Generate Gold sequence from set of sequences
Communications Toolbox / Comm Sources / Sequence Generators
The Gold Sequence Generator block generates a binary sequence with small periodic crosscorrelation properties from a bounded set of sequences. For more information, see Gold Sequences.
This block can output sequences that vary in length during simulation. For more information about variablesize signals, see VariableSize Signal Basics (Simulink).
These icons shows the block with all ports enabled.
oSiz
— Current output sizeCurrent output size, specified as a scalar or a twoelement row
vector. The second element of the vector must be
1
.
Example: [10 1]
indicates the current output column
vector will be of size 10by1.
To enable this port select the Output variablesize
signals parameter and set Maximum output
size source to Dialog
parameter
.
Data Types: double
Ref
— Reference input signalReference input signal, specified as a scalar, column vector.
To enable this port select the Output variablesize
signals parameter and set Maximum output
size source to Inherit from reference
input
.
Data Types: double
Rst
— Reset signalReset signal, specified in one of these forms.
When the output size is variable specify as a scalar.
Otherwise, specify as a scalar or a 2D column vector with a length equal to Samples per frame.
The output signal resets for nonzero Rst input values. For more information, see Reset Behavior
To enable this port, select the Reset on nonzero input parameter.
Data Types: double
Out
— Output signalOutput signal, returned as a binary column vector. At least one element of the Initial states (1) or Initial states (2) vector must be nonzero in order for the block to generate a nonzero sequence.
Data Types: double
Preferred polynomial (1)
— First sequence polynomial'z^6 + z + 1'
(default)  polynomial character vector  binary row vector  integer row vectorFirst sequence polynomial, specified in one of these forms.
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 Preferred polynomial (2) parameter. 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.
Initial states (1)
— Initial states for first sequence polynomial[0 0 0 0 0 1]
(default)  binary vectorInitial states of the shift register for first sequence polynomial of the preferred pair, specified as a binary vector with length equal to the degree of Preferred polynomial (1).
Note
For the block 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.
Preferred polynomial (2)
— Second sequence polynomial'z^6 + z^5 + z^2 + z + 1'
(default)  polynomial character vector  binary row vector  integer row vectorSecond sequence polynomial, specified in one of these forms.
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 second generator polynomial must equal the degree of the first generator polynomial specified by the Preferred polynomial (1) parameter. 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.
Initial states (2)
— Initial states for second sequence polynomial[0 0 0 0 0 1]
(default)  binary vectorInitial states of the shift register for second sequence polynomial of the preferred pair, specified as a binary vector with length equal to the degree of Preferred polynomial (2).
Note
For the block 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.
Sequence index
— Sequence index0
(default)  integer scalar in the range [–2,
2^{n}–2]Sequence index of the output sequence from the set of sequences, specified as an integer scalar in the range [–2, 2^{n}–2]. n is the degree of the preferred polynomials. For more information, see Gold Sequences.
Shift
— Offset of Gold sequence0
(default)  integer scalarOffset of Gold sequence from the initial time, specified as an integer scalar.
Output variablesize signals
— Option to output variablelength signalsoff
(default)  on
Select this parameter to enable variablelength output sequences during simulation. When you clear this parameter, the block outputs fixedlength sequences. When you select this parameter, the block can output variablelength sequences. For information about variablesize signals, see VariableSize Signal Basics (Simulink).
Maximum output size source
— Maximum output size sourceDialog parameter
(default)  Inherit from reference port
Select how to specify the maximum sequence output size.
Dialog parameter
— Select this value to
configure the block to use the Maximum output size
parameter setting as the maximum permitted output sequence length. The
oSiz input port specifies the current size of the
output signal, and the block output inherits the sample time from the input
signal. The input value of oSiz must be less than or
equal to the Maximum output size parameter.
Inherit from reference port
— Select this value
to enable the Ref input port and configure the block to
inherit the sample time, maximum size, and current output size from the
variablesized signal at the Ref input port. These set
the maximum permitted output sequence length.
To enable this parameter, select Output variablesize signals.
Maximum output size
— Maximum output size[10 1]
(default)  vector of the form [n 1]Specify the maximum output size for the block. n is a positive scalar.
Example: [10 1]
specifies a 10by1 maximum size for the output
signal.
To enable this parameter, and set Maximum output size source
to Dialog parameter
.
Data Types: double
Sample time
— Output sample time1
(default)Positive scalars specify the time in seconds between each
sample of the output signal. If you set the Sample time to
1
, the output signal inherits the sample time from downstream. For
information on the relationship between the Sample time and
Samples per frame parameters, see
Sample Timing.
To enable this parameter do not select Output variablesize signals.
Samples per frame
— Samples per frame1
(default)  positive integerSamples per frame, specified as a positive integer indicating the number of samples per frame in one channel of the output data. For information on the relationship between Sample time and Samples per frame, see Sample Timing.
To enable this parameter do not select Output variablesize signals.
Reset on nonzero input
— Reset output signaloff
(default)  on
Select this parameter to enable the Rst port. When a nonzero value is input at the Rst port, the internal shift registers are reset to the original values of the Initial states (1) and Initial states (2) parameters.
Output data type
— Output data typedouble
(default)  boolean
 Smallest unsigned integer
Output data type, specified as boolean
,
double
, or Smallest unsigned
integer
.
When set to Smallest unsigned integer
, the
output data type is selected based on the settings used in the Hardware Implementation Pane (Simulink) of
the Configuration Parameters dialog box of the model. If ASIC/FPGA is
selected in the Hardware Implementation pane, the output data type is the
ideal minimum onebit size (ufix(1)). For all other selections, it is an
unsigned integer with the smallest available word length large enough to fit
one bit, usually corresponding to the size of a char (uint8).
Data Types 

