273
1 INTRODUCTION
Because of physical and technological constraints the
sampling of a signal value (that is its sample) at a
zeroth time is not possible. In other words, the
sampling time of a single signal sample is always
greater than zero. Only in an abstract idealized case
this time can be assumed to be equal to zero, and then
the signal samples related with this case are
considered as ideal ones. Moreover, we say that then
the sampling operation is performed in an ideal way.
Let us denote here the sampling time of a single
signal sample by
τ
and the sampling period by T,
respectively. Obviously, the following relation:
T
τ
≤
must hold between these quantities; but, as we know,
practical reasons require that this inequality is much
more sharper. That is we have
T
τ
. And, this
could suggest that we can assume approximately
0
τ
≅
(in the sense that the following three cases:
0
τ
≠
but very small with the sampling of signal
values modelled as an operation of periodic cutting
out a signal fragment,
0
τ
→
but always greater than
zero (
0
τ
>
) with the sampled signal spectrum
normalization with respect to
τ
, and
0
τ
=
with a
preceding normalization of the sampled signal
spectrum with respect to
τ
– do not considerably
differ from each other) in the analyses involving
signal samples.
The above-mentioned believing is however
misleading; usually, we arrive at different outcomes
in these three cases. This is strongly manifested in
calculations of the sampled signal spectrum for these
kinds of modelling, and analyzed in detail in (Borys
A. 2020a).
In this paper, we discuss, from another
perspective, the differences that occur in calculation of
the sampled signal spectrum between the following
two cases: the first one, in which signal samples are
“smeared” on a time interval
τ
(the case of
0
τ
≠
but very small), and the second one, in which the
signal sampling is performed in an ideal way (the case
of
0
τ
=
). We consider them in the context of the
Shannon’s proof of the reconstruction formula
(Shannon C. E. 1949).
Modelling of the “smearing effect” occurring in a
signal sampling process presented in this paper
differs, however, considerably from the one used in
(Borys A. Sept. 2020). Here, the “smeared” samples
are not modelled as short signal impulses (as in (Borys
A. Sept. 2020)), but as numbers obtained as a result of
performing periodically a local averaging of these
Modelling of Non-ideal Signal Sampling via Averaging
O
peration and Spectrum of Sampled Signal Predicted
by this
Model
A. Borys
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: In this paper, a novel model of a non-ideal signal sampling via a local, periodic averaging
operation is present-ed. The spectrum of a sampled signal predicted by this model is also analysed as well as
compared with a one following from another model.
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 16
Number 2
June 2022
DOI: 10.12716/1001.16
.02.10
274
impulses. So, in effect, we can expect receiving
outcomes not identical with those we arrived at in
(Borys A. Sept. 2020). And, really, it is so as we show
in the next sections.
Moreover, in view of the above, a question arises
as to which of the aforementioned approaches to
modelling of the “sample smearing” effect describes
in a better way a real, non-ideal signal sampling. This
problem is, however, not addressed in this paper
because of a lack in the literature of reliable data on
behavior of real A/D converters in the “transient”
interval
τ
. First, necessary measurements will have
to be carried out and data collected; we hope that
some researchers will be interested in performing this
task.
The remainder of this paper is structured as
follows. The next section presents a modelling of the
“sample smearing” effect via a periodical local
averaging – in the context of the proof (Shannon C. E.
1949) of the reconstruction formula that takes into
account a non-ideal sampling. But, section 3 contains a
comparison of the results which are achieved with the
use of the aforementioned models. The paper ends
with some conclusions.
2 SHANNON’S PROOF OF RECONSTRUCTION
FORMULA TAKING INTO ACCOUNT NON-
IDEAL SIGNAL SAMPLING AND POSSIBLE
DEFINITIONS OF SAMPLED SIGNAL
SPECTRUM
Let us take into account a bandlimited signal
( )
xt
of a continuous time
t
and denote a maximal
frequency present in its spectrum by
m
f
. So, as well
known (Marks II R. J. 1991), (Vetterli M., Kovacevic J.,
Goyal V. K. 2014), (Oppenheim A. V., Schafer R. W.,
Buck J. R. 1998), (Bracewell R. N. 2000), (McClellan J.
H., Schafer R., Yoder M. 2015), (Brigola R. 2013), (So
H. C. 2019), (Wang R. 2010), (Ingle V. K., Proakis J. G.
2012), it is possible to sample this signal and then
reconstruct it perfectly if the sampling period T fulfils
the following condition:
12
sm
Tf f= ≥
, (1)
where
s
f
stands for the sampling frequency (rate).
