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1 INTRODUCTION
Analog signal sampling is an operation performed on
a signal (function) of a continuous time t. Its purpose is
to acquire a sequence of samples of this signal for
performing, for instance, a digital filtering with the use
of a signal processor. The operation of sampling is
performed by A/D (analog-to-digital) signal
converters. But the signal at the output of these
electronic devices can be viewed in two ways. First, as
a discrete function with the values of its dependent
variable being the sample values taken at signal
sampling instants, and its independent variable
running through the indices (from a set of integers) of
the time instants mentioned. That is it can be perceived
as a signal in which the physical time t is not important;
here, only the information about the order in which the
signal samples occur is needed. And this form of the
signal is used in digital signal processing. But,
secondly, the signal at the output of an A/D converter
can be also viewed as a sampled signal that is a
function (or object) that depends upon the physical
time t. Then it represents a function (object) with an
independent variable t, which means a continuous
time (whose values belong to the set of real numbers).
Note now that the Fourier transform (spectrum) of
a sampled signal, which we have at the output of an
A/D converter (and which we view as a function of a
continuous time t), can be calculated according to the
classical definition, i.e. as a Fourier integral. Or if we
perceive a sampled signal as an object called the
weighted Dirac comb (with an independent variable t),
its Fourier transform can be obtained via application of
the corresponding rules that are used in the case of
Dirac distributions.
In the first case, in which one decides to describe the
waveform of a continuous time at the output of an A/D
converter as a function of this time, the best description
of it is via a step function [1], [2]. This is the best way
when we model the sampling operation as a one that is
performed perfectly (ideally). Then the waveform
mentioned above and denoted below by xSTEP(t) is
expressed as
( ) ( ) ( )
k
xSTEP t x kT rect t kT
=−
=−
(1)
where x(t) is an analog signal applied at the input of an
A/D converter. Further, x(kT)’s in (1) with k’s belonging
to the set ℤ of integers stand for its samples. And T
means the period of sampling. Finally, rect() in (1) is a
Behavior of Formula for Spectrum of Sampled Signal
when Sampling Period Tends to Zero
A. Borys
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: This paper is devoted to consideration of behaviour of two formulas used in the literature to describe
the spectra of sampled signals in the case of value of the sampling period going to zero. It is shown here that they
do not lead to the same result, as we would expect. Only one of them gives an outcome compatible with reality.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 3
September 2025
DOI: 10.12716/1001.19.03.21
876
standard rectangular function that is given by the
following formula:
( ) )
1 0, 0rect t for t T and otherwise=
(2)
It has been shown [2] that the Fourier transform, i.e.
the spectrum of the function (1), is given by
( )
( )
( ) ( )
sinc exp
STEP s
k
X f X f kf fT j fT
=−
= − −
(3)
where XSTEP(f) stands for the Fourier transform of
xSTEP(t) and X(f-kfs)’s are the frequency-shifted Fourier
transforms of x(t). Furthermore, fs=1/T is the sampling
frequency,
1j =−
, and sinc() means the standard
function defined as
( )
( )
sin
0
sinc
10
x
for x
x
x
for x
=
=
(4)
Consider now the second approach [3]–[15] to
modelling the waveform of a continuous time at the
A/D converter output with the use of the weighted
Dirac comb. Then this waveform, denoted here by
xD(t), is expressed as
( ) ( ) ( ) ( ) ( )
DT
k
x t t x t x kT t kT
=−
= = −
(5)
where the object
T(t), which is dependent upon time t,
is called the Dirac comb and defined by
( )
T
k
t kT
=−
=−
(6)
In (6),
(t-kT)’s,
k
, mean the time-shifted Dirac
deltas.
It has been shown in the literature [3]–[15] that the
Fourier transform of (5), i.e. its spectrum, is given by
( )
( )
1
Ds
k
X f X f kf
T
=−
=−
(7)
where XD(f) stands for the Fourier transform of xD(t) .
