729
1 INTRODUCTION
Any sampling of an analog signal carried out with the
sampling rate called the Nyquist rate (Landau H. J.
1967) is said to be critical (Korohoda P., Borgosz J.
1999). Its reconstruction from its samples obtained as
mentioned can lead to non-unique results as, for
example, in the case of a cosinusoidal signal of any
phase. In this paper, we consider in very great detail
this particular case.
The analysis of the case of critical sampling of the
cosinusoidal signal of any phase and, then, its
recovery from the samples so obtained is particularly
challenging because of two ambiguities that meet
each other. These are the following ones: Dirac delta
impulses (Dirac P. A. M. 1947) occurring in the
spectrum of a cosinusoidal signal and undefined, in
principle (see, for example, (Hoskins R. F. 2009) for
more details), values of the transfer function of an
ideal rectangular reconstruction filter at the transition
edges from its zero to non-zero values, and vice versa.
Moreover, the critical sampling itself can be a source
of ambiguities. So, altogether, the problem becomes
extremely difficult and troublesome. However, we
show in this paper that using even a relatively simple
mathematics this problem can be successfully and
transparently solved.
We start our considerations in this paper with the
same definition of a rectangular window function that
was used in (Borys A., Korohoda P. 2017); it
originates from (Marks II R. J. 1991). And, note further
that this function was denoted there as
( )
xΠ
and is
given by
( )
1 for 0
1 2 for 1 2
0 for 1 2
x
xx
x
<
Π= =
>
, (1)
where x denotes a variable, which can stand for time t
or frequency f , or for any other variable.
Highlighting Problems Occurring in Analysis of Critical
S
ampling of Cosinusoidal Signal
A. Borys
Gdynia Maritime University, Gdynia, Poland
P
. Korohoda
AGH University of Science and Technology, Kraków, Poland
ABSTRACT: When the sampling of an analog signal uses the sampling rate equal to exactly twice the value of a
maximal frequency occurring in the signal spectrum, it is called a critical one. As known from the literature, this
kind of sampling can be ambiguous in the sense that the reconstructed signal from the samples obtained by
criti-cal sampling is not unique. For example, such is the case of sampling of a cosinusoidal signal of any phase.
In this paper, we explain in very detail the reasons of this behavior. Furthermore, it is also shown here that
manipulating values of the coefficients of the transfer function of an ideal rectangular reconstruction filter at the
transition edges from its zero to non-zero values, and vice versa, does not eliminate the ambiguity mentioned
above.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 14
Number 3
September 2020
DOI: 10.12716/1001.14.0
3.27
730
Afterwards, we will also use a version of the
function
( )
x
Π
, which is modified at the points
12
x
= −
and
12x =
(see section 3).
Further, observe in (1) that the “edge” points there
assume the value which is a half of the “bottom” and
“top” values. That is
( )
( )
12 12Π− =Π =
.
( )
1 0 2 12=+=
. And, this resembles the very well-
known Dirichlet condition (Brigola R. 2013)
formulated for a Fourier series with discontinuities of
the first type. So, we could interpret this observation
as a strengthening just the above choice of
( )
12Π− =
( )
12 12=Π=
. As we will see in section
3, really, it contains an element of truth. We will show
there that this is the best choice from the point of view
of the signal reconstruction though this does not
mean that it leads at the same time to a perfect signal
reconstruction.
By the way, note also that sometimes the term rect
is used in the literature for denoting the function
given by (1). Furthermore, the definition of this
function is expressed in terms of the Heaviside step
function (t) (Brigola R. 2013) as
( )
xΠ=
(x + 1/2)
−
(x
−
1/2) . (2)
The sinc function used in this paper is defined in
the following way:
(
)
( )
sin
for 0
sinc
1 for 0
x
x
x
x
x
≠
=
=
. (3)
A specific object
( )
x
δ
, which we use in our
analysis presented in the next two sections, is called,
after P. A. M. Dirac, the Dirac function or Dirac delta
impulse (Dirac P. A. M. 1947). As well-known, it is
not an ordinary function. In a simplified way, to
facilitate its understanding by engineers, it is very
often expressed in papers and textbooks as an object
satisfying the following three relations:
( )
( )
( )
( )
1
= for 0
:
=0 for 0 .
x dx
xx
x
xx
δ
δ
δ
δ
∞
−∞
=
∞=
⇒
≠
∫
(4)
The remainder of the paper is organized as
follows. Section 2 introduces an example of a
cosinusoidal signal of any phase being subject of
considerations and analysis presented in this paper.
