Discrete prolate spheroidal (Slepian) sequences
[___] = dpss(___,
interpolates from DPSSs of length
Ni in the database
[___] = dpss(___,"trace") prints the
method used to compute the DPSSs in the command window. Possible methods include
direct, spline interpolation, and linear interpolation.
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
dpsssave function with the time-half-bandwidth product, sequence length, and concentration ratios as inputs to create a database of Slepian sequences.
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")
seq_length — Sequence length
Sequence length, specified as a positive integer.
time_halfbandwidth — Time-half-bandwidth product
Time-half-bandwidth product, specified as a positive scalar. This argument
must be less than
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
interp_method — Interpolation method
Ni — Length of DPSSs
Length of DPSSs, specified as a positive integer.
dps_seq — Slepian sequences
Slepian sequences, returned as a matrix with a number of rows equal to
seq_length and columns equal to
lambda — Frequency-domain energy concentration ratios
Frequency-domain energy concentration ratios, returned as a column vector of length equal to the number of Slepian sequences.
Discrete Prolate Spheroidal Sequences
The discrete prolate spheroidal or Slepian sequences derive from the following time-frequency concentration problem. For all finite-energy sequences index limited to some set , which sequence maximizes this ratio
where Fs is the sample rate and . 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 . 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/Fs, where Fs is the sample rate.
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.
Introduced before R2006a