Kasami Sequence Generator

Generate Kasami sequence from set of Kasami sequences

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Libraries:
Communications Toolbox / Comm Sources / Sequence Generators

Description

The Kasami Sequence Generator block generates a sequence from a set of Kasami sequences. The Kasami sequences are a set of sequences that have good cross-correlation properties. For more information, see Kasami Sequences.

This block can output sequences that vary in length during simulation. For more information about variable-size signals, see Variable-Size Signal Basics (Simulink).

These icons show the block with oSiz, Ref and Rst ports enabled.

Examples

Kasami Spreading for Multiuser System in Multipath Channel

This model simulates Kasami sequence spreading for two users in a multipath transmission environment. This is similar to a mobile channel environment where the signals are received over multiple paths. Each path can have different amplitudes and delays. The receiver combines the independent paths coherently using diversity reception to realize gains from the multipath transmissions received. The modeled system does not simulate fading effects and the receiver gets perfect knowledge of the number of paths and their respective delays.

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Ports

Input

oSiz — Current output size

scalar | vector

Current output size, specified as a scalar or a vector of the form [n,1], where n is the number of elements in the output sequence.

Example: [10 1] a specifies a current output column vector of size 10-by-1.

Dependencies

To enable this port, set the Maximum output size source parameter to Dialog parameter.

Data Types: double

Ref — Reference input signal

scalar | column vector

Reference input signal, specified as a scalar, or a column vector.

Dependencies

To enable this port, set the Maximum output size source parameter to Inherit from reference input.

Data Types: double

Rst — Reset signal

scalar | vector

Reset signal, specified as a scalar or vector, depending on the output size.

• When the output size is variable, specify this port as a scalar.

• Otherwise, specify this port as a scalar or a column vector of length equal to the Samples per frame parameter value.

The output signal resets for nonzero Rst input values. For more information, see Reset Behavior.

Dependencies

To enable this port, select the Reset on nonzero input parameter.

Data Types: double

Output

Out — Output Signal

vector

Output signal, returned as a binary-valued column vector. At least one element of the Initial states parameter vector must be nonzero for the block to generate a nonzero sequence.

Data Types: double

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Generator polynomial — Generator polynomial

'z^6 + z + 1' (default) | polynomial character vector | string scalar | binary-valued row vector | integer-valued row vector

Specify the generator polynomial, which determines the connections in the shift register that generates the sequence, as one of these options.

• A polynomial character vector or string scalar that includes the number 1 (for example, 'z^4 + z + 1'). For more information, see Representation of Polynomials in Communications Toolbox.

• A binary-valued row vector that lists the coefficients of a polynomial in order of descending powers. The first and last entries must be 1. The length of this vector must be one more than the degree of the generator polynomial.

• An integer-valued row vector containing the exponents of the nonzero terms of a polynomial in order of descending powers. The last entry must be 0.

For 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.

Initial states — Initial states

[0 0 0 0 0 1] (default) | binary-valued scalar | binary-valued row vector

The initial states of the shift register generate the sequence. If you specify a binary-valued row vector, the length must equal the degree of the generator polynomial specified by the Generator polynomial parameter. If you specify a binary-valued scalar, the block expands the scalar to a row vector of length equal to the degree of the generator polynomial. All entries of the resulting vector equal the scalar.

Sequence index(es) — Sequence index

0 (default) | integer | vector of the form [k m]

Specify an integer or vector of the form [k m] to select a Kasami sequence of interest from the set of possible sequences. Two classes of Kasami sequences exist: those obtained from a small set and those obtained from a large set. You can choose a Kasami sequence from the small set by setting this parameter to an integer in the range [0, 2^n/2–2]. You can choose a sequence from the large set by setting this parameter to a vector of the form [k m]. k must be an integer in the range [–2, 2ⁿ–2], and m must be an integer in the range [–1, 2^n/2–2]. For more information, see Sequence Index.

Shift — Sequence offset from starting point

0 (default) | integer

Specify the offset of the Kasami sequence from the initial time.

