473
1 INTRODUCTION
The signal spectrum is its image or, in other words, its
representation in the frequency domain. And, it is
generally agreed among researchers and engineers
that all the non-pathological, deterministic, and non-
periodic signals exploited in the area of signal
processing possess such the representation. It is a
Fourier transform of a given signal.
Similarly, all the periodic signals can be
represented by their Fourier series. So, their frequency
images can be considered as discrete spectra, what is
visualized in Fig. 1.
Figure 1. Visualization of the discrete spectrum of an
example periodic signal.
In Fig. 1,
( )
ck
means the k-th coefficient in the
Fourier series of a periodic signal considered.
Note that in general the coefficient
( )
ck
is a
complex number. It is connected with the frequency
0
kf
, what makes possible to treat the set of all
( )
ck
’s
as a set of values of a certain function of
f
(spectrum). Because of this reason
( )
ck
is used here
to denote also, in short, this function. Moreover,
f
and
0
f
in Fig. 1 stand for the frequency variable and
Definition of Sampled Signal Spectrum and Shannon’s
Proof of Reconstruction Formula
A. Borys
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The objective of this paper is to show from another perspective that the definition of the spectrum
of a sampled signal, which is used at present by researchers and engineers, is nothing else than an arbitrary
choice for what is possibly not uniquely definable. To this end and for illustration, the Shannon’s proof of
reconstruction formula is used. As we know, an auxiliary mathematical entity is constructed in this proof by
performing periodization of the spectrum of an analog, bandlimited, energy signal. Admittedly, this entity is
not called there a spectrum of the sampled signal - there is simply no need for this in the proof – but as such it is
used in signal processing. And, it is not clear why just this auxiliary mathematical object has been chosen in
signal processing to play a role of a definition of the spectrum of a sampled signal. We show here what are the
interpretation inconsistences associated with the above choice. Finally, we propose another, simpler and more
useful definition of the spectrum of a sampled signal, for the cases where it can be needed.
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the International Journal
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Volume 16
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September 2022
DOI: 10.12716/1001.16.03.08
474
the so-called fundamental frequency (fundamental
harmonic) in the spectrum of this signal.
The common belief among researchers and
engineers is that both kinds of the spectra mentioned
above, i.e. the continuous spectrum related with the
Fourier transform and the discrete one connected with
the Fourier series, fit to each other in some way.
However, this is only an illusion. Why? Because of
many reasons. But, probably, the most important one
follows from the fact that the spectrum of a periodic
signal, say
( )
p
xt
, where
t
means a continuous
time variable, cannot be calculated in a “normal way”
via the Fourier transform.
What we understand by the “normal way of
calculation” mentioned above is illustrated in (1)
below:
( ) ( ) ( )
( ) ( )
( ) ( )
( )
2
32
2
2
exp 2
..... exp 2
exp 2 .....
.
pp
T
p
T
T
p
T
PART
X f x t j ft dt
x t j ft dt
x t j ft dt
Xf
−
−
−
−
= − =
+ + − +
+ − + =
= =
(1)
In (1),
0
1Tf=
denotes the period of the
periodic signal
( )
p
xt
,
( )
p
Xf
its Fourier transform
(if exists?),
1j =−
, and the function
( )
PART
Xf
is
one of the identical components of an infinite sum
there. Obviously, this component cannot be
identically equal to zero for all frequencies. Therefore,
there are bands of frequencies for which (1) results in
infinite values. And, this is the interpretation of what
is expressed in (1).
Further, the above description of
( )
p
Xf
shows
that by no means it can resemble that what is
presented in Fig. 1. So, because of this fact, it seems
that only reasonable conclusion here is the following
one: signal spectra obtained as their Fourier
transforms are not compatible with those following
from the Fourier expansions (having the form of the
one visualized in Fig. 1). And, obviously, this is a
huge problem in cases where both the periodic and
non-periodic signals occur together in a system or
circuit, in a mixed form. One of the representative
examples here are the operations of sampling of an
analog signal and its reconstruction from a sequence
of its discrete values.
As we know very well, there is a theory (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),
(Jenkins W. K. 2009) (to mention only a few of
excellent textbooks on fundamentals of digital signal
processing), which overcomes the problem sketched
above. But, this theory applies non-physical objects
called Dirac distributions (named also Dirac
generalized functions or Dirac deltas) (Schwartz L.
1950-1951), (Dirac P. A. M. 1947).
With application of this theory, (1) results in a
solution, which is a sum of Dirac deltas multiplied by
real numbers – these numbers are the corresponding
coefficients of the Fourier expansion of
( )
p
xt
. So, as
a consequence, the image of the discrete spectrum of a
periodic signal – as it is visualized in Fig. 1 – must be
then modified accordingly. Then, in case of our
example signal, it has the form presented in Fig. 2 and
is denoted by
( )
,pD
Xf
.