Among others, the effects appearing during recovery
of the cosinusoidal signal sampled critically are
discussed here. In section 3, a lemma that regards the
form of a reconstructed signal as well as the best form
of the transfer function for a reconstruction filter to be
applied is proven. Finally, section 4 concludes the
paper.
2 PRELIMINARY MATERIAL REGARDING
CRITICAL SAMPLING AND RECONSTRUC-
TION OF COSINUSOIDAL SIGNAL
As we know, the sampling rate, denoted here by
s
f
,
is the inverse of the distance between successive
signal samples,
t∆
. In what follows, we also use the
capital letter T, in the sense of a period, for denoting
t∆
equivalently. By using the letter T, we simply
underline the fact that
t∆
is a sampling period.
Further, let
m
f
be a maximal frequency in the
spectrum of an analog signal considered. Then, we
will call
11
2
scr m
cr cr
ff
tT
= = =
∆
, (5)
the critical sampling rate for a given signal. Note also
that a critical distance between the successive signal
samples (i.e. a critical sampling period), as defined in
(5), is equal to
( )
1 12
cr cr scr m
tT f f∆= = =
.
Consider now the following cosinusoidal signal
( )
( ) cos 2
m
xt ft
πϕ
= −
, (6)
where
m
f
and
ϕ
are its frequency and phase,
respectively. For simplicity, we assumed here that the
amplitude of this signal is equal to one, and t
means a continuous time. Furthermore, note that
m
f
in (6) is, at the same time, the maximal frequency in
the spectrum of this signal.
Sometimes, it is convenient to rewrite (6) in a way
that expresses this signal as a delayed
( )
cos 2
m
ft
π
signal. That is in the following form:
( )
( )
( ) cos 2
md
xt f t t
π
= −
, (7)
where the delay is given by
2
d
m
t
f
ϕ
π
=
. (8)
So, we see from (8) how the signal phase is related
with a delay “embedded” in the signal given by (6).
Further, we will call sampling of (6) as a critical
one, when the sampling rate,
s
f
, equals
2
m
f
, as (5)
requires. Note that equivalently the signal given by
(6) was called a critical one in (Borys A., Korohoda P.
2017), when its sampling was performed with the rate
2
sm
ff=
. Note also that the latter quantity is called
the Nyquist rate in some papers and textbooks
(Landau H. J. 1967), (Vetterli M., Kovacevic J., Goyal
V. K. 2014).
See now that the Fourier transform of the
cosinusoidal signal given by (5) has the following
form:
( ) ( ) ( ) ( )
1
exp
2
mm m
Xf f f f f jff
δδ ϕ
= ++ − −
, (9)
731
where
(
)
δ
⋅
means the Dirac delta impulse defined
in (4).
Next, by applying the sifting property of the Dirac
delta impulse in (9), we obtain
(
)
( )
(
)
( ) ( )
1
exp
2
exp .
m
m
Xf f f j
ff j
δϕ
δϕ
=++
+− −
(10)
Further, using the Euler formula to
( )
exp j
ϕ
and
(
)
exp j
ϕ
−
occurring on the right-hand side of
equality (10), we get an equivalent form of the latter,
i.e.
(
)
( )
( ) ( )
( ) ( ) ( )
1
cos
2
1
sin
2
mm
mm
Xf f f f f
j ff ff
ϕδ δ
ϕδ δ
= ++ − −
+ +− −
. (11)
The so-called Dirac comb
Ш ()
T
t
is defined as
(
)
Ш ()
T
n
t t nT
δ
∞
=−∞
= −
∑
, (12)
see, for example, (Marks II R. J. 1991), (Osgood B.
