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 is
a matrix with seq_length rows and round(2*time_halfbandwidth) columns. time_halfbandwidth must
be strictly less than seq_length/2.

[dps_seq,lambda] = dpss(seq_length,time_halfbandwidth) returns
the frequency-domain energy concentration ratios 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 sampling
frequency. lambda is a column vector equal in length
to the number of Slepian sequences.

[...] = dpss(seq_length,time_halfbandwidth,num_seq) returns
the first num_seq Slepian sequences with time half
bandwidth product time_halfbandwidth ordered by
their energy concentration ratios. If num_seq is
a two-element vector, the returned Slepian sequences range from num_seq(1) to num_seq(2).

[...] = 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. Valid options for 'interp_method' are 'spline' and 'linear'.
The interpolation method uses the Slepian sequences in the database
with time half bandwidth product time_halfbandwidth and
length closest to seq_length.

[...] = 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.

Construct the first four discrete prolate spheroidal sequences of length 512. Specify a time half bandwidth product of 2.5. Plot the sequences and find the concentration ratios.

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 the
following ratio:

where F_{s} is
the sampling frequency 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.

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 sampling
frequency.