The Line Spectral Frequency Model of a Finite-Length Sequence

The line spectral frequency (LSF) of a causal finite length sequence is a frequency at which the
spectrum
of the sequence
annihilates
or the magnitude spectrum has a
spectral null
. A causal finite-length sequence with (
L
+ 1) samples having exactly
L
-LSFs, is referred as an Annihilating (AH) sequence. Using some spectral properties of finite-length sequences, and some
model parameters
, we develop spectral decomposition structures, which are used to translate any finite-length sequence to an equivalent set of AH-sequences defined by LSFs and some complex constants. This alternate representation format of any finite-length sequence is referred as its
LSF-Model
. For a finite-length sequence, one can obtain multiple
LSF-Models
by varying the model parameters. The
LSF-Model
, in time domain can be used to synthesize any arbitrary causal finite-length sequence in terms of its characteristic AH-sequences. In the frequency domain, the
LSF-Model
can be used to obtain the spectral samples of the sequence as a linear combination of spectra of its characteristic AH-sequences. We also summarize the utility of the
LSF-Model
in practical discrete signal processing systems.