Digital Signal Processing Reference
In-Depth Information
sampling irregularities are similar to the iterative algorithm discussed above. The
basic difference is in the estimation of the Fourier coefficients for a group of signal
components. While standard DFTs are used for this in the case of the described
iterative algorithm, cross-interference coefficients are calculated and the effect of
the cross-interference is taken into account in the case of the algorithm adapted
to sampling irregularities. That changes the situation significantly. The precision
of Fourier coefficient estimation is substantially increased at each iteration cycle,
which leads to faster convergence to the final spectral estimates. Figure 20.3
illustrates this kind of adapted iterative signal processing.
It can be seen from Figure 20.3 that the estimation process develops quickly
in this case. A more detailed comparison of this type of algorithm with both
previously considered ones follows. Note that the illustrated spectrum analysis is
actually only a part of the whole signal processing process. Each adaptation cycle
contains calculations of direct and inverse DFTs. Therefore the signal waveform
is also repeatedly estimated with growing precision.
For calculations carried out during the process of this adaptation, at each itera-
tion cycle they typically cover a relatively small number of signal components at
arbitrary frequencies. Under these conditions, the cross-interference coefficients,
at separate adaptation cycles, are usually calculated for the particular group of
peaks in the signal spectrum that is processed for this cycle. This means that it is
then not necessary to calculate and use the matrix of the cross-interference coef-
ficients characterizing the respective nonuniform sampling point process used to
digitize the signal as described in Chapter 18. Direct on-line calculations of the
cross-interference coefficients are then much more productive and the embedded
systems shown in Figure 20.1 are based on this concept.
In cases where it is essential to achieve high operational speed, the discussed
adaptation process could be realized in accordance with the scheme given in
Figure 18.6. The used nonuniform sampling point process then has to be de-
composed into a number of periodic processes with pseudo-randomly skipped
sampling points. The signal sample values obtained at time instants defined by
each particular sampling point substream can then be processed separately. In this
way the sequential performance of a large number of required computational op-
erations could be replaced by parallel calculations of reduced complexity carried
out in parallel.
20.2.4 Comparison of Algorithm Performance
While all three types of the algorithms discussed above are applicable for adapt-
ing signal waveform reconstruction to the irregularities of the sampling process,