You can use an external signal to reset the values of the internal shift register to the initial state by selecting the Reset on nonzero input parameter. This selection creates an input port for the external signal in this block. The way the block resets the internal shift register depends on whether its output signal and reset signal are sample-based or frame-based. For an example, see Reset Behavior.

Output variable-size signals — Option to output variable-length signals

off (default) | on

Select this parameter to enable variable-length output sequences during simulation. When you clear this parameter, the block outputs fixed-length sequences. When you select this parameter, the block can output variable-length sequences. For information about variable-size signals, see Variable-Size Signal Basics (Simulink).

Maximum output size source — Maximum output size source

Dialog 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 variable-sized signal at the Ref input port. These set the maximum permitted output sequence length.

Dependencies

To enable this parameter, select Output variable-size 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 10-by-1 maximum size for the output signal.

Dependencies

To enable this parameter, select Output variable-size signals and set Maximum output size source to Dialog parameter.

Sample time — Output sample time

1 (default) | -1 | positive scalar

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.

Dependencies

To enable this parameter clear Output variable-size signals.

Samples per frame — Samples per frame

1 (default) | positive integer

Specify the number of samples per frame in one channel of the output data. For information on the relationship between the Sample time and Samples per frame parameters, see Sample Timing.

Reset on nonzero input — Option to reset output signal

off (default) | on

Select this parameter to enable the Rst input port. Use that port to specify an input signal that resets the internal shift registers to the original values of the Initial states parameter value.

Output data type — Output data type

double (default) | boolean

Specify the output data type of the block.

Block Characteristics

Data Types	Boolean | double
Multidimensional Signals	no
Variable-Size Signals	yes

More About

Sample Timing

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 one sample every second. If you increase Samples per frame to 10, then the block outputs a 10-by-1 vector every 10 seconds. This timing ensures that the equivalent output rate is not dependent on the Samples per frame parameter.

Kasami Sequences

Two sets of Kasami sequences exist: the small set and the large set. The large set contains all of the sequences in the small set. Only the small set is optimal in the sense of matching Welch's lower bound for correlation functions.

Kasami sequences have a period of N = 2ⁿ – 1, where n is a nonnegative even integer. Let u be a binary sequence of length N, and let w be the sequence obtained by decimating u by 2^n/2 + 1. This piecewise function defines the small set of Kasami sequences. T is the left shift operator, m is the shift parameter for w, and $$\oplus$$ denotes addition modulo 2.

$$K_{\text{s}}(u,n,m)=\{\begin{array}{ll}u\hfill & m=-1\hfill \\ u\oplus T^{m}w\hfill & m=0,\mathrm{...},2^{n/2}-2\hfill \end{array}$$

The small set contains 2^n/2 sequences.

For mod(n, 4) = 2, this piecewise function defines the large set of Kasami sequences. Let v be the sequence formed by decimating the sequence u by 2^{(n/2 + 1)} + 1. k and m are the shift parameters for the sequences v and w, respectively.

$$K_{\text{L}}(u,n,k,m)=\{\begin{array}{ll}u\hfill & k=-2;m=-1\hfill \\ v\hfill & k=-1;m=-1\hfill \\ u\oplus T^{k}v\hfill & k=0,\mathrm{...},2^n-2;m=-1\hfill \\ u\oplus T^{m}w\hfill & k=-2;m=0,\mathrm{...},2^{n/2}-2\hfill \\ v\oplus T^{m}w\hfill & k=-1;m=0,\mathrm{...},2^{n/2}-2\hfill \\ u\oplus T^{k}v\oplus T^{m}w\hfill & k=0,\mathrm{...},2^n-2;m=0,\mathrm{...},2^{n/2}-2\hfill \end{array}$$

The sequences described in the first three rows of K_L correspond to the Gold sequences for mod(n, 4) = 2. For a description of Gold sequences, see the comm.GoldSequence System object™ reference page. However, the Kasami sequences form a larger set than the Gold sequences alone.

