873

1 INTRODUCTION

It seems that most of the researchers working in the

area of signal processing believe that this highly

celebrated and commonly used [3–5] expression

( ) ( )

X f X f k T



=−

=−



(1)

for describing the spectrum aliasing and folding

effects in the case of an ideal signal sampling is fully

correct – despite receiving in [2] a strong evidence that

just the opposite might be valid. Here, we present for

the skeptics a short and very simple proof of this what

has been shown in longer considerations in [2]. We

hope that this very transparent proof, which is

presented here, will convince them.

In (1),

( )

means the spectrum of an energy,

bandlimited signal

( )

with

used to denote a

maximal frequency present in this spectrum;

( )

X f k T−

is this spectrum shifted by

on the

frequency

axis. Further,

stands for the

sampling frequency used in sampling the signal

( )

in an ideal way, where

is a continuous time

variable. Moreover,

Tf=

, where

is a sampling

period satisfying the Nyquist condition

T f f=

[4]. And,

( )

means the spectrum

of the signal

( )

sampled ideally (denoted here as

( )

This is another trial of the author of this paper to

convince researchers that (1) is not a relevant formula

for a correct description of the spectrum of a sampled

signal, when the sampling operation is carried out in

an ideal way. To do this, a simple proof is

constructed, which, however, needs some

introductory material. This material has been

presented and notation introduced in [2], but at this

moment is not available for the reader. Therefore, we

start here with presenting it first.

Any sampled signal can be modeled in two ways

in the continuous time domain, as illustrated in Fig. 1

(upper curve) and in Fig. 2.

As we see in Fig. 1 (upper curve), a graphical

description of the sampled signal, denoted here as

( )

, consists of a series of the weighted Dirac deltas

occurring uniformly on the continuous time axis in

the distance of

from each other. And, this is the

Simple Proof of Incorrectness of the Formula

Describing Aliasing and Folding Effects in Spectrum of

Sampled Signal in Case of Ideal Signal Sampling

A. Borys

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: A simple proof of the incorrectness of the formula, which is used in the literature nowadays, for

description of the aliasing and folding effects in the spectrum of a sampled signal in the case of an ideal signal

sampling, is given in this paper. By the way, it is also shown that such the effects cannot occur at all, when the

signal sampling is considered to be performed perfectly.

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Volume 15

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DOI: 10.12716/1001.15.04.20

874

first way of modeling; it is used in the literature; see,

for example, [3–5].

The second possible way of modeling is

graphically illustrated in Fig. 2.

Figure 1. Example sampled signal representation (upper

curve) in form of a series of weighted Dirac deltas occurring

uniformly on the continuous time axis in the distance of T

from each other, and its un-sampled version (lower curve).

Here, the sampling operation is assumed to be carried out

ideally. Moreover, we note that the figure exploits the same

signal, which was also discussed in [1, 2].

Figure 2. Graphical illustration for a sampled signal

representation of the un-sampled signal shown in Fig. 1

(lower curve) in form of a series of time-dependent signal

samples occurring uniformly on the continuous time axis in

the distance of T from each other. Here, similarly as in Fig.

1, the sampling operation is assumed to be performed

ideally.

Now, before going into further details, we would

like to underline that both the ways of modeling of

the sampled signal, the one illustrated in Fig. 1 (upper

curve) as well as the second depicted in Fig. 2,

concern an ideal sampling operation (that is a one

performed ideally). And, this is important in the

considerations presented here; the case of a non-ideal

sampling will be reported elsewhere.

The sampled signal in Fig. 2 is denoted as

( )

,KT

;

it is not identical with the signal

( )

in Fig. 1. So,

these two ways of modeling of the sampled signal are

evidently different. Whereas its more natural

description (that is a true one) is, obviously, the one

presented in a graphical form in Fig. 2. Why? Simply

because it consists of a series of true signal sample

values occurring at appropriate time instants. And

nothing more (on the contrary to the case shown in

Fig. 1 (upper curve)).

Thus, in this context, a question of why the

sampling signal modeling presented in Fig. 2 is not

used in the literature – is a quite legitimate one.

The answer to this question is simple: the signal

illustrated in Fig. 2 neither possesses the Fourier

transform, nor it can be expanded in a Fourier series.

In other words, this signal has no representation in the

frequency domain – via a conventional understanding

of the signal spectrum. And, this is, obviously, a very

serious obstacle for its usage in the signal processing.