Further, assume that, as well known, the sampling
operation of
( )
xt
cannot be performed in an ideal
way; simply, A/D converters that provide perfect
signal samples by sampling a signal “pointwise on the
time axis” do not exist. All the real A/D converters
need some time, denoted here as
τ
, to carry out the
sampling operation. And, a value of the parameter
τ
depends, obviously, upon the design principles and
electronics that are used in construction of a given
A/D converter. Therefore, the values that appear at
outputs of A/D converters must be treated as
“deformed” or “smeared” ones – in comparison to the
wished perfect samples.
Further, note also that the aforementioned
behavior is usually modelled in the literature by a
local convolution (with the signal being sampled)
(Marks II R. J. 1991) or as a local signal averaging; see,
for example, (Vetterli M., Kovacevic J., Goyal V. K.
2014). (By the way, note that the latter can be also
expressed as a convolution, see, for instance, (Borys A.
2020b).)
In this paper, we use a description of the non-ideal
signal sampling that follows from modelling it by a
local signal averaging. And, this seems to be a
reasonable approach, as shown in the literature, using
many convincing arguments, see, for example, (Borys
A. 2020b), (Borys A. 2020c), (Strichartz R. 1994). Here,
we exploit a slightly modified version of that
presented in (Borys A. 2020b). This modification
regards the instant of “delivering” a “smeared” value
of a sample. Namely, in modelling of a measuring
process, this instant must be the one at which a result
of a local signal averaging process “departs” this
process. But, unlike this, in a non-ideal sampling, the
result of a local signal averaging has to be “glued” to
the instant of beginning the averaging process.
For illustration, consider Fig. 1.
Figure 1. Illustration to non-ideal sampling: representation
of an example, not ideally sampled signal (upper curve) in
form of a series of smeared samples of the signal (forming
narrow impulses) shown below it (lower curve). Figure
taken from (Borys A. 2020a).
In Fig. 1,
( )
,ST
xt
means a „smeared” sampled
signal in which every sample does not represent a
single value, but it builds up an impulse of width
τ
.
So, it can be viewed as “a kind of smearing of a
discrete sample value on an interval
τ
”. And, this
way of modelling of the non-ideal signal sampling has
been used in (Borys A. 2020a). But, as opposed to the
approach applied in (Borys A. 2020a), we assume here
that the non-ideal signal sampling delivers not
impulses at an A/D output but discrete sample values
– as in an ideal case. However, now, they are modified
by the smearing process of the ones that would have
been obtained in an ideal sampling. And, they are
“glued” to the instants of virtual appearances of the
latter ones. Furthermore, the aforementioned signal
sample smearing process is modelled here as a local
signal averaging.
So, as a result of performing these two operations
described above, we get
275
( ) ( )
( )
( )
(
( )
)
( )
( ) ( )
,, ,
,, ,
from
to
AV av
,
AT ST ST
k
ST ktT ktT
k
x t x t x kT
x kT t x kT t
λ
λ τδ δ
∞
=−∞
∞
=−∞
= = =
=+=
∑
∑
(2)
where the symbol AV stands for an operator that
transforms an infinite train of impulses as in Fig. 1
into an infinite train of single values as shown in Fig.
2 – according to these two rules given in a descriptive
form above. The result of this operation is the signal
( )
,AT
xt
. And, the next symbol, “small av”, means
performing a local averaging around an indicated
“smeared” sample (that is on a given impulse of
( )
,ST
xt
); the result of this operation is denoted here
by
( )
x kT
. And finally,
(
)
,ktT
t
δ
in (2) means a
time-shifted Kronecker time function (Borys A.
2020a), (Borys A. 2020d).
Figure 2. Illustration to transformation of the signal
( )
,ST
xt
shown in Fig. 1 to the signal
( )
,AT
xt
. Note that the lengths
of “posts” in Fig. 2 are not equal to the values of ideal signal
samples
( )
x kT
. They differ from them and equal the
values of
(
)
x kT
.
Here, to model the local signal averaging, we use
its description presented in detail in (Borys A. 2020b)
and (Strichartz R. 1994). So, along these lines, we
write
(
) ( ) ( )
( )
( ) ( )
fromav to
,
kT
kT
x kT x t kT x t kT
x a kT d
τ
τ
λ τλ λ
+
= = =+=
= +−
∫
(3)
where the function
( )
at
is assumed to have the
following form (Borys A. 2020b):
( )
1 for 0 and 0 elsewhere
at t
ττ
= <<
. (4)
Note further that, because of a sifting character of
the function
( )
at
given by (4) – in the interval from
0 to
τ
, (3) can be rewritten as
( ) ( ) ( )
x kT x a kT d
λ τλ λ
∞
−∞
= +−
∫
. (5)
So, we conclude from (5) that the averaged
(smeared sample) value
( )
x kT
can be expressed as
a convolution of the signal
( )
xt
with an impulse
response
( )
at
τ
+
, and that is calculated for the
instant
kT
. (For the needs of our further derivations,
we assume that this convolution exists for all
t
∈
,
where
denotes the set of real numbers.)
