dpss

Discrete prolate spheroidal (Slepian) sequences

Syntax

dps_seq = dpss(seq_length,time_halfbandwidth)

[dps_seq,lambda] = dpss(seq_length,time_halfbandwidth)

[___] = dpss(seq_length,time_halfbandwidth,num_seq)

[___] = dpss(seq_length,time_halfbandwidth,interp_method)

[___] = dpss(___,Ni)

[___] = dpss(___,"trace")

Description

dps_seq = dpss(seq_length,time_halfbandwidth) returns the first round(2*time_halfbandwidth) discrete prolate spheroidal (DPSS) or Slepian sequences of length seq_length.

[dps_seq,lambda] = dpss(seq_length,time_halfbandwidth) returns the frequency-domain energy concentration ratios lambda of the column vectors in dps_seq. The ratios represent the amount of energy in the passband [–W,W] to the total energy from [–F_s/2,F_s/2], where F_s is the sample rate.

[___] = dpss(seq_length,time_halfbandwidth,num_seq) returns the first num_seq Slepian sequences with a time-half-bandwidth product equal to time_halfbandwidth. The function returns the sequences in order of their energy concentration ratios.

[___] = dpss(seq_length,time_halfbandwidth,interp_method) uses interpolation to compute the DPSSs from a user-created database of DPSSs. Create the database of DPSSs with dpsssave and ensure that the resulting file, dpss.mat, is in the MATLAB^® search path.

[___] = dpss(___,Ni) interpolates from DPSSs of length Ni in the database dpss.mat.

[___] = dpss(___,"trace") prints the method used to compute the DPSSs in the command window. Possible methods include direct, spline interpolation, and linear interpolation.

Examples

Generate Set of Slepian Sequences

Construct the first four discrete prolate spheroidal sequences of length 512. Specify a time-half-bandwidth product of 2 and include the "trace" flag to print the algorithm used to compute the sequences.

seq_length = 512; 
time_halfbandwidth = 2;
num_seq = 4;
[dps_seq,lambda] = dpss(seq_length,time_halfbandwidth,num_seq,"trace");

Computing the DPSS using direct algorithm...

Plot the sequences and display the concentration ratios.

plot(dps_seq)
title("Slepian Sequences, N = 512, NW = 2")
axis([0 512 -0.15 0.15])
legend("1st","2nd","3rd","4th")

concentration_ratios = lambda'

concentration_ratios = 1×4

    0.9999    0.9976    0.9594    0.7218

Use the dpsssave function with the time-half-bandwidth product, sequence length, and concentration ratios as inputs to create a database of Slepian sequences.

dpsssave(time_halfbandwidth,dps_seq,lambda); 

Compute discrete prolate spheroidal sequences of length 256 from dpss.mat using spline interpolation. Plot the results.

spline_dps_seq = dpss(256,time_halfbandwidth,"spline");
plot(spline_dps_seq)
title("Interpolated Slepian Sequences, N = 256, NW = 2")
axis([0 512 -0.15 0.15])
legend("1st","2nd","3rd","4th")

Input Arguments

seq_length — Sequence length

positive integer

Sequence length, specified as a positive integer.

Data Types: single | double

time_halfbandwidth — Time-half-bandwidth product

positive scalar

Time-half-bandwidth product, specified as a positive scalar. This argument must be less than seq_length/2.

Data Types: single | double

num_seq — Number of sequences

positive integer | two-element vector

Number of sequences to return, specified as a positive integer or two-element vector. If you specify num_seq as a two-element vector, then the output Slepian sequences range from num_seq(1) to num_seq(2).

Data Types: single | double

interp_method — Interpolation method

"Spline" | "linear"

Interpolation method, specified as "spline" or "linear". The interpolation method uses the Slepian sequences in the database with a time-half-bandwidth product equal to time_halfbandwidth and length closest to seq_length.

Data Types: char | string

Ni — Length of DPSSs

positive integer

Length of DPSSs, specified as a positive integer.

Data Types: single | double

Output Arguments

dps_seq — Slepian sequences

matrix

Slepian sequences, returned as a matrix with a number of rows equal to seq_length and columns equal to round(2*time_halfbandwidth).

lambda — Frequency-domain energy concentration ratios

column vector

Frequency-domain energy concentration ratios, returned as a column vector of length equal to the number of Slepian sequences.

More About

Discrete Prolate Spheroidal Sequences

The discrete prolate spheroidal or Slepian sequences derive from the following time-frequency concentration problem. For all finite-energy sequences $$x[n]$$ index limited to some set $$[N_1,N_1+N_2]$$, which sequence maximizes this ratio

$$\lambda =\frac{{\displaystyle \underset{-W}{\overset{W}{\int }}|X(f){|}^2}df}{{\displaystyle \underset{-Fs/2}{\overset{Fs/2}{\int }}|X(f){|}^{2}df}}$$

where F_s is the sample rate and $$\left|W\right|<Fs/2$$. Accordingly, this ratio determines which index-limited sequence has the largest proportion of its energy in the band [–W,W]. For index-limited sequences, the ratio must satisfy the inequality $$0<\lambda <1$$. The sequence maximizing the ratio is the first discrete prolate spheroidal or Slepian sequence. The second Slepian sequence maximizes the ratio and is orthogonal to the first Slepian sequence. The third Slepian sequence maximizes the ratio of integrals and is orthogonal to both the first and second Slepian sequences. Continuing in this way, the Slepian sequences form an orthogonal set of bandlimited sequences.

Time Half Bandwidth Product

The time half bandwidth product is NW where N is the length of the sequence and [–W,W] is the effective bandwidth of the sequence. In constructing Slepian sequences, you choose the desired sequence length and bandwidth 2W. Both the sequence length and bandwidth affect how many Slepian sequences have concentration ratios near one. As a rule, there are 2NW – 1 Slepian sequences with energy concentration ratios approximately equal to one. Beyond 2NW – 1 Slepian sequences, the concentration ratios begin to approach zero. Common choices for the time half bandwidth product are: 2.5, 3, 3.5, and 4.

You can specify the bandwidth of the Slepian sequences in Hz by defining the time half bandwidth product as NW/F_s, where F_s is the sample rate.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• All inputs must be constants. Expressions or variables are allowed if their values do not change.

Version History

Introduced before R2006a