2014). It is a useful object in signal processing and
telecommunications theories. Among others, it is used
to express the operation of signal sampling as a Dirac
comb modulation by a given analog signal to be
sampled. In other words, the operation of sampling
can be modeled as a multiplication of the Dirac comb
by this signal. That is in the following way:
( ) ( )
() ()Ш ()
sT
n
x t x t t x nT t nT
δ
∞
=−∞
=⋅= −
∑
, (13)
where
()
s
xt
denotes a continuous-time sampled
version of the signal
()xt
. So, applying (6) in (13),
we arrive at
( ) ( )
( )
( )
( )
( ) cos 2
cos 2 .
sm
n
md
n
x t f nT t nT
f nT t t nT
π ϕδ
πδ
∞
=−∞
∞
=−∞
= − −=
= −−
∑
∑
. (14)
The signal given by (14) can be converted into the
frequency domain. For doing this, note at the first step
that applying the Fourier transform definition to (13)
gives
( ) ( )
ss s
n
X f f X f nf
∞
=−∞
= −
∑
, (15)
where
( )
s
Xf
means the Fourier transform of
()
s
xt
.
In the next step, introducing (11) into (15) results
in
(
)
( )
{
( )
(
) (
)
( )
(
)
}
1
cos
2
sin
.
s s sm
n
sm sm
sm
X f f f nf f
f nf f j f nf f
f nf f
ϕδ
δ ϕδ
δ
∞
=−∞
= −+ +
+ −− + −+ −
− −−
∑
(16)
It has been shown, see (Borys A., Korohoda P.
2017) and (Borys A., Korohoda P. 2020), that (16) can
be simplified when the signal given by (6) is sampled
critically. That is in the case of applying
2
s scr m
ff f= =
, as given by (5). Then, we get
( ) (
) ( )
( )
( ) ( )
( )
2 cos 2 1
2 cos 2 1 .
sm m
n
mm
n
Xf f f n f
f fnf
ϕδ
ϕδ
∞
=−∞
∞
=−∞
= −− =
= −+
∑
∑
(17)
The signal reconstruction or its recovering
performed in the frequency domain means
multiplication of the Fourier transform of the sampled
signal, denoted here by
(
)
s
Xf
, by the transfer
function of the so-called interpolation filter, say
( )
Hf
, to get a Fourier transform of an original un-
sampled signal (Marks II R. J. 1991), (Osgood B. 2014).
In other words, the above means carrying out the
following operation:
(
)
(
)
(
)
s
HfX f Xf
=
. (18)
Further, it has been shown in (Marks II R. J. 1991)
that the transfer function
( )
Hf
of the interpolation
filter has the form
( )
1
ss
f
Hf
ff
= Π
, (19)
where the function
(
)
Π⋅
is defined in (1). Note that
another form of
(
)
Hf
is also used in the literature
for the interpolation filter. It differs, however, only
slightly from the one given by (19) and (1), and is
used, for example, in (Osgood B. 2014). The difference
between these transfer functions mentioned above
regards only two points
2
s
ff= −
and
2
s
ff=
,
where the transfer function of Marks equals
( )
12
s
f
,
but the one of Osgood is equal to 0. So, a legitimate
question arises at this point whether the above
difference can have any influence on the result of
multiplication
( ) ( )
s
HfX f
on the left-hand side
of (18) at
2
s
ff= −
and at
2
s
ff=
.
In what follows, we will consider only the latter
point because the situation at the former one is exactly
a mirror image of that at
2
s
ff=
.
