737

1 INTRODUCTION

We are talking about a critical sampling of an analog

signal when the sampling frequency used for

performing this operation equals exactly twice the

value of a maximal frequency occurring in the

spectrum of this signal. Note that equivalently we can

talk in this case about a critical sampling frequency

(rate).

It is well known (Marks II R. J. 1991), (Korohoda

P., Borgosz J. 1999), (Osgood B. 2014), (Borys A.,

Korohoda P. 2017) that signal sampling performed at

the critical sampling rate can provide some unwanted

problems when carrying out the inverse operation.

That is in reconstructing (restoring) the analog signal

from its samples.

In this paper, we illustrate the problems

mentioned above on an example of a perfect

recovering a cosinusoidal signal of any phase being

sampled critically. We do this through performing

very detailed analysis of that what really happens

when the cosinusoidal signal is sampled with the

Nyquist rate and afterwards reconstructed from its

samples.

We show in this paper that recovering both the

original cosinusoidal signal amplitude and its phase

is not possible. But, what is possible then? It is

possible to recover one of these quantities under the

assumption that the second one is known. However,

carrying out some additional calculations is also then

needed.

In this paper, we show also that transfer functions

of the recovering filters used in the cases of critical

and non-critical sampling are not identical. Their form

is derived here.

The remainder of the paper is organized as

follows. Section 2 introduces an example of a

cosinusoidal signal of any phase, which is discussed

throughout this paper. Also, in this section, a

thorough analysis of the effects appearing during

recovery of the cosinusoidal signal sampled critically

Impossibility of Perfect Recovering Cosinusoidal Signal

of Any Phase Sampled with Nyquist Rate

A. Borys

Gdynia Maritime University, Gdynia, Poland

P. Korohoda

AGH University of Science and Technology, Kraków, Poland

ABSTRACT: In this paper, a problem of a perfect recovering cosinusoidal signal of any phase being sampled

critically is considered. It is shown that there is no general solution to this problem. Its detailed analysis

presented here shows that recovering both the original cosinusoidal signal amplitude and its phase is not

possible at all. Only one of this quantities can be recovered under the assumption that the second one is known.

And even then, performing some additional calculations is needed. As a byproduct, it is shown here that a

transfer function of the recovering filter that must be used in the case of the critical sampling differs from the

one which is used when a cosinusoidal signal is sampled with the use of a sampling frequency greater than the

Nyquist rate. All the results achieved in this paper are soundly justified by thorough derivations.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Number 3

September 2020

DOI: 10.12716/1001.1

4.03.28

738

is presented. In Section 3, we consider the

reconstruction formula and the form of a transfer

function occurring in it for the case of occurrence of

Dirac deltas in the signal spectrum together with its

critical sampling. In the next section, complementary

results are presented for the case of non-critical

sampling. Finally, Section 5 concludes the paper.

2 A SIMPLE EXAMPLE ILLUSTRATING THE

PROBLEM

Let us start consideration of the problem we are

discussing in this paper with an example. And, to this

end, let us take into account the sampling of a co-

sinusoidal signal of the form

( )

( ) cos 2

xt ft

πϕ

= +

, (1)

where

and

are its frequency and phase,

respectively. For simplicity, the amplitude of this

signal is assumed here to be equal to one, and t in

(1) means a continuous time.

So, sampling of (1) will be critical, when the

sampling rate,

, being the inverse of the distance

between samples, T, is equal to

f Tf= =

. (2)

Moreover, see that the Fourier transform of the

cosinusoidal signal given by (1) has the following

form:

( )

(

) (

)

(

)

exp

mm m

Xf f f f f jff

δδ ϕ



= ++ −



, (3)

where

( )

⋅

means the so-called Dirac delta impulse

(Dirac P. A. M. 1947), (Marks II R. J. 1991), (Osgood B.

2014), which is also called the Dirac delta function

(improperly) or the Dirac distribution (properly) in

the literature.

Next, using the sifting property of the Dirac delta

impulse in (3), we get

( ) ( ) ( )

( ) ( )

exp

exp .

