Digital Signal Processing Reference
In-Depth Information
approaches to processing transformed nonuniformly sampled signals have been
developed. Some of the methods and techniques used for this type of signal pro-
cessing and their potential are discussed in this chapter.
It is worth paying a lot of attention to calculating raw DFT of nonuniformly
sampled signals, as the estimates obtained in this way play a vital role in the
subsequent digital processing of the transformed signals. Such DFT support all
kinds of digital signal filtering and other signal processing procedures, including
signal waveform reconstruction, and the quality of the final signal processing
results often depends on the performance of the involved DFT.
15.1 Problems Related to Sampling Irregularities
To achieve the capability of processing signals in a broad frequency range not
limited by half of the used sampling rate, sampling must be nonuniform. However,
once the signal digitizing is based on nonuniform sampling, processing of the
signals then has to be organized with the specifics of this technique taken into
account. Apparently it is essential to obtain a clear insight into the details of
processing this type of digital signal.
15.1.1 Alternative Approaches to DFT
General considerations of this have been given in Chapter 3. If the sampling point
process satisfies condition (3.13), then it follows from Equation (3.14) that
E
lim
N
a
i
E
lim
N
b
i
=
a
i
and
=
b
i
.
⇒∞
⇒∞
This means that asymptotically there should be no bias error and the statistical
estimation errors should tend to
c
2
N
(
c
is the amplitude of the signal component
being estimated). Therefore, for large
N
/
interval systematic bias and statistical
error at least should be small. That would be so if the side effect caused by the
nonorthogonality of the basis functions and the cross-interference effect due to
it did not exist. This effect certainly cannot be ignored. If these considerations
are taken into account, the following randomized estimation schemes can be
suggested:
,
1. The signal
x
(
t
) is sampled pseudo-randomly at predetermined time instants
{
. The values of cos(2
π
f
i
t
k
) and sin(2
π
f
i
t
k
) are calculated beforehand, stored
in a memory and then later used for estimation. To reduce estimation errors to
an acceptable level by averaging, a relatively large data block is processed. The
errors due to the cross-interference remain, fact that has to be taken into account
at further processing of the obtained raw estimates of the Fourier coefficients.
t
k
}