Obviously, the difference mentioned above has no
influence when
( )
20
ss
Xf =
. Because then for
2
s
ff=
we obtain
( )
12 0 0⋅=
and
00 0⋅=
,
respectively, in the cases mentioned above. That is we
get then the same value. However, note that the
situation changes completely when
( )
20
ss
Xf ≠
. In
this case, we arrive at
( ) ( )
12 2 0
ss
Xf⋅≠
and
( )
0 20
ss
Xf⋅=
, accordingly. That is we obtain then
two different values. And, obviously, this can lead to
732
getting two different solutions of the recovery
problem.
In what follows, we will show that the problem
considered in this paper of sampling critically and
recovering afterwards a cosinusoidal signal belongs
just to such a category of problems, where
(
)
20
ss m
Xf f
= ≠
. Further, we will also judge
whether the Marks’s filter description or the Osgood’s
one is appropriate in this case.
Let us now return to the problem of recovering the
cosinusoidal signal spectrum
( )
Xf
from the
spectrum of its critically sampled version
( )
s
Xf
given by (17). And, for solving this problem, we will
use the left-hand side expression in (18), hoping that it
will yield a correct result. That is indicating
(
)
Xf
,
as (18) suggests, to be correct, but we are not sure of
this. Therefore, we denote below the result of
multiplication indicated in (18) by
( )
r
Xf
.
So, applying (19) with
2
sm
ff=
and (17) on the
left-hand side of (18), we get
(
)
( )
(
)
(
)
( )
( ) ( )
1
2 cos
22
21
1
cos .
2
rm
mm
m
n
mm
f
Xf f
ff
fnf
ff ff
ϕ
δ
ϕδ δ
∞
=−∞
=Π⋅
⋅ −+ =
= ++ −
∑
(20)
Observe that we obtained this simple result in the
last line of (20) due to the fact that all the “peaks” of
Dirac deltas occurring under the summation symbol
in (20), except two, are multiplied by zeros coming
from the function
( )
( )
2
m
ffΠ
. Only “peaks” of
(
)
m
ff
δ
+
and
( )
m
ff
δ
−
meet nonzero values,
( )
( )
2 12
mm
ffΠ− =
and
( )
( )
2 12
mm
ffΠ=
,
respectively.
The final result in (20) seems to be a reasonable
outcome though it does not give the expected result
(11). In the next section, we will show that it has a
physical justification - despite not resulting in (11). It
has a strong practical confirmation contrary to the
solution we would have received using the
description of the interpolation filter transfer function
( )
Hf
as in (Osgood B. 2014) with the “edge” points
( )
0
Osg m
Hf−=
and
( )
0
Osg m
Hf=
.
Note that then the equivalent of (20) would have
the following form:
( )
0
rOsg
Xf
≡
, where
( )
rOsg
Xf
denotes just the version of (20) with the Osgood’s
(Osgood B. 2014) function
( )
Osg s
ffΠ
there. The
latter function differs from
( )
s
ffΠ
given by (1)
only in two points,
( )
12
s
ff= −
and
( )
12
s
ff=
.
Consequently, this leads to
( )
0
Osg m
Hf−=
and
( )
0
Osg m
Hf=
when
2
sm
ff=
, as used above.
Obviously,
( )
0
rOsg
Xf≡
, after applying the
inverse Fourier transform to it, provides an identically
zero signal - as the following:
( ) ( )
exp( 2 )
0
rOsg rOsg
x t X f j ft df
C CC
π
∞
−∞
∞
−∞
= =
= =−≡
∫
(21)
shows. And, it is difficult to accept that the signal
(
)
0
rOsg
xt
≡
describes an un-sampled cosinusoidal
signal in a reasonable fashion.
Let us come back to consideration of (20) and
transform it to the time domain. The inverse Fourier
transform of
( )
r
Xf
gives
( )
( )
(
)
cos cos 2
rm
x t ft
ϕπ
=
. (22)
Comparison of (22) with (6) shows how the
reconstructed signal
(
)
r
xt
differs from the original
analog one,
( )
xt
. In the next section, we will look for
the cause of this.