Xf f f j

ff j

δϕ



= + −+





+−



(4)

Note further that by applying the Euler formula to

( )

exp j

−

and

( )

exp j

in (4) we arrive at an

equivalent form of the latter, i.e.

( ) ( ) ( ) ( )

( ) ( ) ( )

cos

sin

Xf f f f f

j ff ff

ϕδ δ



= ++ − −





− +− −



. (5)

In the literature, the operation of signal sampling

is modeled as a modulation of the so-called Dirac

comb (Marks II R. J. 1991), (Osgood B. 2014), i.e.

( )

comb ( )

t t nT

∞

=−∞

= −

∑

, (6)

by a given analog signal to be sampled. In other

words, the above operation can be expressed as a

multiplication of the Dirac comb by this signal. That is

( ) ( ) ( )

() ()comb

x t x t t x nT t nT

∞

=−∞

=⋅= −

∑

, (7)

where

()

means a continuous-time sampled

version of the signal

()xt

. So, we get from (7)

( ) ( )

( ) cos 2

x t f nT t nT

π ϕδ

∞

=−∞

= +−

∑

. (8)

in the case of (1).

The equivalent of (7) in the frequency domain is

given by

( ) { } { } ( )

{ }

( )

() () comb

ss T

X f x t xt t

f X f nf

∞

=−∞

==⊗=

= −

∑

  

(9)

where

{}

⋅

stands for performing the Fourier

transformation of a signal occurring in braces of this

symbol. Moreover, the symbols

⊗

and

denote

the convolution operation and sampling rate,

respectively.

Looking at the result (9), we see that the Fourier

transform of a sampled signal consists of an infinite

sum of periodically shifted Fourier transforms of its

un-sampled version multiplied by the sampling

frequency

fT=

. For details regarding derivation

of (9), see, for example, (Marks II R. J. 1991), (Osgood

B. 2014).

Further, applying (5) in (9) leads to

( ) ( )

{

( )

( ) ( ) ( )

( )

}

cos

sin

s s sm

sm sm

X f f f nf f

f nf f j f nf f

f nf f

ϕδ

δ ϕδ

∞

=−∞



= −+ +





+ −− − −+ −





− −−



∑



(10)

In the next step, note that the general expression

(10) can be highly simplified in the case of the critical

sampling of the signal (1). That is with the use of

ff=

, as given by (2). Then, the following:

739

( )

21 2

m mm

f nf f

f n f ff

f nf f f

f nf f

∞

=−∞

∞

=−∞

∞

′

= − =−∞

∞

′

→ =−∞

− +=

= −− −+=

′

= − −+=

= −−

∑

(11)

holds. Further, application of (11) in (10) gives

( ) ( ) (

)

( )

( ) ( )

( )

2 cos 2 1

2 cos 2 1 .

sm m

Xf f f n f

f fnf

ϕδ

∞

=−∞

∞

=−∞

= −− =

= −+

∑



(12)

What does it mean the operation of recovering a

signal in the frequency domain? See that it can be

simply formulated as finding a filter possessing the

transfer function, say

(

)

, which fulfills the

following:

(

) (

)

HfX f Xf

=

. (13)

So, applying (9) in (13) gives

( )

Xf Xf

f X f nf

∞

=−∞

= =

−

∑

. (14)

Further, in the next step, one can check that the

function

( )

rect







, (15)

with an auxiliary function

( )

rect

⋅

defined by

( )

rect 1 for and 0 for

xx x=≤>

, (16)

is a solution in (14), when the function

( )

well-defined (that is it is a function, not a

distribution). Note also that such a filter as that one

given by (15) is called an interpolation filter (Marks II

R. J. 1991).

Let us now introduce (12) and (15) with

ff=

into (13). This leads to

(

)

( )

(

)

(

)

( ) ( ) ( )

rect 2 cos

cos .

