Instrumentation & Measurement Magazine 24-4 - 12

Technology
Table 1 - Main aspects of TI, DBI and ATI technologies
Highlights
Time interleaving (TI)
Digital bandwidth
interleaving (DBI)
Asynchronous time
interleaving (ATI)
Pioneering technology exploiting channel
resources sharing
Introduces heterodyning to support
unprecedented bandwidth performance
Deploys compact and low noise architectures
capable of improving DSO bandwidth
noise contributions such that the related power contribution is
halved. Yet, the ATI performance in terms of bandwidth lags
behind that offered by DBI.
The main aspects characterizing the overviewed technologies
are summarized in Table 1. The common trait, shared
by the three of them, lays in the combination of more channel
resources, which represents an established design approach
since the deployment of the first time-interleaved ADCs in
the early 1980s. The role of the digital signal processing is
fundamental to enabling the combination and granting the
realization of more ideal acquisition channels [5]. In particular,
digital processors allow implementing streamline
calibration, which is accomplished by means of equalization
filtering, largely preferable to analog adjustments. Also,
digital circuits have superior tolerance to component variations
and substantially avoid performance degradation due
to aging. It is worth noticing, finally, that both DBI and ATI
technologies inherit much of the technical accomplishments
of time-interleaved designs that they tailor to their own
architectures.
Informative product notes aimed at explaining DSOs
principles never illustrate how the internal digital processors
effectively grant the combination of channel resources
[6], [7]. The product managers limit presentation of the DSOs
to a mere description of the basic ideas, just as briefly summarized
in this section, without diving into the math of some
analytical results that would definitely provide better insight
[8]-[10]. In the following, the attention is paid to DSOs'
architectures that combine linear and time invariant channels.
The presented framework can be suitably extended to
describe the architectures implementing DBI or ATI, where
the channels include time variant effects due to different
modulations, which is however far from the intent of this
contribution.
General Framework for Streamline
Calibration Design
Theoretical Notes
Time interleaved architectures that combine channel resources
in DSOs can be described using a common model, theorized
by A. Papoulis and named generalized sampling expansion
(GSE), [11]. According to GSE, a deterministic signal u(t) band
12
Fig. 1. The input band limited signal is split on M channels and sampled, on
each channel, at a sampling rate 1/M the Nyquist rate; M reconstruction filters
allow perfect signal reconstruction.
IEEE Instrumentation & Measurement Magazine
June 2021
Drawbacks
Bandwidth limited by the baseband nature of
the digitizing system
Uses asymmetrical channels requiring
synchronization and massive processing that
impact complexity and cost of the DSO
Promising alternative technology: the
bandwidth performance is intermediate
between TI and DBI solutions
limited to (-B, B) that is passed through M linear and time invariant
channels, with frequency responses Hl
(nTs), acquired at sample rate fs
(f) l = 1, ...,
M, can be reconstructed from the samples of the M signals
vl
= 2B/M, provided that the
outputs of the channels are independent from each other. Independence
can be obtained by deploying on the channels
filters characterized by different frequency responses, or simply
by introducing mutual time-offsets between the sampling
instants of the otherwise identical channels, as in basic timeinterleaved
architectures [12]-[14]. Reconstruction is achieved
by interpolating the samples from each channel with suitable
functions, yl
(t), required to be band limited to (-B, B), and summing
the M resulting signals, as expressed by:

M 
u t     

ln
1 
where Ts = M/2B is the sampling period, reciprocal of the
sample rate fs
v nT y t nT 
l sl s
(1)
. Said differently, reconstruction is obtained by
filtering the sampled signals with suitable filters and summing
the filtered outputs. A schematic of a multi-channel system
that samples the output of the different channels, represented
in the frequency domain with Vl
(f) l = 1, ..., M, and uses the filters
characterized by the frequency responses Yl (f) l = 1, ..., M,
for reconstruction, is shown in Fig. 1.
In order to derive the frequency response of the reconstruction
filters, it is convenient to describe sampling as
multiplication with the ideal pulse train,

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