3 THE LEMMA AND ITS PROOF
For strengthening the validity of the results presented
in (20) and (22) as well as for giving a physical
justification to them, another way of their obtaining
seems to be meaningful. In particular, showing
another way of achieving the result (22) in the time
domain would be advisable and helpful. So, to start
with the latter, let us rewrite the signal given by (6) in
the following form:
( )
( )
( ) ( )
( ) cos 2 cos sin 2 sin
mm
xt ft ft
π ϕπϕ
= +
(23)
Note now that using (23) allows us to express the
values of samples of the signal (6) sampled critically
as
( ) ( ) ( ) (
)
( ) cos cos sin sin ,
, .... , 2, 1, 0, 1, 2, ...., .
x nT n n
n
πϕ πϕ
= +
= −∞ − − ∞
(24)
In (24) as well as in what follows, the subscript cr
at
cr
T
is dropped for simplicity of notation.
However, if a need appears to use the variable T in its
original meaning defined in the beginning of section
2, this will be indicated.
Further, observe that we have the following:
( )
( )
sin 0 for any
, .... , 2, 1, 0, 1, 2, ....,
1 for even values of
and cos
1 for odd values of .
n
n
n
n
n
π
π
=
= −∞ − − ∞
=
−
(25)
in (24). So, applying this in the latter, we get
( )
( )
( )
cos for even values of
cos for odd values of .
n
x nT
n
ϕ
ϕ
=
−
(26)
733
Description of the series of signal samples in the
form given by (26) has been used in the analyses
presented in (Korohoda P., Borgosz J. 1999) and
(Borys A., Korohoda P. 2017).
In what follows, let us use the interpolation
formula in the time domain (Marks II R. J. 1991),
(Vetterli M., Kovacevic J., Goyal V. K. 2014)
( )
( )
( )
sinc
r
n
x t x nT t nT
T
π
∞
=−∞
= −
∑
. (27)
to recover (reconstruct) a signal
()xt
from its
samples
( )
, ,.., 1,0,1,.., ,x nT n = −∞ − ∞
where T
means a sampling period (which, in particular, can
assume the value following from the condition of
critical sampling).
Substituting (26) into (27) leads to the following
form:
( ) ( ) ( ) ( )
cos 1 sinc
n
r
n
x t t nT
T
π
ϕ
∞
=−∞
=−−
∑
. (28)
of the reconstructed cosinusoidal signal that was
sampled critically, with T in (28) meaning now the
critical sampling period.
In the next step, observe that with the substitution
of
2 1, ,.., 1,0,1,.., ,nk k= + = −∞ − ∞
(28) can be
rewritten as
( ) ( ) ( )
( ) ( )
( )
( )
cos sinc 2
sinc 2 cos
sinc 2 sinc 2 .
r
k
n
x t t kT
T
t kT T
T
t nT t nT T
TT
π
ϕ
π
ϕ
ππ
∞
=−∞
∞
=−∞
= −−
− −− = ⋅
⋅ − − −−
∑
∑
(29)
So, now, if we expect (22) and (29) to give the same
result we must postulate the following:
( )
( ) ( )
sinc 2
sinc 2 cos 2
n
m
t nT
T
t nT T f t
T
π
π
π
∞
=−∞
−−
− −− =
∑
(30)
to hold, where
21
m
fT=
. In what follows, we will
show that the equality (30) is really satisfied. We will
do this by formulating a formal lemma regarding this
issue, and proving it afterwards.
Lemma. The expression on the left-hand side of
(30) can be reduced to
( )
cos tT
π
.
Proof. Note that we can treat the expression on the
left-hand side of (30) as a function of a variable t.
And, for simplicity of notation, let us denote it a
function
( )
vt
. For further simplification of our
considerations, it will be convenient to introduce a
normalized time variable
( )
2tT
τ
=
in
( )
vt
. As a
result, we get then a function, say
( )
h
τ
, of a
normalized variable
τ
. Precisely, we get the
following:
( ) ( ) ( )
( )
( )
( )
2 sinc 2
sinc 2 1 2 .
n
h vT n
n
τ τ πτ
πτ
∞
=−∞
= = −−
− −−
∑
(31)
Looking at (31), it is easy to recognize that the
function
( )
h
τ
is a periodic function with a period
equal to 1.