Xf f

fnf

ff ff

ϕδ δ

∞

=−∞



= ⋅





⋅ −+ =



= ++ −



∑



(17)

Looking at (17), we see that the result obtained

( )

Xf

differs from the expected one, that is from

( )

Xf

given by (3) or equivalently by (4). And, for

this reason, it is denoted differently, by

( )

Xf

Further, note also that the inverse Fourier transform

( )



is given by

( ) ( )

{

}

( ) ( )

2cos cos 2

rr m

x t X f ft

ϕπ

−

= =

, (18)

where

{ }

1−

⋅

stands for performing the inverse

Fourier transformation of a Fourier transform in

braces of this symbol.

Comparison of (18) with (1) shows that the

reconstructed signal

( )

evidently differs from the

original analog one,

( )

. In what follows, we will

look for the cause of this. So, to this end, let us first

check whether the transfer function given by (15),

which was derived from (13) and (14), is a correct one

in our case.

3 TRANSFER FUNCTION IN RECONSTRUCTION

FORMULA IN CASE OF OCCURRENCE OF

DIRAC DELTAS IN SIGNAL SPECTRUM AND

CRITICAL SAMPLING

Consider now in more detail (13), which is the

reconstruction formula expressed in the frequency

domain, when both the signals

( )

Xf

and

( )

Xf

in it contain Dirac deltas.

Further, we continue in this section analysis of the

example introduced in the previous section about a

critical sampling of an analog cosinusoidal signal.

And, note now that because of a critical character of

the latter operation it is highly advisable to check

correctness of calculations at edges of the

characteristics of

( )

. That is for the left-hand

side edge occurring at

ff f=−=−

and for

its right-hand side counterpart at

ff f= =

. For

this purpose, we modify slightly

( )

given by

(15) by introducing there a coefficient c, which we

assume to be unknown at first instance. So, we

rewrite then (15) as

( )

rect







, (19)

where c means a real number and

( )

stands for

a modified

( )

Substituting

( )

Xf

given by (12),

( )

given by (19), and

( )

Xf

given by (5) into (13), with

ff=

, leads to

740

( )

( ) ( ) ( )

rect 2 cos

cos

sin .

fnf

ff ff

j ff ff

ϕδ δ

∞

=−∞



⋅





⋅ −+ =



= ++ − −





− +− −



∑

(20)

In the next step, note that performing the

operations indicated in the expression on the right-

hand side of (20) results in

( )

(

)

( )

( ) ( ) ( )

rect 2 cos

cos .

fnf

c ff ff

ϕδ δ

∞

=−∞



⋅





⋅ −+ =



=⋅ ++ +



∑

(21)

Next, introducing (21) into (20), we get finally

( ) ( ) ( )

cos

sin .

c ff ff

ff ff

j ff ff

ϕδ δ



⋅ ++ + =





= ++ − −





− +− −



(22)

Observe now that there is no real number c that

fulfills equation (22), in general, for all phases

the cosinusoidal signal given by (1). However, for a

particular value of

radians, equation (22)

simplifies to

( ) ( )

( )

c ff ff

ff ff

δδ



⋅ ++ + =





= ++ −



(23)

and the resulting equation (23) is identically satisfied

for

12c =

Observe also that our previous results (17) and (18)

will be slightly modified if we take into account the

coefficient

12c =

in (19) instead of

1c =

; the latter

follows from (14) and (15). Then, we will have

( ) ( ) ( ) ( )

cos

rc m m

Xf ff ff

ϕδ δ



= ++ −





(24)

and

( ) ( )

{ }

( ) ( )

( )

cos cos 2

cos 2

rc rc m

x t X f ft

A ft

ϕπ

−

= = =



, (25)

respectively, where

( )

means a modified

version of

( )

Xf

and

( )

is the inverse

Fourier transform of the former one. Moreover,

means an amplitude of the cosinusoidal signal in (25).

Now, see finally that for

radians the

original signal

( )

given by (1) and its recovered

version

( )

given by (25) are identical. That is in

this particular case we have to do with a perfect

reconstruction. And, this is a very strong argument

for the use of the transfer function (19) with

12c

instead of the one given by (15) in our further

considerations.