Moreover, note that the above convolution can be
equivalently calculated in the frequency domain as a
product of the following Fourier transforms:
( )
Xf
and
( )
( )
( ) ( )
exp 2at A f j f
τ πτ
+=
with
( )
⋅
denoting a standard Fourier transform (of a given
function),
( )
Af
standing for this transform for
( )
at
, and
1j
= −
. Furthermore, by carrying out a
few standard calculations involving properties of
Fourier transformation, we obtain
( ) ( ) ( )
( )
( ) ( )
( ) ( )
( )
( )
sinc
sin
exp exp ,
Yf Xf at Xf f
f
jf Xf jf
f
ττ
πτ
πτ πτ
πτ
= ⋅ += ⋅
⋅=
(6)
where
( )
Yf
means the aforementioned product,
but the definition of the function
(
)
sinc f
τ
used
here follows from a comparison of the first and the
second line of (6).
Let us now come back to the assumed band-
limitedness of the signal
(
)
xt
. It means that its
Fourier transform
( )
Xf
has nonzero values only in
the range
,
mm
ff<− >
(that is this is a support of
this function). Further, because of this reason the
function
( )
Yf
given by (6) is also bandlimited to
the interval
,
mm
ff<− >
. So, it allows its
periodization (i.e. obtaining a repetition of this
function in form of a Fourier series). But, because of
the reasons explained in (Borys A. 2020e), we extend
here the support of
( )
Yf
to the interval
2, 2
ss
ff<− >
, and consequently construct a
corresponding Fourier series with a fundamental
frequency
0 s
ff=
(not
2
ms
ff≤
).
In what follows now, we proceed similarly as in
(Borys A. 2020e). That is we perform first
periodization of the function
(
)
Yf
to a periodic
one, say,
( )
p
Yf
, and expand it in a Fourier series.
Next, we express coefficients of this series through the
samples
( )
y kT
of the function
( ) ( )
( )
1
yt Y f
−
=
, where
( )
1
−
⋅
stands for the
inverse Fourier transform. And, in the next step, we
introduce them into the aforementioned Fourier
series. In this way, we get the discrete time Fourier
transform (DTFT) McClellan J. H., Schafer R., Yoder
M. 2015), (Vetterli M., Kovacevic J., Goyal V. K. 2014),
(Wang R. 2010), (Ingle V. K., Proakis J. G. 2012) of the
sampled (discrete) signal
( )
y kT
, which is equal to
( )
p
Yf
. And, we call it a spectrum of this signal, say,
SPECT1 (it forms a first of its possible definitions).
Next, we recall at this point that a different definition
of the spectrum of the aforementioned sampled signal
is also possible (Borys A. 2020f); it is named the
modified DTFT in (Borys A. 2020f) (in short, DTFTm).
Furthermore, we know from (Borys A. 2020e) that the
following:
SPECT2 DTFTm= =
( )
Yf=
holds,
where SPECT2 stands (in short) for the second
possible spectrum definition – of the sampled signal
( )
y kT
.
Both these spectra occur in the “inverse part” of
the Shannon’s proof; that is in
276
(
) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
exp 2
exp 2
exp 2
ex
DTFT
DT
p 2 and so on
Fm
.
T
s
s
s
s
f
p
f
f
f
y t Y f j ft df
Y f j ft df
y kT j ft df
y kT
j ft df
π
π
π
π
∞
−∞
−
−
∞
−∞
= =
= =
= =
= ⋅
⋅=
∫
∫
∫
∫
(7)
And, we see that the way they occur in (7) is
exactly the same as in its counterpart in (Borys A.
2020e). So, we conclude, similarly as in (Borys A.
2020e), that the “clever” Shannon’s proof of the
reconstruction formula does not provide, also here,
any tool for resolving the question of which of them:
( )
( )
1
SPECT1 DTFT
p
k
Y f Y f kT
T
∞
=−∞
= = = −
∑
or
( )
SPECT2 DTFTm Yf= =
is a correct one? This is so, as already found in (Borys
A. 2020e), because the Shannon’s proof does not need,
in fact, to use such a notion as the sampled signal
spectrum.