The correlation functions for the sequences take on the values

{–t(n), –s(n), –1, s(n) – 2, t(n) – 2},

where

$$\begin{array}{c}t(n)=1+2^{(n+2)/2}\text{, when }n\text{ is even and}\\ s(n)=\frac{1}{2}\left(t(n)+1\right).\end{array}$$

Polynomials for Generating Kasami sequences

Kasami sequences have a period of N = 2ⁿ – 1, where n is a nonnegative even integer. This table lists some of the polynomials that you can use to generate the set of Kasami sequences.

n	N	Polynomial	Set
4	15	'z^4 + z +1'	Small
6	63	'z^6 + z +1'	Large
8	255	'z^8 + z^4 + z^3 + z^2 +1'	Small
10	1023	'z^10 + z^3 +1'	Large
12	4095	'z^12 + z^6 + z^4 + z +1'	Small

Sequence Index

The Sequence index(es) parameter specifies the shifts of the sequences v and w used to generate the output sequence. You can specify the parameter in one of these two ways.

• To generate sequences from the small set, when n is even, you can specify Sequence index(es) as an integer m in the range [–1, 2^n/2 – 2]. This table describes the output sequences corresponding to Sequence index(es) values.

  Sequence index(es) Value	Range of Indices	Output Sequence
  -1	m = –1	u
  m	m = 0, 2n/2 – 2	$$u\oplus T^{m}w$$

• To generate sequences from the large set for mod (n, 4) = 2, where n is the degree of the generator polynomial (set by the Generator polynomial parameter), you can specify Sequence index(es) as a vector of the form [k m]. In this case, the output sequence is from the large set k is an integer in the range [–2, 2ⁿ – 2], and m is an integer in the range [–1, 2^n/2 – 2]. This table describes the output sequences corresponding to Sequence index(es) values.

  Sequence index(es) Value	Range of Indices	Output Sequence
  [-2 -1]	k = –2 m = –1	u
  [-1 -1]	k = –1 m = –1	v
  [k –1]	k = 0, 1, ..., 2n – 2 m = –1	$$u\oplus T^{k}v$$
  [-2 m]	k = – 2 m = 0, 1, ..., 2n/2 – 2	$$u\oplus T^{m}w$$
  [-1 m]	k = –1 m = 0, ..., 2n/2 –2	$$v\oplus T^{m}w$$
  [k m]	k = 0, ..., 2n –2 m = 0, ..., 2n/2 –2	$$u\oplus T^{k}v\oplus T^{m}w$$

Reset Behavior

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 Kasami Sequence Generator block outputs [1 0 0 1 1 0 1 1] when no reset exists. This table shows the effect on the Kasami Sequence Generator block output for the parameter values indicated.

Reset Signal	Reset Signal Settings	Kasami Sequence Generator block	Reset Signal and Output Signal
No reset	Sample time is 1 Samples per frame is 1 Rst is [0 0 0 0 0 0 0 0]	Sample time is 1 Samples per frame is 1 Out is [1 0 0 1 1 0 1 1]
Scalar reset signal	Sample time is 1 Samples per frame is 1 Rst is [0 0 0 1 0 0 0 0]	Sample time = 1 Samples per frame is 1
Vector reset signal	Sample time is 1 Samples per frame is 8 Rst is [0 0 0 1 0 0 0 0]	Sample time is 1 Samples per frame is 8

For the no-reset 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 variable-sized outputs, the block supports only scalar reset signal inputs.

References

[1] Peterson, W. Wesley, and E. J. Weldon. Error-Correcting Codes. 2d ed. MIT Press, 1972.

[2] Proakis, John G. Digital Communications. 4th ed. New York: McGraw Hill, 2001.

[3] 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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced before R2006a

R2020a: Existing models automatically update this block to current version

Starting in R2020a, Simulink no longer allows you to use the Kasami Sequence Generator block version available before R2015b.

Existing models automatically update to load the Kasami Sequence Generator block version announced in Source blocks output frames of contiguous time samples but do not use frame attribute. For more information on block forwarding, see Maintain Compatibility of Library Blocks Using Forwarding Tables (Simulink).