By the way, note that using another method it has

been shown in (Borys A., Korohoda P. 2017) that

reasonable results of analysis of the example

discussed in this paper are achieved only when the

value of the coefficient

c =

in (19).

Let us now come back to consideration of (22) for

phases

≠

radians. It can be easily shown that

there is no such real or complex-valued coefficient c

that satisfies equation (22). Here, we propose to

overcome this problem in a way presented below.

And, afterwards, we check whether results achieved

are reasonable.

We start with ignoring the imaginary component

in (22). That is we ignore the existence of

( )

( ) ( )

sin

j ff ff

ϕδ δ



− +− −



(26)

in (22). Or, equivalently, we ignore the fact that the

imaginary components on both side of (22) do not

compensate each other. That is we ignore that the

following:

( ) ( ) ( )

sin

j ff ff

ϕδ δ



− +− −



≠



(27)

holds. And, what remains then? It remains consider-

ation of the real parts on both sides of (22) exclusively.

That is consideration of the following

( ) ( ) ( )

cos

cos ,

c ff ff

ff ff

ϕδ δ



⋅ ++ + =





= ++ −



(28)

where it is assumed that the coefficient c is real-

valued.

Observe now that for

(

)

cos 0

≠

equation (28)

simplifies to (23). Therefore, its solution

c =

identical with the previous one as well as are the

related solutions (24) and (25) describing the form of

the reconstructed signal.

Let us now take a closer look at the recovered

signal given by (25). Obviously, it is not identical with

the original one given by (1). So, the signal recovery

process in the case of

≠

and

( )

cos 0

≠

at the

same time, as described above, is not perfect.

However, not all is lost here. Namely, proceeding

heuristically as shown below allows us to arrive at the

original form of the signal (1) when we have the result

741

(25). Simply, considering the amplitude of the

cosinusoidal signal given by (25), we can write

(

)

arccos

. (29)

With this in mind, we can modify the previous

recovery algorithm for the case of

≠

and

( )

cos 0

≠

at the same time - as follows.

1 Perform all the operations to get

( )

as given

by (25). Observe that frequency of the original

signal is correctly recognized in

( )

. However,

signal amplitude there is equal to

(

)

cos

instead of 1.

2 Calculate the phase

from (29).

3 In (25), replace the amplitude

( )

cos

with 1 and

add also there the calculated (in point II above)

value of

to the cosine function argument

. It results finally in getting (1).

The algorithm sketched in points I, II, and III

above can be viewed as an improved signal recovery

algorithm for the case of recovery of cosinusoidal

signals of any phase. As seen, its application leads

then to a perfect signal reconstruction.

Note however that all this presented above works

correctly only under the assumption of knowing the

amplitude of a cosinusoidal signal before sampling.

And, in this context, we recall that we assumed for

simplicity in our example the signal given by (1) that

amplitude is 1. Obviously, without this knowledge,

the quantity

in (25) cannot be fully attributed to

( )

cos

. Then, the recovered quantity

is a

product of an unknown amplitude of the un-sampled

cosinusoidal signal and of a value of the cosine

function of its phase. That is it is a product of two

unknowns. Hence, in this case, nothing can be said

about the amplitude and phase of the original

cosinusoidal signal.

Furthermore, it follows from the detailed analysis

performed in this section that there is no such a

“recovery” transfer function

( )

in sense of

equation (13), which allows a perfect recovery of a

cosinusoidal signal of any phase sampled with

Nyquist rate. As shown, this fact follows from the

impossibility to satisfy equation (22).

4 TRANSFER FUNCTION IN RECONSTRUCTION

FORMULA IN CASE OF OCCURRENCE OF

DIRAC DELTAS IN SIGNAL SPECTRUM AND

NON-CRITICAL SAMPLING

It is worthy to complete our derivations presented in

the previous section by considering also the case of a

non-critical sampling of the cosinusoidal signal given

by (1). That is we will sample now the signal given by

(1) with a sampling frequency fulfilling the following

inequality:

f Tf= >

. (30)