However, the people came up with a way of

circumventing this. They simply do the following

with the signals such as the one shown in Fig. 2:

multiply the signal sample values occurring at the

appropriate time instants by the time-shifted Dirac

deltas (shifted to the appropriate instants of the signal

sampling). Thereby, they get such a signal as the one

presented in Fig. 1 (upper curve). And, this signal

possesses the spectrum; it is expressed by the

expression on the right-hand side of (1). That is the

spectrum of the ideally sampled signal,

( )

, is

assumed to be equal to

( )

, where the latter

means the Fourier transform of the signal

( )

. Or,

in other words, the spectrum

( )

is identified

with the spectrum

( )

. Is this legitimate? The

answer to this question is negative in this paper.

By the way, note that in view of what was said

above the following formula:

( ) ( )

X f X f k T



=−

=−



(2)

is a correct version of the one given by (1).

What we need to formulate our proof of the

incorrectness of the formula (1) are the analytical

descriptions of the signals

( )

, and

( )

,KT

And, let us start with

( )

. To this end, see that it

can be expressed analytically as a signal

( )

multiplied by the so-called Dirac comb

( )



[3–5].

That is as

( ) ( ) ( )

x t t x t



=

, (3)

where the Dirac comb

( )



is defined as follows:

( ) ( )

t t kT





=−

=−



(4)

with

( )

, ., 1,0,1,.,t kT k



− = −

meaning the time-

shifted Dirac deltas (distributions or impulses) [3–5].

Next, consider the analytical description of the

signal

( )

. Because of the reasons discussed just

before, we conclude that in the case of an ideal

sampling we have

( ) ( )

,s K T

x t x t=

(5)

Once again, this follows simply from the fact that a

true sampled signal looks like the one visualized in

Fig. 2, not as the signal depicted in Fig. 1 (upper

curve).

The third signal,

( )

,KT

, which in fact – due to

(5) – represents the true sampled signal, can be

described analytically as shown in [2] – via the

Kronecker functions and the Kronecker comb.

In [2], a basic Kronecker time function

( )

0,tT



has been defined as

0, 0,

1 if 0 with defined

as a real number (or, in other

words, when this real-valued

number assumes the integer

value 0)

0 otherwise .

r t T

r t T r













(6)

875

Accordingly, a time-shifted Kronecker time

function

( )

,k t T



has been also defined in [2] – as

the following function:

1 if with defined

as a real number (or, in other

words, when this real-valued

number assumes the integer

value )

0 otherwise .

k r k t T

k r t T r













(7)

And, note that it follows from (7) that the function

( )

,k t T



is a function

( )

0,tT



but shifted now on

the continuous time axis t by k time units T – to the

right if k > 0 , and to the left when k < 0.

Using the time-shifted Kronecker time function

( )

,k t T



, it is easily to define the so-called Kronecker

comb [2]. It is denoted here by

( )

,KT



; and, it is

defined as

( ) ( )

,,K T k t T





=−



(8)

where the first index K at

( )

,KT



stands for the name

of Kronecker, but the second one, T, means a

repetition period. This comb is illustrated in Fig. 3.

Figure 3. Visualization of the Kronecker comb given

analytically by (8).

Note now that using (8) we can describe

analytically such a signal

( )

,KT

as depicted in Fig. 2

in the following form:

( ) ( ) ( )

,,K T k t T

x t x kT t





=−



, (9)

where, similarly as before, the first index K at

( )

,KT

stands for the name of Kronecker, but the second one,

T, means a sampling period.

Further, see that the following:

( ) ( ) ( )

( ) ( ) ( ) ( )

K T k t T

k t T K T

x t x kT t

t x t t x t







=−



=−

=  = 



(10)

then also holds. Hence, we can write

( ) ( ) ( )

,,K T K T

x t t x t



=

(11)

The remainder of this paper consists of one section.

It contains a proof of the incorrectness of the formula

(1) that describes the aliasing and folding effects in the

spectrum of the sampled signal sampled in an ideal

way. Moreover, it is also shown there that the signal

( ) ( )

,s K T

x t x t=

does possess the spectrum and the

following:

( ) ( ) ( )

,s K T

X f X f X f==

(12)

holds, instead of (1). In (12),

( )

,KT

means the

spectrum of the signal

( )

,KT

2 PROOF OF THE INCORRECTNESS OF THE

FORMULA (1)

The most important for the proof presented below is

to notice that the following:

( ) ( ) ( ) ( )

,K T T T

x t t x t t



 = 

. (13)

holds. And, what we need in addition here is the

assumption of the existence of the spectrum of the

signal

( )

,KT

From the previous section, we know that the

Fourier transform of

( )

,KT

does not exist.

However, it does not mean at the same time that its

spectrum does not exist, too. Why? Because the

spectrum of a signal can be defined in a broader sense;

not simply as a (direct) Fourier transform of the

signal. And, we use this possibility in this paper.

