legendreseq

R2026b

Legendre sequence with perfect periodic autocorrelation function

Since R2024a

collapse all in page

    Syntax

    S = legendreseq(N)

    Description

    S = legendreseq(N) generates a length-N Legendre sequence with perfect periodic autocorrelation function (PACF) S. Sidelobe levels of a PACF are zero. Legendre sequences are also known as quadratic residue sequences. The input N specifies the length of the generated sequence. N must be an odd prime number.

    • If N is a prime of a form N = 4n – 1, where n is a positive integer, the generated sequence is biphase. That is, the sequence elements take two distinct values.

    • If N is a prime of a form N = 4n – 3, where n is a positive integer, the generated sequence is three-phase. The third value appears only once in the first element of the sequence.

    example

    Examples

    collapse all

    Open Live Script

    Generate a phase-coded waveform with a perfect periodic autocorrelation function (PACF). The number of chips (N) is 61. The chip width (tau) is 1e-6 and the modulation period is T*tau sec. The sampling frequency (fs) is 10/tau. Plot the periodic ambiguity function to verify that the PACF has a zero sidelobe level.

    Use a legendre sequence for the code.

    N = 61;
s = legendreseq(N);

    Set the chip width and sampling freqyency.

    tau = 1e-6;
T = tau*N;
fs = 10/tau;

    Create a custom phase-coded waveform System object™ and generate waveform samples. Set the PRF to be equal to the inverse of the modulation period.

    waveform = phased.PhaseCodedWaveform(SampleRate=fs,Code="Custom", ...
    CustomCode=s,ChipWidth=tau,PRF=1/T);
x = waveform();

    Plot the periodic ambiguity function and show that it has zero sidelobe.

    pambgfun(x,fs,Cut="Doppler")

    Figure contains an axes object. The axes object with title Periodic Ambiguity Function, 0 MHz Doppler Cut, xlabel Delay tau (µs), ylabel Periodic Ambiguity Function contains an object of type line.

    Input Arguments

    collapse all

    Code sequence length, specified as an odd-valued positive integer.

    • If N is a prime of the form N = 4n – 1, where n is a positive integer, the generated sequence is biphase. The sequence elements take two distinct values.

    • If N is a prime of the form N = 4n – 3, where n is a positive integer, the generated sequence is three-phase. The third value appears only once in the first element of the sequence.

    Example: 11

    Data Types: double

    Output Arguments

    collapse all

    Code sequence, returned as a length-N complex-valued vector.

    Data Types: double
    Complex Number Support: Yes

    References

    [1] Levanon, N. and E. Mozeson. Radar Signals. Hoboken, NJ: John Wiley & Sons, 2004.

    [2] Farnett, Edward C., George H. Stevens, and Merrill Skolnik. "Pulse compression radar." In Radar Handbook Second Edition (1990).

    [3] Bomer, L., and Markus Antweiler. "Perfect three-level and three-phase sequences and arrays.", IEEE Transactions on Communications 42, no. 234 (1994): 767–772.

    Extended Capabilities

    expand all

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

    Version History

    Introduced in R2024a

    See Also

    | | | | | |