Further details and explanations concerning the
above, the interested reader finds in (Borys A. 2020e).
3 COMPARISON OF RESULTS PROVIDED BY
TWO MODELS THAT TAKE INTO ACCOUNT
FINITE DURATION OF GETTING SIGNAL
SAMPLE
As already said, the method presented here of taking
into account a finite duration of getting a sample – in
a model of the non-ideal signal sampling – is not the
only one possible. One can, for example, model also a
train of non-ideal samples of a signal as a train of
impulses, as illustrated in Fig. 1 (upper curve). That is
as a signal
( )
,ST
xt
denoted there. And, there is no
problem with calculation of its spectrum, as shown in
(Borys A. 2020a). Moreover, it has been shown in
(Borys A. 2020a) that
( ) ( )
,ST k
k
X f aX f kT
∞
=−∞
= −
∑
, (8)
where
( )
X ⋅
and
( )
,ST
Xf
stand for the Fourier
transforms (spectra) of the signals
( )
xt
and
( )
,ST
xt
, respectively. And, the coefficients
k
a
in (8)
are given by
( ) ( )
exp sinc
k
a jkT kT
T
τ
πτ πτ
=−⋅
. (9)
We remark at this point that the detailed
derivations and explanations concerning (8) and (9)
have been provided in (Borys A. 2020a). They are not
repeated here because of a lack of space as well as to
avoid accusation of auto-plagiarism. Moreover, the
reference (Borys A. 2020a) is well available.
For performing comparisons between the sampled
signal spectra foreseen in the discussed models – in a
clear way, let us denote now by
SPECT
=
( )
( )
1
k
T X f kT
∞
=−∞
= −
∑
the spectrum which one
obtains in the highly celebrated and commonly used
model (see, for example, (Marks II R. J. 1991), (Vetterli
M., Kovacevic J., Goyal V. K. 2014), (Oppenheim A.
V., Schafer R. W., Buck J. R. 1998), (Bracewell R. N.
2000), (McClellan J. H., Schafer R., Yoder M. 2015),
(Brigola R. 2013), (So H. C. 2019), (Wang R. 2010),
(Ingle V. K., Proakis J. G. 2012)) that uses Dirac deltas
in description of the sampled signal in the continuous
time domain. Further, let us denote by
( )
,
SPECT0
ST
Xf=
given by (8).
In what follows, we remark that:
1. The form of SPECT0 for positive values of the
parameter
τ
is identical with the one of SPECT,
except the coefficients which multiply the shifted
spectra
( )
X f kT
−
. They are given by (9) in the
first case and are identically equal to
1 T
for all
the indices k in the second one.
2. Because of the oscillatory-damping character of the
magnitude of the coefficients
k
a
(see (9)) the
same character has also the magnitude of SPECT0.
So, this is also the character of the aliasing and
folding effects in the sampled signal spectrum via
this model. Obviously, it differs substantially from
the case of SPECT.
3. An expectation that the problem of modelling
properly a non-ideal behavior of getting samples
in the signal sampling process can be uniquely
resolved by describing it through local signal
averaging operations on short time intervals
τ
, as
discussed in this paper, turned out to be only a
vain hope. At least with regard to the spectrum of
the sampled signal. Furthermore, note that the
averaging operation does not also provide a
description of the sampled signal in the time
domain in an idealized case, in which
0
τ
→
, as a
train of sample values multiplied by Dirac deltas
(as a highly celebrated and commonly used model
(Marks II R. J. 1991), (Vetterli M., Kovacevic J.,
Goyal V. K. 2014), (Oppenheim A. V., Schafer R.
W., Buck J. R. 1998), (Bracewell R. N. 2000),
(McClellan J. H., Schafer R., Yoder M. 2015),
(Brigola R. 2013), (So H. C. 2019), (Wang R. 2010),
(Ingle V. K., Proakis J. G. 2012) foresees). To see
this, consider (4) with
0
τ
→
in it. And, using
arguments presented, for example, in (Strichartz R.
1994) for this case, we can write, then,
( ) ( )
at t
δ
→
, where
( )
t
δ
means a Dirac delta.
So, applying this result in (5) allows us to express
the latter as
( )
( ) ( )
( ) ( )
x kT x kT x kT d x kT
λδ λ λ
∞∞
−∞ −∞
→ = −= ⋅
∫∫
( )
kT d
δ λλ
⋅−
.
