2)/2}\hfill & n\text{odd}\hfill \end{array}$$
The object uses two PN sequence generators to generate a preferred pair of sequences. The object then XORs these sequences to produce the output Gold sequence, as shown in this figure.
The FirstPolynomial
and
SecondPolynomial
properties determine the preferred pair of sequences and the feedback connections
for the shift registers used by the PN sequence generators to generate their output.
For more details on PN sequence generation, see the Simple Shift Register Generator section on the
comm.PNSequence
System object reference page.
This table provides examples of preferred pairs.
Degree of Generator Polynomials (n)  Pair of Sequences Period (N)  FirstPolynomial Property Value  SecondPolynomial Property Value 

5  31 
[5 2 0]

[5 4 3 2 0]

6  63 
[6 1 0]

[6 5 2 1 0]

7  127 
[7 3 0]

[7 3 2 1 0]

9  511 
[9 4 0]

[9 6 4 3 0]

10  1023 
[10 3 0]

[10 8 3 2 0]

11  2047 
[11 2 0]

[11 8 5 2 0]

The FirstInitialConditions
property and SecondInitialConditions
property (or secondinitcond
input argument) are values that specify the initial
values of the shift registers corresponding to FirstPolynomial
and
SecondPolynomial
,
respectively.
Note
For the object to generate a nonzero sequence, at least one element of one of the initial conditions vectors must be nonzero. Specifically, the initial state of at least one of the shift registers must be nonzero.
You can offset the starting point of the Gold sequence by setting the Shift
property to a nonzero
value. For an example that adjusts the shift setting, see Generate Gold Sequences with Various Shift Values.
Preferred Pairs of Sequences
Preferred pairs of sequences, u and v, comprise the set of Gold sequences G(u, v).
For a pair of sequences, u and v, of period N = 2^{n}–1 to be a preferred pair, they must satisfy these requirements:
n is the degree of the generator polynomials specified by the
FirstPolynomial
andSecondPolynomial
properties.n is not divisible by
4
.v = u[q], where
q is odd.
q = 2^{k}+1 or q = 2^{2k}–2^{k}+1.
v is obtained by sampling every qth symbol of u.
$$\mathrm{gcd}(n,k)=\{\begin{array}{cc}1& n\equiv 1\mathrm{mod}2\\ 2& n\equiv 2\mathrm{mod}4\end{array}$$
For more details on PN sequence generation, see the Simple Shift Register Generator section on the
comm.PNSequence
System object reference page.
Sequence Index
The sequence index specified by the Index
property specifies which
Gold sequence in the set G(u,
v) is output.
The set of available Gold sequences is
$$G(u,v)=\{u,v,u\oplus v,u\oplus Tv,u\oplus {T}^{2}v,\mathrm{...},u\oplus {T}^{N1}v\}$$
u and v are the two preferred PN sequences, T is the operator that shifts vectors cyclically to the left by one place, and $$\oplus $$ represents addition modulo 2. G(u,v) contains N+2 Gold sequences of period N = 2^{n}–1.
The range of Index
is [–2,
2^{n}–2], where
n is the degree of the generator polynomials specified by the
FirstPolynomial
and
SecondPolynomial
properties. The index values 2
and 1
correspond to the first and second preferred PN sequences as generated by FirstPolynomial
and SecondPolynomial
, respectively. This table shows the correspondence
between the sequence index and the output sequence.
Index Property Value  Output Sequence 

–2  u 
–1  v 
0 
$$u\oplus v$$ 
1 
$$u\oplus Tv$$ 
2 
$$u\oplus {T}^{2}v$$ 
...  ... 
2 ^{n}
–2

$$u\oplus {T}^{{2}^{n}2}v$$ 
References
[1] Proakis, John G. Digital Communications. 5th ed. New York: McGraw Hill, 2007.
[2] Gold, R. “Maximal Recursive Sequences with 3Valued Recursive CrossCorrelation Functions (Corresp.).” IEEE Transactions on Information Theory 14, no. 1 (January 1968): 154–56. https://doi.org/10.1109/TIT.1968.1054106.
[3] Gold, R. “Optimal Binary Sequences for Spread Spectrum Multiplexing (Corresp.).” IEEE Transactions on Information Theory 13, no. 4 (October 1967): 619–21. https://doi.org/10.1109/TIT.1967.1054048.
[4] Sarwate, D.V., and M.B. Pursley. “Crosscorrelation Properties of Pseudorandom and Related Sequences.” Proceedings of the IEEE 68, no. 5 (1980): 593–619. https://doi.org/10.1109/PROC.1980.11697.
[5] Dixon, Robert C. Spread Spectrum Systems: With Commercial Applications. 3rd ed. New York: Wiley, 1994.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
See System Objects in MATLAB Code Generation (MATLAB Coder).
Version History
Introduced in R2008a
See Also
Objects
Functions
Blocks
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