Multidimensional Signals 

VariableSize Signals 

The time between output updates is equal to the product of the
Samples per frame and Sample time parameter
values. For example, if Sample time and Samples per
frame each equal 1
, the block outputs a sample every
second. If you increase Samples per frame to 10, then the block outputs
a 10by1 vector every 10 seconds. This ensures that the equivalent output rate is not
dependent on the Samples per frame parameter.
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 parameter. 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, or between shifted versions of them, 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 block uses two PN sequence generators to generate the preferred pair of sequences. The block then XORs these sequences to produce the output Gold sequence, as shown in this figure.
The Preferred polynomial (1) and Preferred polynomial (2) parameters 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 PN Sequence Generator block reference page.
This table provides examples of preferred pairs.
Degree of Generator Polynomials (n)  Pair of Sequences Period (N)  Preferred Polynomial (1) parameter value  Preferred Polynomial (2) parameter 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 Initial states (1) and Initial states (2) parameters are vectors specifying the initial values of the registers corresponding to Preferred polynomial (1) and Preferred polynomial (2), respectively.
Note
At least one element of the initial states vectors ( Initial states (1) or Initial states (2)) must be nonzero in order for the block to generate a nonzero sequence. Specifically, the initial state of at least one of the registers must be nonzero.
You can shift the starting point of the Gold sequence with the Shift parameter, which is an integer representing the length of the shift.
You can use an external signal to reset the values of the internal shift register to the initial state by selecting Reset on nonzero input. This creates an input port for the external signal in the block. The way the block resets the internal shift register depends on whether its output signal and the reset signal are scalar or vector. For more information, see Reset Behavior.
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 Preferred polynomial (1) and Preferred polynomial (2) parameters.
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.
$$\mathrm{gcd}(n,k)=\{\begin{array}{cc}1& n\equiv 1\mathrm{mod}2\\ 2& n\equiv 2\mathrm{mod}4\end{array}$$
The Sequence index parameter specifies which Gold sequence in the set G(u, v) is output. The range of Sequence index is [–2, –1, 0, 1, 2, ..., 2^{n}–2], where n is the degree of the generator polynomials specified by the Preferred polynomial (1) and Preferred polynomial (2) parameters. This table shows the correspondence between Sequence index and the output sequence.
The sequence index specified by the Sequence index parameter 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 Sequence index is [–2,
2^{n}–2], where
n is the degree of the generator polynomials specified by the
Preferred polynomial (1) and Preferred polynomial
(2) parameters. The index values 2
and
1
correspond to the first and second preferred PN sequences
as generated by Preferred polynomial (1) and
Preferred polynomial (2), respectively. This table shows
the correspondence between the sequence index and the output sequence.
Sequence Index  Output Sequence 

–2  u 
–1  v 
0 
$$u\oplus v$$ 
1 
$$u\oplus Tv$$ 
2 
$$u\oplus {T}^{2}v$$ 
...  ... 
2 ^{n} –2 
$$u\oplus {T}^{{2}^{n}2}v$$ 
Before you can reset the generator sequence, you must select the
Reset on nonzero input parameter to enable the
Rst input port. Suppose that the Gold Sequence Generator block outputs
[1 0 0 1 1 0 1 1]
when no reset exists. This table shows the effect
on the Gold Sequence Generator block output for the parameter values indicated.
Reset Signal  Reset Signal Settings  Gold Sequence Generator block  Reset Signal and Output Signal 

No reset 



Scalar reset signal 



Vector reset signal 


For the noreset case, the block outputs the sequence without resetting it.
For the scalar and vector reset signal cases, the block inputs the reset signal [0
0 0 1 0 0 0 0]
to the Rst port. Because the fourth bit of
the reset signal is a 1
and Sample time is
1
, the block resets the sequence output at the fourth bit.
For variablesized outputs, the block supports only scalar reset signal inputs.
The Gold Sequence Generator Reset Behavior example demonstrates the reset behavior in a Simulink^{®} model.
Behavior changed in R2020a
Starting in R2020a, Simulink no longer allows you to use the Gold Sequence Generator block version available before R2015b.
Existing models automatically update to load the current Gold Sequence Generator block version. For more information on block forwarding, see Maintain Compatibility of Library Blocks Using Forwarding Tables (Simulink).
[1] Proakis, John G. Digital Communications. 3rd ed. New York: McGraw Hill, 1995.
[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.
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