comm.GoldSequence

Generate Gold sequence from set of sequences

Description

The comm.GoldSequence System object™ generates a binary sequence with small periodic cross-correlation properties from a bounded set of sequences. For more information, see Gold Sequences.

To generate a Gold sequence from set of sequences:

1. Create the comm.GoldSequence object and set its properties.

2. 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

Syntax

goldseq = comm.GoldSequence

goldseq = comm.GoldSequence(Name=Value)

Description

goldseq = comm.GoldSequence creates a Gold sequence generator System object. This object generates a pseudorandom Gold sequence of binary numbers.

goldseq = comm.GoldSequence(Name=Value) sets properties using one or more name-value arguments. Enclose each property name in quotes. For example, 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 | binary-valued row vector | integer-valued 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.

• Binary-valued 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.

• Integer-valued 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) = z⁸ + z² + 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) | binary-valued scalar | binary-valued vector

Initial conditions used for the shift register of the first preferred PN sequence generator when the simulation starts, specified as a binary-valued scalar or binary-valued 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 | binary-valued row vector | integer-valued 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.

• Binary-valued 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.

• Integer-valued 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 the SecondInitialConditions property.

• 'Input port' — Specify PN sequence generator initial conditions by using the secondinitcond 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) | binary-valued scalar | binary-valued vector

Initial conditions used for the shift register of the second preferred PN sequence generator when the simulation starts, specified as a binary-valued scalar or binary-valued 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, 2ⁿ–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 variable-size outputs

false (default) | true

Enable variable-size outputs, specified as a numeric or logical 0 (false) or 1 (true). To enable variable-size 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-by-1.

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 = 2ⁿ – 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' |

Data type of the output, specified as 'double', 'logical', or 'Smallest unsigned integer'.

To use the Smallest unsigned integer option, you must have the Fixed-Point Designer™ product.

Data Types: string | char

Usage

Syntax

outSequence = goldseq()

outSequence = goldseq(secondinitcond)

outSequence = goldseq(outputsize)

outSequence = goldseq(resetseq)

outSequence = goldseq(secondinitcond,outputsize)

outSequence = goldseq(outputsize,resetseq)

Description

outSequence = goldseq() outputs a frame of the Gold sequence in a column vector based on the configured object.

outSequence = goldseq(secondinitcond) uses 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.

outSequence = goldseq(outputsize) uses outputsize as the output size.

To enable this syntax, set the VariableSizeOutput property to true.

outSequence = goldseq(resetseq) uses resetseq as the reset signal.

To enable this syntax, set the ResetInputPort property to true and the SecondInitialConditionsSource property to 'Property'.

outSequence = goldseq(secondinitcond,outputsize) specifies inputs for the second PN sequence initial conditions and the output size.

To enable this syntax, set the SecondInitialConditionsSource property to 'Input port', the ResetInputPort property to false, and set the VariableSizeOutput property to true.

outSequence = goldseq(outputsize,resetseq) specifies inputs for the output size and a reset signal.

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 binary-valued scalar or binary-valued 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 variable-size 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 Variable-Size 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)

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
reset	Reset internal states of System object

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 7-sample 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 7-sample 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 7-sample 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 15-sample 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 15-sample 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 0s. Display consecutive 15-sample 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 15-sample 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 cross-correlation 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 = 2ⁿ – 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

$$G(u,v)=\{u,v,u\oplus v,u\oplus Tv,u\oplus T^{2}v,\mathrm{...},u\oplus T^{N-1}v\}$$

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 cross-correlation 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 = 2ⁿ–1 to be a preferred pair, they must satisfy these requirements:

• n is the degree of the generator polynomials specified by the FirstPolynomial and SecondPolynomial 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.

• $$\gcd (n,k)=\{\begin{array}{cc}1& n\equiv 1\mathrm{mod}2\\ 2& n\equiv 2\mathrm{mod}4\end{array}$$

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

$$G(u,v)=\{u,v,u\oplus v,u\oplus Tv,u\oplus T^{2}v,\mathrm{...},u\oplus T^{N-1}v\}$$

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 = 2ⁿ–1.

The range of Index is [–2, 2ⁿ–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	$$u\oplus T^{2}v$$
...	...
2n –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 3-Valued Recursive Cross-Correlation 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