Many algorithms used for digital signal processing, parameter estimation and digital measurement
technology have unsatisfactory accuracy due to the digitalization, unsuitable sampling conditions
and approximately mathematical methods etc. The calculation errors of these approximate algorithms
are so difficult to be determined theoretically that they cannot easily be compensated by conventional
correction methods. Therefore, selfcorrection (SC) algorithms are proposed to improve the accuracy
of digital signal processing and parameter determination.
A selfcorrection algorithm (Fig. 1) consists of three main parts, i.e., initialization,
selfcalibration, and errors correction.
In the initialization original output parameters Y_{m}^{*}
(m=1,2,..., M) are estimated by an approximate algorithm under correction (AUC) using input signals
X_{k}(i) (k=1,2,..., K; i=0,1,..., N1), where K and N denote the numbers
of input signals and the samples number of each input signal. These output parameters deviate from
their true values due to the inaccuracy of the AUC. The parameters Y_{m}^{*}
serve for the selfcalibration of the approximate algorithm to determine the calculation errors and
for the errors correction.
Fig. 1 selfcorrection algorithm of digital signal processing
Fig. 2 Selfcalibration of the algorithm under correction
To determine the calculation errors of the AUC, the original output parameters Y
_{m}^{*} are assigned as reference parameters for the selfcalibration
(Fig.2). Using the reference parameters Y_{m}^{*} reconstruction
data X_{rk}(i) (k=1,2,..., K; i=0,1,..., N1) are generated with
a corresponding model. Reference output parameters Y_{rm}^{*}
(m=1,2,..., M) are then estimated by the same AUC with the use of the reconstruction data X
_{rk}(i). The errors of the parameters calculated by the AUC can be determined by
if the reconstruction data X_{rk}(i) can accurately be calculated
with the corresponding model. After the selfcalibration errors correction is made by
The calculation errors of the AUC can thus be selfcorrected without any error analysis and
external reference.
Remark 1: The selfcorrection algorithm mentioned above bases on an accurately realizable
signal reconstruction without considering any signal noise. This condition limits the application
areas of the selfcorrection algorithm. For instance, this algorithm can be applied to a discrete
Fourier transform for processing spectral limited signals but not for spectral unlimited
signals, because a spectral unlimited signal cannot accurately be reconstructed with the inverse
Fourier transform due to the spectral leakage of higher frequency components.
Remark 2: The calculation errors of an approximate algorithm depends on the input signals
X_{k}(i). The reconstruction data X_{rk}(i)
are not the same of the input signals X_{k}(i), because the reference
parameter Y_{m}^{*} deviate from the true parameters. Therefore the
errors determined by the selfcalibration are not the true errors caused by the approximate algorithm (AUC).
This is why a recursive/iterative selfcorrection algorithm has been proposed.
For processing signals interfered with noise, however, only the first few selfcorrections are useful
for the accuracy improvement. The further iterative selfcorrections are not necessary for processing
practical signals. In most cases up to 3 selfcorrections are enough for the errors correction of an
approximate algorithm. As example Fig. 3 shows an algorithm that uses two selfcorrections.
Fig.3 Scheme of an iterative selfcorrection algorithm
In this case the reference parameters Y_{m,1} for the second selfcalibration
are more accurate to the true parameters than the reference parameters Y_{rm}^{*}
for the first selfcalibration, so that the errors DY_{m,2
} are more accurate to the true errors caused by the AUC in the initialization. The final output
parameters can be calculated by
where Y^{*}_{rm,1} and Y^{*}_{rm,2}
are the reference output parameters in the first and second selfcalibrations, respectively.
Generally, the final output parameters of an iterative selfcorrection algorithm are written by
with J as the number of iterative selfcorrections and Y^{*}_{rm,j}
as the reference output parameters in the jth selfcalibration.
Remark 3: The residual errors of the selfcalibration algorithm depends on the original
parameter errors caused by the approximate algorithm. A larger residual error is caused by a larger original error.
All original errors of calculated parameters must be less than 70 %, otherwise errors convergence of
the selfcalibration algorithm cannot be guaranteed. Therefore, it is useful to reduce the original
errors by using other methods in advance if the original errors are larger than the limit value.
These algorithms are applied to the discrete FourierSeries/Transformation
(DFT) of periodic signals sampled asychronously and to
parameter estimation of damped oscillation signals.
