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The Gold Sequence Generator block generates a Gold sequence. Gold sequences form a large class of sequences that have good periodic cross-correlation properties.
This block can output sequences that vary in length during simulation. For more information about variable-size signals, see Variable-Size Signal Basics in the Simulink^{®} documentation.
The Gold sequences are defined using a specified pair of sequences u and v, of period N = 2^{n} - 1, called a preferred pair, as defined in Preferred Pairs of Sequences below. The set G(u, v) of Gold sequences is defined by
$$G(u,v)=\{u,v,u\oplus v,u\oplus Tv,u\oplus {T}^{2}v,\mathrm{...},u\oplus {T}^{N-1}v\}$$
where T represents the operator that shifts vectors cyclically to the left by one place, and $$\oplus $$ represents addition modulo 2. Note that G(u,v) contains N + 2 sequences of period N. The Gold Sequence Generator block outputs one of these sequences according to the block's parameters.
Gold sequences have the property that the cross-correlation 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 Gold Sequence Generator block uses two PN Sequence Generator blocks to generate the preferred pair of sequences, and then XORs these sequences to produce the output sequence, as shown in the following diagram.
You can specify the preferred pair by the Preferred polynomial [1] and Preferred polynomial [2] parameters in the dialog for the Gold Sequence Generator block. These polynomials, both of which must have degree n, describe the shift registers that the PN Sequence Generator blocks use to generate their output. For more details on how these sequences are generated, see the reference page for the PN Sequence Generator block. You can specify the preferred polynomials using either of the following formats:
A vector that lists the coefficients of the polynomial in descending order of powers. The first and last entries must be 1. Note that the length of this vector is one more than the degree of the generator polynomial.
A vector containing the exponents of z for the nonzero terms of the polynomial in descending order of powers. The last entry must be 0.
For example, the vectors [5 2 0] and [1 0 0 1 0 1] both represent the polynomial z^{5} + z^{2} + 1.
The following table provides a short list of preferred pairs.
n | N | Preferred Polynomial[1] | Preferred Polynomial[2] |
---|---|---|---|
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. These parameters must satisfy these criteria:
All elements of the Initial states[1] and Initial states[2] vectors must be binary numbers.
The length of the Initial states[1] vector must equal the degree of the Preferred polynomial[1], and the length of the Initial states[2] vector must equal the degree of the Preferred polynomial[2].
The Sequence index parameter specifies which sequence in the set G(u, v) of Gold sequences the block outputs. The range of Sequence index is [-2, -1, 0, 1, 2, ..., 2^{n}–2]. The correspondence between Sequence index and the output sequence is given in the following table.
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 | $$u\oplus {T}^{{2}^{n}-2}v$$ |
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 Gold Sequence Generator block. The way the block resets the internal shift register depends on whether its output signal and the reset signal are sample-based or frame-based. The following example demonstrates the possible alternatives. See Example: Resetting a Signal for an example.
The requirements for a pair of sequences u, v of period N = 2^{n}–1 to be a preferred pair are as follows:
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
$$\mathrm{gcd}(n,k)=\{\begin{array}{cc}1& n\equiv 1\mathrm{mod}2\\ 2& n\equiv 2\mathrm{mod}4\end{array}$$
Vector specifying the polynomial for the first sequence of the preferred pair.
Vector of initial states of the shift register for the first sequence of the preferred pair.
Vector specifying the polynomial for the second sequence of the preferred pair.
Vector of initial states of the shift register for the second sequence of the preferred pair.
Integer specifying the index of the output sequence from the set of sequences.
Integer scalar that determines the offset of the Gold sequence from the initial time.
Select this check box if you want the output sequences to vary in length during simulation. The default selection outputs fixed-length signals.
Specify how the block defines maximum output size for a signal.
When you select Dialog parameter, the value you enter in the Maximum output size parameter specifies the maximum size of the output. When you make this selection, the oSiz input port specifies the current size of the output signal and the block output inherits sample time from the input signal. The input value must be less than or equal to the Maximum output size parameter.
When you select Inherit from reference port, the block output inherits sample time, maximum size, and current size from the variable-sized signal at the Ref input port.
This parameter only appears when you select Output variable-size signals. The default selection is Dialog parameter.
Specify a two-element row vector denoting the maximum output size for the block. The second element of the vector must be 1 For example, [10 1] gives a 10-by-1 maximum sized output signal. This parameter only appears when you select Output variable-size signals.
Period of each element of the output signal.
Determines whether the output is frame-based or sample-based.
The number of samples in a frame-based output signal. This field is active only if you select Frame-based outputs.
When selected, you can specify an input signal that resets the internal shift registers to the original values of the Initial states parameter
The output type of the block can be specified as boolean, double or Smallest unsigned integer. By default, the block sets this to double.
When the parameter is set to Smallest unsigned integer, the output data type is selected based on the settings used in the Hardware Implementation pane 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 one-bit size, i.e., 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 (e.g., uint8).
[1] Proakis, John G., Digital Communications, Third edition, New York, McGraw Hill, 1995.
[2] Gold, R., "Maximal Recursive Sequences with 3-valued Recursive Cross-Correlation Functions," IEEE Trans. Infor. Theory, Jan., 1968, pp. 154-156.
[3] Gold, R., "Optimal Binary Sequences for Spread Spectrum Multiplexing, IEEE Trans. Infor. Theory, Oct., 1967, pp. 619-621.
[4] Sarwate, D.V., and M.B. Pursley, "Crosscorrelation Properties of Pseudorandom and Related Sequences," Proc. IEEE, Vol. 68, No. 5, May, 1980, pp. 583-619.
[5] Dixon, Robert, Spread Spectrum Systems with Commercial Applications, Third Edition, Wiley–Interscience, 1994.