Figure 2. Visualization of the discrete spectrum of an
example periodic signal after a model that results in Dirac
deltas in (1).
The arrows in Fig. 2 represent the frequency-
shifted Dirac deltas
( )
0
f kf
−
multiplied by the
corresponding coefficients
( )
ck
of the Fourier
series of the signal
( )
p
xt
. So, the spectra presented
in Fig. 1 and in Fig. 2 are not identical; they are two
different images of the signal
( )
p
xt
in the
frequency domain. But, this is allowed in the theory
that is currently in force.
Note however that the above philosophy allowing
a signal to have two (or more) different spectra can be
a source of confusions and misinterpretations. One
notable example of this kind is a strong belief of
researchers and engineers that there occur aliasing
and folding effects in spectra of sampled signals
(sampled in an ideal way) (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),
(Jenkins W. K. 2009). Note that their mistake in this
case lies in the fact that they draw their conclusions
from the analysis of the second image of the spectrum
of a sampled signal, which is derived from a model
involving Dirac deltas.
An alternative view on this problem is presented
in (Borys A. 2020a). It is based on consideration of the
first possible representation of the sampled signal
spectrum (i.e. without involvement of Dirac deltas)
and leads to quite different conclusions.
The objective of this paper is to show, from
another perspective, that really the spectrum of a
sampled signal (sampled in an ideal way) cannot be
uniquely defined. And, as it does not exist (Borys A.
2020a), (Borys A. 2020b) as a Fourier transform of a
true sampled signal, an extension of its definition is
needed, what can be done in a variety of ways – as
proposed, for example, in (Borys A. 2020b). So, this
takes place in an arbitrary way.
To strengthen this point of view, that is
arbitrariness of the choice mentioned above, we
consider here an example of solving a problem, in
which two definitions of the spectrum of a sampled
(ideally) signal are tacitly used, but not named
explicitly at all. (Note that if these different definitions
475
of the same object were used this would be viewed as
a mistake.)
Our example is the Shannon’s proof of the
reconstruction formula (Shannon C. E. 1949) applied,
here, for obtaining a description of the signal
sampling and sampled signal spectrum. It is
presented and analyzed in the next section. The paper
ends with a concluding remark.
2 SHANNON’S PROOF OF RECONSTRUCTION
FORMULA APPLIED TO DESCRIPTION OF
SIGNAL SAMPLING AND SAMPLED SIGNAL
SPECTRUM
In this section, we consider a case when a signal
( )
yt
of a continuous time
t
is a bandlimited one.
And, we denote the maximal frequency present in its
spectrum by
m
f
. So, this signal can be sampled and
reconstructed perfectly if the sampling period T fulfils
the following Nyquist-Shannon condition:
12
sm
T f f=
, (2)
where
s
f
means the corresponding sampling
frequency. (Note that from now the symbol T will
play here two roles: of a signal repetition time and of
a signal sampling period – the one applicable at the
moment will follow from the context; moreover, the
sampling frequency
s
f
corresponds with the
frequency
0
f
that was defined and used in the
previous section.)
It follows from the above that the Fourier
transform
( )
Yf
of the signal
( )
yt
has nonzero
values only on the segment
,
mm
ff −
of the
frequency axis (that is supported only on this
segment). This property allows to expand it on the
whole frequency axis – in form of a Fourier series.
Obviously, one can take into account a wider range
of frequencies around
( )
Yf
than
,
mm
ff −
,
and treat it as “an extended support” of
( )
Yf
. And,
just this is done in what follows – it makes a slight
modification of the Shannon’s scheme in (Shannon C.
E. 1949). We build up a periodic function
( )
p
Yf
from
( )
Yf
on the following “extended supporting
interval”:
2, 2
ss
ff −
with
s
f
given by (2).
In other words, we perform here a periodization of
( )
Yf
on its “extended support” to get a periodic
function on the whole frequency axis.
Assume now that the function
( )
p
Yf
so
obtained fulfills the conditions (Bracewell R. N. 2000),
(Brigola R. 2013) allowing its expansion in a Fourier
series. That is we get then
( ) ( )
( )
exp 2
exp 2
pk
k
ks
k
Y f a j kTf
a j k f f
=−
=−
==
=
, (3)
where the coefficients
( )
k
a a k=
are given by
( ) ( )
2
2
1
exp 2
s
s
f
k p s
s
f
a Y f j k f f df
f
−
=−
. (4)
In the next step, observe that (4) can be re-written
as
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2
2
1
exp 2
exp 2
exp 2
1
exp 2 .
m
s
m
m
s
m
f
k p s
s
f
f
ps
f
f
ps
f
s
s
a Y f j k f f df
f
Y f j k f f df
Y f j k f f df
Y f j k f f df
f
−
−
−
−
= − +
+ − +
+ − =
=−
(5)
The final result in (5) follows from the fact that the
first and third integrals there are equal to zero,
( ) ( )
p
Y f Y f
in the interval
,
mm
ff −
, and
( )
Yf
is identically zero outside the latter frequency
range.