So, to this end, we define an extended signal

spectrum as follows.

Provisional extended definition of signal spectrum.

If a signal of a continuous time is represented by an

integrable function that possesses a Fourier transform,

then the spectrum of this signal is given by the usual

Fourier transform. But, when a signal of a continuous

time is represented by a non-integrable function

which is a train of single values separated uniformly

by intervals with all values being identically equal to

zero, then its spectrum is defined as a Fourier

transform of a function resulting from transforming

the train of single values (separated uniformly by

intervals with all values being identically equal to

zero) to an integrable function that is close (in some

sense; a few good measures for defining this can be

defined) to this train. Making this provisional

definition a precise one will follow from the results

presented at the end of this paper.

Note now that the second part of the above signal

spectrum definition can be viewed as a generalization

of its first part, which builds up an usual signal

spectrum definition. And, in this regard, we can see

here a very good analogy with the notions of

functions and generalized functions (i.e. distributions,

as, for example, Dirac delta), where the latter ones are

generalizations (in some sense) of the former ones, but

still remaining nonconventional objects (when we

compare them with ordinary functions).

To see this analogy in more detail, let us start with

the following observation: both the Dirac delta as well

as the spectrum

( )

,KT

of the signal

( )

,KT

not exist in a conventional sense. The first one does

not exist as a function; however, nowadays, nobody

doubts that it at all exists. And, similarly, we know

876

that

( )

,KT

does not exist as an usual Fourier

transform. But, it does not mean, at the same time,

that this signal spectrum does not exist at all. It exists

via the extended definition of the signal spectrum

formulated above.

The second observation regards a way of how the

Dirac delta and the spectrum

( )

,KT

“reveal

themselves” in the world of functions and the world

of spectra of signals possessing Fourier transforms,

respectively. Or how they “cooperate” with these

corresponding worlds?

For illustration, let us start with the Dirac delta.

And, in what follows, we use its definition that

exploits the notion of a functional and the so-called

test functions [6, 7]. Further, let us assume that

( )



stands here for a test function [6, 7]. With this, we

define the Dirac distribution as such an object (a

generalized function) that is characterized by a

functional, say, D , which, when applied to any test

function

( )



results in

( )

0Dt



Note that this – with

( )

in place of

( )



– is

expressed symbolically in the following way:

( ) ( ) ( )

0x t t dt x





−



in the signal processing literature

(although, it has no strict mathematical meaning [6,

7]). Further, it is worth noting that both the above

equations express the so-called sifting property of the

Dirac delta. Further, the second equation with

( )





in it is used by engineers as a symbolic definition of

the operator (functional) D.

So, we see from the above that the Dirac delta

“reveals itself” in the world of functions simply

through its definition (which is nothing else than its

highly celebrated sifting property).

Now, note that we have a similar situation in the

case of the spectrum

( )

,KT

. To see this, let us recall

the second part of the extended definition of the

signal spectrum formulated before and invoke a

corresponding operator, say, R (transforming a non-

integrable function of a continuous time (of the type

mentioned) into another one, say,

( )

that is,

however, an integrable function and possesses a

Fourier transform) – for performing this task. So,

according to the aforementioned definition, we can

write

( ) ( )

( )

,r K T

x t R x t=

(14)

and

( ) ( )

( )

K T K T

X f R x t

x t X f

(15)

where

( )



stands for the usual Fourier transform.

So, through (15),

( )

,KT

“reveals itself” in the

world of spectra of signals possessing Fourier

transforms. Moreover, (15) shows that

( )

,KT

exists as a “well-defined” function (in the sense of

being integrable) and can be convolved with other

spectra (because

( )

can).

Note now that using (13) and the above result

regarding the existence of

( )

,KT

we can write

( ) ( ) ( )

( )

K T T T

X f d X f

X f k T

  



−



=−

 − = =

=−





(16)

for the frequency domain. In (16),

( )

f

means the

Fourier transform of the Dirac comb given by (4);

moreover, it is itself a Dirac comb. So, the

( )

f

has

the following form [3–5]:

( ) ( )

( )

f f kf







=−

 = −



(17)

In the next step, see that after taking into account

(17) in (16) and performing all the needed operations

there, we can rewrite (16) in the following form:

( ) ( )

( )

K T T

X f k T X f

X f k T



=−



=−

− = =

=−



(18)

And, finally, (18) shows that (12) must hold. This

also means that

( ) ( ) ( )

X f X f X f==

must hold. In

other words,

( )

is not given by (1).

Furthermore, it allows to make precise our

provisional extended definition of the signal

spectrum. Simply, the operator R associated with it

must be so chosen that

( ) ( )

x t x t=

holds.

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