Instrumentation & Measurement Magazine 24-4 - 16

References
[1] P. Pupalaikis et al., " Technologies for very high bandwidth realtime
oscilloscopes, " in Proc. IEEE Bipolar/BiCMOS Circuits and
Technical Mtg.(BCTM), pp. 128-135, 2014.
[2] J. Muller et al., " Method and apparatus for artefact signal
reduction in systems of mismatched interleaved digitizers, " U.S.
Patent No. US 7 386 409 B2, June 10, 2008.
Fig. 5. Relationship between inputs and outputs cross-correlations for
discrete linear systems: hl
channel, g(n) the desired response, cl
the identity system.
z z1 2
(n) represents the impulse response of the l-th
(n) the required correction filter, and δ(n)
r n r nr n
 cc

12
x x1 2
    
the identity system, it holds:
   
r n cn rhh n
l
h g
  ll
l
(n) unaltered.
(10)
Now, considering the scenario depicted in Fig. 5, where hl
(n) represents the impulse response of the l-th channel, g(n) the
reference response, cl
(n) the required correction filter, and δ(n)
(11)
since the cross-correlation of the impulse response of the correction
filter with the impulse response of the identity system
returns cl
If both actual hl
(n) and desired g(n) are defined for
n = 0, ..., N - 1, the length of their cross-correlation is 2N - 1.
To identify the correction filter, the N-length central part of
the cross-correlation sequence 
r    with k highest integer
not greater than N/2, is singled out. It can thus be extracted
from (10) the linear algebraic system:
hg r k N 1
ll lh g
h g
,
r kr k
1 , , hg
l

r  Rc
h g
l
ll
ll
hh l
(12)
where Rhh is an N-size square matrix whose coefficients are
defined as:  
Rhhl pq r N p q
 
h h
ll
l ,2 , and cl
is an N-length
column vector containing the coefficients of the desired correction
filter. The correction filter is straightforwardly obtained
by inverting (12):
cRl  r l
hh h g

1
ll
(13)
Conclusions
Streamline calibration approaches for digital storage oscilloscopes
that use time-interleaved channels are never illustrated
in detail in product notes or demos, although the performance
of the oscilloscopes largely benefits from these solutions. This
brief contribution provides a reference theoretical framework
and shows practical issues customarily faced by the DSO designers
and technicians who work at the production level. It
also highlights the actual trend in digitizing instrument design,
where the use of digital processors allows refining and
improving analog system performance, leading to the realization
of more ideal acquisition channels.
16
rn , which is less disturbed
T
hgl
by tailing effects and corresponds to the column vector
  
[3] D. G. Knierim, " Test and measurement instrument including
asynchronous time-interleaved digitizer using harmonic mixing, "
U.S. Patent Application No. US 2012/0299579 A1, November 29,
2012.
[4] " Techniques for Extending Real-Time Oscilloscope
Bandwidth, " Tektronix White Paper, Lit.Num. 55W-29371-3,
March 2015. [Online]. Available: www.tek.com/dl/55W29371-2ATIWhitepaper.pdf.
[5]
J. J. Pickerd, " DSP in high performance oscilloscopes, " Tektronix
White Paper, Lit. Num. 55W-17589-0.
[6] M. D'Apuzzo and M. D'Arco, " A wideband DSO channel based
on three time- interleaved channels, " JINST, vol. 11, P08003, 2016.
[7] M. D'Apuzzo and M. D'Arco, " Sampling and time-interleaving
strategies to extend high speed digitizers bandwidth, "
Measurement, vol. 111, pp. 389-396, 2017.
[8] P. Monsurrò, A. Trifiletti, L. Angrisani, and M. D'Arco,
" Streamline calibration modelling for a comprehensive design
of ATI-based digitizers, " Measurement, vol. 125, pp. 386-393, Sep.
2018.
[9] M. D'Arco, E. Napoli, and L. Angrisani, " A time base option for
arbitrary selection of sample rate in digital storage oscilloscopes, "
IEEE Trans. Instrum. Meas., vol. 69, no. 6, pp. 3936-3948, Jun. 2020.
[10] L. Angrisani, M. D'Arco, G. Ianniello, and M. Vadursi, " An
efficient pre-processing scheme to enhance resolution of bandpass
signals acquisition, " IEEE Trans. Instrum. Meas., vol. 61, no.
11, pp. 2932-2940, Nov. 2012.
[11] A. Papoulis, " Generalized sampling expansion, " IEEE Trans.
Circuits and Syst., vol. CAS-24, no. 11, pp. 652-654, Nov. 1977.
[12] J. L. Brown, " Multichannel sampling of low pass signals, " IEEE
Trans. Circuits and Syst., vol. CAS-28, no. 2, pp. 101-106, Feb. 1981.
[13] Y. C. Eldar and A. V. Oppenheim, " Filterbank reconstruction of
bandlimited signals from nonuniform and generalized samples, "
IEEE Trans. Signal Process., vol. 48, no. 10, pp. 2864-2875, Oct. 2000.
[14] S. Maymon and A. V. Oppenheim, " Quantization and
compensation in sampled interleaved multichannel systems, "
IEEE Trans. Signal Process., vol. 60, no. 1, pp. 129-138, Jan. 2012.
[15] M. D'Arco, E. Napoli, and E. Zacharelos, " Digital circuit for
seamless resampling ADC output streams, " Sensors MDPI, vol.
20, p. 1619, 2020.
[16] L. Angrisani, M. D'Arco, P. Monsurrò, and A. Trifiletti, " Two
novel architectures for 4-channel mixing/filtering/processing
digitizers, " Measurement, vol. 142, pp. 138-147, Aug. 2019.
[17] L. Angrisani, R. S. Lo Moriello, and M. D'Apuzzo, " New proposal
for uncertainty evaluation in indirect measurements, " IEEE Trans.
Instrum. Meas., vol. 55, no. 4, pp. 1059-1064, 2006.
[18] L. Angrisani, F. Bonavolontà, M. D'Apuzzo, R. S. Lo Moriello, and
M. Vadursi, " A compressive sampling based method for power
measurement of band-pass signals, " in Proc. IEEE IMTC 2013, pp.
102-107, 2013.
IEEE Instrumentation & Measurement Magazine
June 2021
https://www.tek.com/document/whitepaper/techniques-extending-real-time-oscilloscope-bandwidth https://www.tek.com/document/whitepaper/techniques-extending-real-time-oscilloscope-bandwidth

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