Further, note that the result achieved in (5) can be
also expressed in the following way:
( ) ( )
( )
( ) ( )
( )
1
exp 2
exp 2 ,
ks
s
a Y f j k f f df
f
T Y f j kT f df
−
−
= − =
=−
(6)
and that this is a form of the inverse Fourier transform
of
( )
Yf
calculated at the time point
kT−
. So, it
allows us to write
( )
k
a T y kT= −
. (7)
And substituting (7) into (3) gives
( ) ( ) ( )
exp 2
p
k
Y f T y kT j kTf
=−
=−
. (8)
Note now that (8) can be rewritten in form of the
so-called 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), (Oppenheim A. V.,
Willsky S., Nawab S. H. 1996) of the discrete signal
( )
y kT
– which is obtained by sampling the signal
( )
yt
with the rate
1
s
fT=
and which shows
only the samples of
( )
yt
(that is without “any
interest in what happens in the intervals between the
successive moments of sampling”). To see this, let us
introduce an auxiliary index
kk
=−
in (8). This
results in
( ) ( ) ( )
( ) ( )
exp 2
exp 2 .
p
k
k
Y f T y k T j k Tf
T y k T j k Tf
−
=
=−
= − =
=−
(9)
476
Omitting afterwards the prime symbol at
k
in
(9), we obtain
( ) ( ) ( )
exp 2
p
k
Y f y kT j kTf
=−
=−
(10)
with
( ) ( )
y kT y kT T=
. So, finally, we see that the
right-hand side of (10) constitutes really a definition of
the DTFT of the signal
( )
y kT
, see (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), (Oppenheim A. V., Willsky S.,
Nawab S. H. 1996).
On the other hand, we know from the literature
(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),
(Jenkins W. K. 2009) that
( )
p
Yf
is also identified, at
the same time, with the spectrum (i.e. called the
spectrum) of the sampled signal, say
( )
,DT
yt
,
modelled as a generalized function of a continuous
time t and consisting of the discrete signal
( )
y kT
mentioned just before with zeros filling the intervals
between the successive points of sampling. So, let us
express this fact as
( )
( )
( ) ( )
( )
,
,S DTFT PECT1
D T p
y t Y f y kT==
(11)
where
( )
( )
,
SPECT1
DT
yt
denotes one of the
possible definitions of the spectrum of the sampled
signal
( )
,DT
yt
that uses the notion of DTFT in the
sense as explained above.
As well known, the identity between
( )
( )
,
SPECT1
DT
yt
and
( )
p
Yf
is also manifested in
the literature (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), (Jenkins W. K. 2009) in another
way, namely by writing the following:
( )
( )
( )
( )
,
1
SPECT1
.
DT
k
p
y t Y f k T
T
Yf
=−
= − =
=
(12)
Observe now that the derivations and
relationships presented above suggest that, probably,
the expression on the right-hand side of (10) has been
named the DTFT because it incorporates samples of
an analog signal, a sum replacing an integral, and
exponential functions of the type:
( )
exp 2j kTf
−
–
all of them connected with each other into a whole
resembling an usual Fourier transform. And, as
shown in (10), when we go from the right to the left
there, this DTFT equals the auxiliary periodic function
( )
p
Yf
calculated in the Shannon’s proof. But, the
Shannon’s proof does not need to define the spectrum
of the sampled signal
( )
,DT
yt
.
Let us examine however correctness of the
definition of the spectrum
( )
( )
,
SPECT1
DT
yt
assumed in (11). To this end, assume for a moment
that there exists an inverse operator, say
1
SPECT1
−
,
which enables to obtain the sampled signal
( )
,DT
yt
from its spectrum
( )
( )
,
SPECT1
DT
yt
. Formula (12)
tells us how it could look like, namely as
( ) ( )
( )
( )
1
,
1
1
1
,
DT
k
k
y t Y f k T
T
Y f k T
T
−
=−
−
=−
= − =
=−
(13)
where
( )
1−
stands for the usual inverse Fourier
transform. Further, it is easy to obtain, from (13), the
following:
( ) ( ) ( )
( )
,
1
exp 2
DT
k
y t y t j k T t
T
=−
=
(14)
However, note now that it has been shown in
(Borys A. 2020c) that the operations performed in (13)
cannot be considered as being fully correct within the
classic mathematics (i.e. the one which does not
include such objects like distributions, in particular
Dirac distributions (Dirac P. A. M. 1947)). The reason
of this, detailed explanations, and a remedy to
circumvent the problem have been provided in (Borys
A. 2020c). This material will not be, however, repeated
here because of a lack of space as well as to avoid
accusation of auto-plagiarism. Moreover, the
reference (Borys A. 2020c) is well available.