277
1 But, it does not mean that an object
( ) ( )
x kT
δ
⋅
occurring under the symbol of
integration in the latter equation represents the
sample value
( )
x kT
. As seen, it is a result of
performing the above operation of integration. In
other words, simply,
( ) ( ) ( )
x kT x kT
δ
⋅≠
.
4. Also, we draw the reader’s attention here to the fact
that the averaging procedure in the time domain
(applied in this paper to the smeared sample
impulses and connected then with the spectrum
definition SPECT1) provides a different result
compared to the normalization of the spectrum
( )
,
ST
Xf
with respect to the parameter
τ
(Borys
A. 2020a) (to avoid its vanishing with
0
τ
→
).
The resulting expressions that describe the spectra
of the sampled signal in both cases are similar in
form but not identical. However, because of a lack
of space we do not discuss here this interesting
observation in more detail.
5. Note once again that the scheme of the Shannon’s
proof applied in this paper to the signal sampled
not in an ideal way differs from the one discussed
in (Borys A. 2020e) only in one aspect, namely
( )
Yf
(in this paper) is not a Fourier transform of
the signal to be sampled. It is a “deformed”
spectrum of this signal. And, it follows from (6)
that a level of its deformation can be expressed by,
say, a “deformation” factor
( )
f
β
defined as
( )
( )
(
)
( )
( )
sin
exp
d
Xf f
f jf
Xf f
πτ
β πτ
πτ
= =
, (10)
1 where
( ) ( )
d
X f Yf=
stands for the spectrum
(
)
Xf
that is deformed by the local averaging
operator av. Moreover, note that because of the
band-limitedness of
( )
Xf
(and also of
( )
Yf
)
it has only sense in the frequency interval
,
mm
ff<− >
(outside this range, it should be
assumed to be equal to zero). Further, see from (10)
that both the magnitude and phase of the spectrum
(
)
Xf
get deformed. Here, for illustration, let us
consider only a deformation in the magnitude.
And, we make a few observations:
1. See first that
( )
1f
β
≤
for all possible values
of frequency and parameter
τ
.
2. For
0
τ
=
,
( )
1f
β
=
. That is there occurs no
sampled signal deformation in this case (as it
should be for
0
τ
=
).
3. For illustration, let us assume that we wish to
have the deformation of the sampled signal
spectrum magnitude less than 10% in the worst
case. To determine a condition for the
parameter
τ
that satisfies the above
requirement, we consider the magnitude of
(
)
f
β
given by (10) for positive frequencies f.
And, note that the most critical here is the
frequency
m
f
. Further, assume that the
sampling rate is so chosen that
( )
2 12
s mm
fff T= →=
holds. So, for this
value of
m
ff=
, we have
( )
( )
( )
sinc 2
m
fT
β πτ
=
. And, we require to
satisfy the following:
( )
0,9
m
f
β
≥
; while the
latter is satisfied approximately for
12T
τ
<
.
Obviously, at the same time, this is a condition
we looked for.
4 CONCLUSIONS
The problem of modelling the sample “smearing”
behavior of real A/D converters used in signal
sampling has not received much attention in the
literature. It seems to have been assumed that this
effect is irrelevant – compared to, for example,
(amplitude) quantization errors produced by A/D
converters. As we show in this paper and in a
previous one (Borys A. 2020a), such reasoning is
rather not correct. This is so because the
aforementioned effect has a significant influence on
the sampled signal spectrum – and, this has been
already proven. What remains to be done yet should
concentrate, in our opinion, on finding a detailed
model and adjusting it to the sample “smearing”
behavior of real A/D converters.
Two relevant models has been already proposed, a
one in this paper and the second in (Borys A. 2020a)
(perhaps, there will be also others).
Note that a modelling principle of the first one is
based on performing periodically a local averaging of
impulses of short duration, starting at sampling
instants, and delivering its averages at the ends of the
aforementioned impulses (which, however, are
“glued” to their beginning instants). So, in this case,
the outcomes of the “sample smearing” operations are
numbers. In contrast to this, in the model presented in
(Borys A. 2020a) the impulses mentioned above are
taken to constitute the “smeared” samples (that is
electric “spikes” of duration ).
There is a variety of design principles, techniques,
and circuit schemes for A/D converters. Therefore,
probably, more than only one model for describing
correctly their “sample smearing” behavior will be
needed. And, for checking practical usefulness of
these models many investigations will be also needed.
Moreover, note that there are still open questions of
more general nature as, for example, the one
considered in (Borys A. 2020d). So, we are still far
from a satisfactory solution to the problem of the
sampled signal spectrum.
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