In what follows, it will be also helpful to define
another auxiliary function
( )
g
τ
as follows
( ) ( ) ( )
( )
sinc 2 sinc 2 1 2
g
τ πτ π τ
=−−
. (32)
So, using (32), we can rewrite (31) as
(
)
( )
n
h gn
ττ
∞
=−∞
= −
∑
. (33)
Observe now that as the function
( )
h
τ
is a
periodic one with a period equal to 1 it can be
expressed in the form of a Fourier series
(
)
( )
exp 2
k
k
h c jk
τ πτ
∞
=−∞
=
∑
, (34)
where the Fourier series coefficients
k
c
,
,..,
k = −∞
1, 0,1,.., ,
−∞
are given by
( )
( )
1
0
exp 2
k
c h j kd
τ πτ τ
= −
∫
. (35)
Next, substituting (33) into (35) gives
( ) (
)
1
0
exp 2
k
n
c g n j kd
τ πτ τ
∞
=−∞
= −−
∑
∫
. (36)
Further, let us introduce a new auxiliary variable
pn
τ
= −
in (36) and swap the symbols of integration
and summation there. This leads to
( ) (
)
( )
( ) ( ) ( )
1
1
exp 2
exp 2 exp 2
n
k
n
n
n
n
n
c g p j k p n dp
j kn g p j kp dp
π
ππ
−
∞
=−∞
−
−
∞
=−∞
−
= − +=
=−−
∑
∫
∑
∫
. (37)
Note now that
( )
exp 2 1j kn
π
−=
(38)
holds for any combination of integers k and n. Taking
this into account in (37) as well as the following:
734
(
) ( ) ( ) ( )
11 1nn n
n nn
nn n
dp dp dp dp
′
− − +∞
∞ −∞ ∞
′
=−∞ =∞ =−∞
′
− − −∞
⋅= ⋅= ⋅=⋅
∑∑∑
∫ ∫ ∫∫
, (39)
we can rewrite (37) finally as
(
)
( )
exp 2
k
c g p j kp dp
π
∞
−∞
= −
∫
. (40)
Looking at (40), we see that
k
c
is at the same time
the Fourier transform
( )
Gf
of the function
( )
gp
- calculated at the integer-valued frequency k. That is
for
fk=
. So, in other words, we can write
( )
k
c Gk=
. (41)
In the next step, let us find a Fourier transform of
the signal
(
)
gp
given by (32) with the variable
τ
therein called now
p
. And, to this end, we use the
following transform pair (Marks II R. J. 1991),
(Osgood B. 2014):
( ) ( )
sinc
c
f
πτ
↔ Π
, (42)
where the function
( )
c
fΠ
means a slightly modified
function
( )
fΠ
that was defined in (1) with the
variable f used in place of the variable x. Namely,
here, we define
(
)
c
fΠ
as
( )
1 for 0
for 1 2
0 for 1 2
c
f
fc f
f
<
Π= =
>
, (43)
where the constant c means any real number different
from infinity. Note that in the literature many
different values of c are used and, furthermore, it is
argued that its specific value is not relevant. The most
popular are the following ones:
12c =
, as in (1) -
and used - for example, in (Marks II R. J. 1991);
0c =
as, for instance, in (Osgood B. 2014); as well as
1c =
used, for example, in (Borys A., Korohoda P. 2020).
In the course of this proof, we show that (30) is not
absolutely true. An intermediate result, we arrive at,
will depend upon the value of c. To get the
equivalence of the left- and right-hand sides as
postulated in (30), we will need to make use of some
additional arguments of a physical nature. They will
indicate the choice of
12c =
.
