Digital Signal Processing Reference
In-Depth Information
transforms can be applied. In this case additional estimation errors will result
from sampling irregularities. If this is acceptable, irregularly sampled signals can
be transformed in this way.
14.4
Discrete Unorthogonal Transforms
A signal tra
nsfor
m is considered unorthogonal if the system of functions
Φ
=
{
φ
i
(
t
k
)
,
k
=
1
,
N
}
, applied to perform the transform, is unorthogonal. It is agreed
Φ
that the system
does not contain any f
unct
ions identically equal to zero. Under
2
φ
i
(
t
)
>
=
,
this condition,
m
.
A few typical signal processing applications will now be mentioned where the
signals need to be transformed on the basis of unorthogonal transforms:
0 for all
i
1
1. Signal transforms carried out by means of a system of analog functions, which
are originally orthogonal and become unorthogonal in the course of digitizing,
as when the sampling applied is irregular. As an example to illustrate this
case, refer to Chapter 15, where the discrete Fourier transforms of randomly
sampled signals are discussed.
2. Short-time periodic signal spectrum analysis, when the signals have to be
transformed under the condition that the observation time interval [0,
Θ
]is
shorter than the given period of the signal. In this case the frequencies of
the true signal components are known but the system of the functions
{
φ
(
t
)
}
,
chosen correspondingly, is unorthogonal.
3. Spectrum analysis of quasi-periodic signals, which are actually nonstationary
with a slowly varying period.
4. Spectrum analysis of signals containing components at frequencies irregularly
spaced along the frequency axis.
5. Decomposing signals into true components, which are mutually unorthogonal.
Even this incomplete list of unorthogonal transform applications shows that the
problems approached in this way are significant. In this topic, the most interest
lies in the first application.
Now consider a system of functions
Φ
={
φ
i
(
t
k
)
,
k
=
1
,
N
}
,
i
=
1
,
m
, which
is unorthogonal. Some nonzero coefficients
i
, which are not placed on
the diagonal of the corresponding matrix defined by the equation system (14.12),
definitely exist. For this reason, this equation system cannot b
e red
uced to the
system of equalities (14.20). Hence, the coefficients
α
,
j
=
ij
m
,havetobe
calculated on the basis of the equation system (14.12) and the condition (14.16)
should be met.
{
c
i
}
,
i
=
1
,