Formulas (3) and (7) for the spectrum of a sampled
signal, i.e. of a waveform at the output of a sampling
device differ obviously from each other because they
are obtained with the use of different models
(descriptions) of the sampling (ideal) process. To
determine which one better describes reality, it seems
worth checking how they behave in a limiting case, for
example, when the value of the sampling period T goes
to zero. Do they converge then to the same form? It
seems worth doing such a 'test' because it is justified
and for a very simple reason. When the sampling
period goes to zero, the time waveform observed at the
output of the sampling device becomes more and more
similar to that applied to its input. Hence, the spectrum
of the former signal should become closer and closer to
the spectrum of the latter one, too. Does it really
happen? This question is answered in the next section.
The paper ends with a conclusion.
2 COMPARISON OF SPECTRA OF SAMPLED
SIGNAL VIA (3) AND (7) WHEN SAMPLING
PERIOD TENDS TO ZERO
Let us perform here the test mentioned in the previous
section. To perform it, assume that we are interested
only in the frequency range which we are able to
observe physically (that is, for example, carry out in it
a spectral analysis of the sampled signal). Moreover,
we restrict ourselves here to considering sampling of
only band-limited signals. Then, as well known, if the
Nyquist condition of sampling [3]–[15] is satisfied,
aliasing does not occur. As a consequence of this, if the
sampling frequency fs is high enough, that is the
sampling period T is small enough, then only one
‘nonzero pattern’ of the spectrum of the signal x(t)
occurs in the range of frequencies mentioned above. In
other words, we can express this fact as follows: then
only one ‘nonzero pattern’ of the spectrum of the signal
x(t) is ‘visible’ to us (for the range of frequencies
mentioned); all the other ‘nonzero patterns’ are outside
this range. And, here, we denote it by fob.
It is best to illustrate the above with a suitable
example. The author of this paper has already come up
with such an example and discussed it in one of his
previous publications [16]. So there is nothing left but
to use it here as well; we exploit it in what follows. In
this example, we refer to the radio frequency (RF)
range, i.e. the frequencies of electromagnetic waves
that are used in radio communication. It is assumed
that these are the frequencies whose scope extends
from about 3 kHz to about 3 THz. Therefore, the
maximum range of frequencies covered by a “RF
spectrum nonzero pattern” on the frequency axis will
be equal to approximately 2fob = 6 THz (from -fob to +fob).
Further, by identifying fob with fm, i.e. the maximal
frequency present in the spectrum of the signal x(t),
occurring in the Nyquist condition
1
2
sm
ff
T
=
(8)
for the absence of aliasing effects in the sampled signal
spectrum [3]–[15], we get T 1/(2fob) from the above
inequality. And finally, substituting the value of 2fob =
6 THz mentioned before into the latter inequality, we
obtain T 0,16 ps = 160 fs.
So if we sample any RF signal with the sampling
periods smaller or equal to 160 fs, we will not
experience any periodicity of the sampled signal
spectrum. Because, then, this periodicity will be not
‘visible’ in the observed range of frequencies extending
from 0 Hz to 3 THz.
Thus, based on this example, the formulas (3) and
(7) can be rewritten to the following forms:
( ) ( ) ( ) ( )
sinc exp
STEPob
X f X f fT j fT
=−
, (9)
and
( )
( )
Dob
Xf
Xf
T
=
(10)
for RF signals when they are sampled with sampling
periods smaller or equal to 160 fs. In (9) and (10),
XSTEPob(f) and XDob(f) mean, respectively, XSTEP(f) and
XD(f) valid for just this range of sampling periods.
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In the next step, note that the following relations:
( )
( )
( )
0
lim
STEPob
T
X f X f
→
=
(11)
and
( )
( )
( )
0
lim 0
Dob
T
X f for all X f
→
=
(12)
hold.
Comparison of (11) with (12) shows that the limits
indicated there do not converge to the same result.
Hence, it should be concluded that the modelling of the
signal sampled via a sequence of weighted Dirac deltas
differs (we can say significantly) from the modelling of
this signal by means of a step waveform. However, we
note that the limit in (11) is consistent with our
expectation because in the case of decreasing the value
of the sampling period (T tending to zero), the sampled
signal coincides more and more with the signal x(t),
whose spectrum is just X(f). So we must therefore
conclude that modelling the sampled signal with a step
waveform is closer to reality.
3 CONCLUSION
This paper presents another (new) result
demonstrating the weakness of the approach, in which
the sampled signals are described via sequences of the
weighted Dirac deltas.
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