Instrumentation & Measurement Magazine 24-4 - 14






Yf

11
M
1
Yf


1

 
1  sM s 





Hf M f H f M f
Hf
 
 

 f , of the last column of the matrix in (8), which is the inverse
of the matrix given in (7) particularized with h = 1.
Each system allows determining the reconstruction filters
in a single interval, and summing the solutions of all systems
provides the solution all over (-B, B) namely, for the l-th channel
 
and coincide with the functions, divided by fs
1

Yf Y f

l
lh
h 1
representation prevents that the solutions gained solving the
other systems of the set, corresponding to different values of
the parameter h, produce overlapping contributions in adjacent
intervals; the representations of the reconstruction filters
all over the interval (-B, B) can thus be attained by merely summing
the partial representations.
It can also be shown that there is a shortcut to obtain the
reconstruction filters all over (-B, B). In fact, each system matrix
contains shifted versions of the same functions, such that
the partial representation of the reconstruction filters related
to the h-th interval can be gained selecting the (M - h +1)-th
column in (8) and shifting it rightward by (h-1)fs
plying by h f .
  . The use of the characteristic function in the
M

HM f
11 0
1



and restricted by
1
(8)
1
fs
1 f

propagation paths typically exhibit irregular skews between
channels with respect to the nominal time offsets required
by the interleaving design. Therefore, real architectures use
digital signal processors to implement correction filters that
contrast the aforementioned issues and grant streamline calibration.
These filters constitute a core part of the firmware of
the DSO [15], [16]. The GSE-based framework provides a general
methodology for the identification of the correction filters,
that requires the knowledge of the actual frequency responses
Hl
(f) of the individual channels.
At the production stage, the frequency responses of the individual
channels are separately measured, typically through
sine wave tests, for a number of frequencies forming a uniform
grid, starting from the zero frequency and considering
L further frequencies. The tests exploit both the periodicity
of the calibration stimulus and the time-equivalent sampling
options typically available in DSOs, such that the frequency
response of each channel can even be determined over a bandwidth
greater than the Nyquist band related to the available
maximum real-time sample rate fs
of the adopted ADC. The
measured data are processed, through mirroring operations
that take into account the Hermitian symmetry nature, to obtain
a description of the channel frequency response on a
double-sided interval in terms of 2L + 1 bins. Specifically, measured
data are conveniently represented as a (2L +1)-length
complex sequence:
before multiFrom
an operative point of view, system (7) is more often
solved numerically for given frequencies, and the practitioner
must be careful to use the equation system valid for the given
frequencies, or apply the aforementioned shortcut, which involves
some constraints on
the frequency grid adopted
in the identification task.
A classical application
example of the GSE is given
in Appendix 1.
Practical Issues
Real architectures have to
take into account that the
amplitude frequency responses
of the channels
are far from being identical
to each other: all of them
typically lack of flatness,
especially at high frequency,
where resonances
and notches of several dBs
are often experienced. The
phase frequency responses
are also affected by nonidealities,
since the signal
14
H k fH f







l sl
l sl
  

H k fH f
  
21
fk

L
f Lk 
21
fs
21
L
where the superscript asterisk in the equation on the second
row stands for conjugation.
fs
k   L
k L  L
0, ,
1, ,2
(9)
Fig. 3. Hybrid block-diagram of a multi-channel digitizer where the digital circuital section is highlighted with a gray
background; channels are numbered from top to bottom.
IEEE Instrumentation & Measurement Magazine
June 2021

Instrumentation & Measurement Magazine 24-4

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