In what follows below, we use the main result
from (Borys A. 2020c); it says that the definition of the
DTFT occurring on the right-hand side of (10) must be
modified to
( )
( )
( ) ( )
( )
( )
,
D
TFTm
DTFT
exp 2
for 1 2 and
0 for 1 2 m
k
s
s
y kT y kT j kTf
Tf f f
y kT Tf f f
=−
=−
=
=
(15)
where
( )
DTFTm
stands for the modified DTFT
after the theory presented in (Borys A. 2020c).
Therefore, the middle expression in (12), expressing
the DTFT in an equivalent way, must be modified,
too. Then, it has the following form:
( )
( )
( )
DTFTm y kT Y f=
. (16)
For details of derivation of (16), see (Borys A.
2020c).
Along the same lines as before, let us now identify
it with the spectrum of a sampled signal. That is, let
us write an equivalent of (11) for this case. We get
then
( )
( )
( ) ( )
( )
,
,S DTFTm PECT2
ET
y t Y f y kT==
(17)
where
( )
( )
,
SPECT2
KT
yt
stands for another
possible definition of the spectrum (which exploits the
notion of the DTFTm), and the sampled signal is
477
denoted now by
( )
,ET
yt
. Further, application of the
inverse Fourier transform in (17) gives
( ) ( ) ( )
( )
1
,ET
y t y t Y f
−
==
. (18)
Now, let us “demonstrate the occurrence of
samples ” in these two signals
( )
,DT
yt
and
( )
,ET
yt
that model the sampled signal in the
continuous time domain. To this end, see that the sum
of exponentials in (14) can be expressed as the so-
called Dirac comb multiplied by T (Bracewell R. N.
2000), (Osgood B. 2014). So, this gives
( ) ( ) ( )
( ) ( )
,
comb
,
D T T
k
T
y t y t t
T
y kT t kT
=−
= =
=−
(19)
where the symbol
( )
comb
T
t
stands for the Dirac
comb (Bracewell R. N. 2000), (Osgood B. 2014).
Further, for
( )
,ET
yt
, we need to use the
reconstruction formula (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), (Jenkins W. K. 2009).
Applying it in (18), we arrive at
( ) ( ) ( )
,
sinc
ET
k
y t y kT t T k
=−
=−
, (20)
with the function
( )
sinc x
defined as
( ) ( )
sinc sinx x x
=
for
0x
and 1 for
0x =
.
Observe now that both the signals
( )
,DT
yt
and
( )
,ET
yt
differ from a true image of the sampled
signal considered in the continuous time domain
(Borys A. 2020b). (This image is called in (Borys A.
2020b) a reference representation of the sampled
signal and is denoted by
( )
,RT
xt
there.) So, for
getting
( )
,RT
xt
from
( )
,DT
yt
or
( )
,ET
yt
, an
additional operation is needed (see for more details
(Borys A. 2020b)). In other words, our conclusion at
this point is that the true sampled signal
( )
,RT
xt
cannot be obtained neither by an inverse operation of
the spectrum
( )
SPECT1
nor by an inverse
operation of the spectrum
( )
SPECT2
.
Let us now come back to the Shannon’s proof
(Shannon C. E. 1949), to its second part (which can be
loosely understood as obtaining the continuous time
domain version of a signal from its discrete (digital)
form). In short, it can be expressed in the following
way:
( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
2
2
2
2
2
2
2
2
exp 2
exp 2
exp 2
exp 2
exp 2 exp 2
exp 2
sinc .
DTFT
DTFTm
s
s
s
s
s
s
s
s
f
p
f
f
f
f
k
f
f
k
f
k
y t Y f j ft df
Y f j ft df
y kT j ft df
y kT j ft df
y kT j kTf j ft
df y kT j f kT t df
y kT t T k
−
−
−
−
=−
−
=−
−
=−
==
=
==
=
= −
= − − =
=−
(21)
3 CONCLUDING REMARK
As a concluding remark, let us observe in (21) that
both the spectra of the two possible definitions of the
sampled signal occur implicitly in the Shannon’s
proof of the reconstruction formula, but they are
absolutely superfluous there. So, this underlines their
arbitrariness, artificialness, and rather a limited
usefulness – as we tried to show in this paper.
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