So, using the general formula (9) and (4), we arrive

( ) ( )

exp

ss s

X f f X f nf

f nf f j

δϕ

∞

=−∞

∞

=−∞

= −



= −+ −+





+ −−



∑



(31)

with, do not forget now,

Next, let us come to the operation of recovery of

( )

Xf

given by (4) from

( )

Xf

expressed by

(31). To this end, we will take into account, as before,

the reconstruction formula in the frequency domain

presented in (13). Further, observe that a good

candidate to play a role of

( )

Hf

in (13) is

(

)

Hf

given by (19) because it filters out all the spectral

components outside the range of frequencies

ff f≤

, as wanted. So, applying

( )

Hf

to (31)

gives

( ) ( ) ( ) ( )

( ) ( )

exp

exp .

cs m

H fX f f f j

ff j

δϕ



= + −+





+−





(32)

And, substituting (32) into (13) with

( )



given by (3) results in

( ) ( ) ( ) ( )

exp exp

= exp exp .

ff j ff j

δ ϕδ ϕ



+ −+ − =





+ −+ −



(33)

In the next step, grouping Dirac deltas with the

same arguments in (33), we get

( ) ( ) ( )

1 exp

1 exp 0 .

c j ff

ϕδ

− − ++

+− − =

(34)

Observe now that (34) will be satisfied if and only

if the coefficients multiplying the Dirac deltas in (34)

are equal zero. This follows from the theory of

distributions (Hoskins R. F. 2009). That is from the

fact that

( )

⋅ ⋅=

if and only if

0a =

. So,

applying this in (34) leads to the following conclusion:

c must be equal to 1 in the case considered in this

section. Further, substituting

1c =

in (19) results in

(

) (

)

rect

H f Hf



= =





. (35)

So, we can conclude finally that in the case of

occurrence of Dirac deltas in signal spectrum and

non-critical sampling the transfer function of a

reconstruction filter is equal to

( )

given by (15).

At the end of this section, it is also worthy to draw

attention to the fact that some algebraic operations on

distributions are forbidden (Hoskins R. F. 2009). For

example, the following notation:

742

( )

(

)

( ) ( ) (

) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

exp exp

1 11

exp exp

sm sm

s m m ss

ff j ff j

f nf f j f nf f j

ff j ff j

f ff j ff j f f

δ ϕδ ϕ

∞

=−∞

= =



+ −+ −



= =





−+ −+ −−





+ −+ −

= = ⋅=

+ −+ −

∑



(36)

is highly incorrect, mainly because of the occurrence

of divisions of Dirac deltas. They are forbidden (or

undefined) in the theory of distributions.

5 CONCLUSIONS

The reasons of impossibility to recover both the

original cosinusoidal signal amplitude and its phase

from samples of this signal sampled critically have

been recognized in this paper. They follow from the

very detailed analyses presented.

Furthermore, it has been shown that only when

one of the aforementioned quantities is known in the

process of signal reconstruction, the value of the

second one can be recovered.

Finally, it has been also shown that a transfer

function of the reconstruction filter that must be used

in the case of a critical sampling differs from the one

which is used when a cosinusoidal signal is not

sampled critically.

REFERENCES

Marks II R. J. 1991. Introduction to Shannon Sampling and

Interpolation Theory, Springer-Verlag, New York.

Korohoda P., Borgosz J. 1999. Explanation of sampling and

reconstruction at critical rate, Proceedings of the 6th

International Conference on Systems, Signals, and Image

Processing (IWSSIP), Bratislava, Slovakia, 157-160.

Osgood B. 2014. The Fourier Transform and Its

Applications, Lecture Notes EE261, Stanford University.

Borys A., Korohoda P. 2017. Analysis of critical sampling

effects revisited, Proceedings of the 21st International

Conference Signal Processing: Algorithms, Architectures,

Arrangements, and Applications SPA2017, Poznań, Poland,

131 – 136.

Dirac P. A. M. 1947. The Principles of Quantum Mechanics,

3rd Ed., Oxford Univ. Press, Oxford.

Hoskins R. F. 2009. Delta Functions: An Introduction to

Generalised Functions, Horwood Pub., Oxford.