Instrumentation & Measurement Magazine 24-4 - 15

More detailed considerations can be drawn with reference
to Fig. 3, that describes with less abstraction with respect to Fig.
1 the architecture of a system with M channels, where both the
analog and digital parts are highlighted and the latter are distinguished
by a light gray background. Signals and systems
are described in terms of time-domain impulse responses:
namely, the impulse response of the l-th channel is hl
responding to the inverse Fourier transform of Hl
impulse response of the related digital correction filter is gl
which are related to the function Yl(f). The time-interleaved op(t),
cor(t),
and the
(n),
eration characterizing the ADCs is implemented by means of
Ts
, is digitally up-sampled to the
= 2B, equal to the Nyquist
/M delays, represented with small rings deployed along the
clock propagation path. The data stream produced by each
ADC, driven at a clock rate fs
sample rate of u(n), which is Mfs
rate of the input signal, bandlimited to (-B, B). The upsampling
operation is performed interleaving M - 1 zeros between any
couples of samples, and it is accounted by the circular boxes labelled
with a top oriented arrow and the factor M.
The digital filters gl
channels that can be used both as independent channels to simultaneously
measure different signals or in a combined way
to grant higher performance, for example, in terms of higher
sample rate; the combination can require the use of external accessories
or be enabled with a button on the keypad [17], [18].
In these DSOs, the independent channels are identified at
the production stage and described in terms of their own impulse
responses, i.e., digital finite-length sequences hl
(n) with
(n) can be designed through the procedure
detailed in the following. (It is worth noting, however,
that other approaches to the synthesis of stable and causal
calibration filters, alternative to the proposed one and characterized
by transfer functions with zeros and poles, are also
possible.)
The frequency responses Hl
fined as H ke
l
 
jk
L
2 1
21
l

, k = 0, ..., 2L + 1, imagining that the acquired
samples are related to delayed instances of the input signal,
sampled synchronously. This is convenient because the digital
circuitry synchronously processes the samples from the parallel
channels: namely, at any clock cycle it retrieves a set of M
samples, aligned by a suitable interface buffer. If the frequency
responses of the channels would not be redefined, the samples
released by the ADCs should be positioned at the upsampling
stage differently from what is shown in Fig. 3. Specifically, the
use of a more complex digital circuitry would be necessary to
position each sample on the l-th channel amid a zero sequence,
with M - l zeros on the left and l-1 zeros on the right side.
Numerically solving the GSE equations written for Hl
offers an estimate of Yl (k) on the same 2L + 1 bins. Apply(k)
ing
the inverse discrete Fourier transform to the sequence
Yl
(k), k = 0, ..., 2L, permits determining the sequence yl
n = 0, ..., 2L, that is multiplied with a window wL(n), n = 0, ...,
(n),
2L, to obtain the impulse response of the correction filters as
   

l lL
gn yn w n . The digital filters, gl (n) l = 1, ..., M, are
characterized by a finite impulse response (FIR filters) and
grant a stable output. The choice of the window wL
(n) allows
reducing the undesired effects due to the abrupt truncation implicit
in any finite length sequences. The digital representation
of the signal u(n) is finally obtained by adding point by point
the M data streams filtered by the correction filters.
Special Calibration Approaches
There is a variety of medium/high-end DSOs available on the
market that use calibration filters that are identified without
using the GSE framework. This is the case of DSOs offering
June 2021
(k) given in (9) are first redethe
same length L. Moreover, the preferred identification procedure,
carried out at the production stage, is based on step
response tests. These tests are aimed at measuring the step response
profiles of the individual channels and subsequently
gain the correspondent impulse responses by taking the time
derivative. For step response measurements, once again the
time-equivalent sampling capabilities of the channels are exploited,
such that using repeated stable step stimuli, the step
response profiles can be determined with a time resolution superior
to that granted by real-time sampling; the required time
resolution has to be at least equal to the final sample rate utilized
to combine the channel resources, i.e., Mfs
. Taking the
Discrete Fourier Transform of the impulse responses of the individual
channels the frequency responses Hl
(f) required for
applying the GSE can be obtained and the correction filters calculated
through (8).
In order to assure streamline calibration both when used
independently from each other or in a combined way, the independent
impulse responses hl
(n) are averaged to obtain the
L-length digital sequence h(n). A reference impulse response
g(n) is then chosen and its parameters are identified in order
to gain the best fit to the average impulse response h(n). The
most common models adopted as reference are either (truncated)
Gaussian, or Bessel filters; the first provides the fastest
step response without overshoot and ringing, whereas Bessel
roll-off is suitable to perform some measurements required in
telecommunication and optical systems, such as eye-diagram
measurements. The correction filters cl
(n) for each individual
channel are then determined as those filters that modify the actual
impulse response of the channel hl
to the reference impulse response g(n).
correction filters, it is useful to recall, with the help of Fig. 4,
that the cross-correlation 
rn of the outputs of two linear
zz
12
filters can be determined as convolution between the crosscorrelation
of the inputs, 
xx
impulse responses of the filters, 
12
cc
12
rn , and cross-correlation of the
rn :
(n) and make it adhere
In this case, to illustrate in detail the identification of the
Fig. 4. Relationship between inputs and outputs cross-correlations for
discrete linear systems.
IEEE Instrumentation & Measurement Magazine
15
Instrumentation & Measurement Magazine 24-4

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