Observe now that to calculate the Fourier
transform of
( )
g
τ
given by (32), or
( )
gp
identically equal to
( )
g
τ
with
p
τ
=
, we need also,
besides (42), to use the shifting in time and scaling
properties of the Fourier transform. So, applying this
along with the linearity of the Fourier transformation
to (32), we obtain
( ) ( ) ( )
2 1 exp
1
2
c
Gf f jf
π
Π −−=
. (44)
Next, for the integer-valued frequency k, that is for
fk=
, we get
( )
( )
( )
( ) ( )
2 1 exp
2 1 cos .
1
2
1
2
c
c
kj
k
G k
k
k
π
π
Π −− =
=Π−
=
(45)
And, in the next step by introducing (41) and (45)
into (34), we arrive at
( )
( ) ( ) ( )
2 1 cos e
1
xp
2
2
c
k
h k k jk
τ π πτ
∞
=−∞
=Π−
∑
. (46)
Observe now that potentially only three
components on the right-hand side of (46) can be
nonzero. These are the ones which involve indices
1k = −
,
0k
=
, and
1
k =
.
Note that for all the remaining indices values of
the function
( )
2
c
kΠ
are identically equal to zero
according to (43). So, because of this fact, all the
remaining components in the sum on the right-hand
side of (46), except of these three mentioned above,
equal identically zero.
Taking all the above into account, we can rewrite
(46) as
( )
( ) ( ) ( )
( ) (
)
( ) ( ) ( )
1 2 1 cos exp 2
0 1 cos 0
1 2 1 cos exp 2 .
1
2
1
2
1
2
c
c
c
hj
j
τ π πτ
π πτ
=Π− − +
Π− +
+Π − −
. (47)
Observe that (47) can be further simplified because
(
)
cos 0 1
=
, but
( )
cos 1
π
= −
. Introducing this into
(47) leads to
( ) ( ) ( )
( ) ( )
1 2 exp 2
1 2 exp 2 .
c
c
hj
j
τ πτ
πτ
=Π− +
+Π −
(48)
Substituting next
( ) ( )
12 12
cc
cΠ− =Π =
, which
follows from (43), into (48) gives
( ) ( ) ( )
( )
exp 2 exp 2
2 cos 2 .
hc j j
c
τ πτ πτ
πτ
= +− =
= ⋅
(49)
Finally, taking into account in (49) that
( )
2tT
τ
= =
m
ft=
, we arrive at
( )
( )
( )
( ) ( )
(
)
( )
2
exp 2 exp 2
2 cos 2 =2 cos .
mm
m
h tT t
c j ft j ft
c ft c tT
τν
ππ
ππ
= = =
= +− =
=⋅⋅
(50)
Note now that (50) does not provide an
unambiguous result. This is so because of the fact that
the coefficient c in (50) can be chosen arbitrarily. That
is c can be any real number, as mentioned before.
However, it cannot be
∞
(because the latter does not
belong to the set of real numbers). Also, in other
735
words, (50) shows that the function
( )
t
ν
, denoting
the left-hand side of (30), and the function
(
)
cos tT
π
are “equivalent to each other” in the
sense that only a proportionality constant factor stays
between them; this factor is equal to
2c
. Obviously,
if we choose
12c =
in (50), we obtain a perfect
equivalence between these functions. That is we get
then the equality in (30) as postulated therein. But, a
question still remains how to justify, in the context of
our problem, the choice of
12c =
. To do this, we
have to recourse to the arguments of a physical
nature. And, let start with the following observation:
the period
21
mm
TTf= =
of the un-sampled
periodic function
( )
xt
given by (6) is preserved in
its recovered version
( )
r
xt
, see (29), (30), and (50).
Therefore, it is natural also to postulate preservation
of the amplitude of the above periodic function in its
recovered version. In what follows, we will do this.
First, see that
( )
xt
given by (6) can be rewritten
as
( )
( )
( ) cos 2
expression , ,
m
m
xt A f t
A ft
πϕ
ϕ
=⋅ −=
= ⋅
, (51)
where A means an amplitude assumed to be equal to
1, for simplicity; it is associated with the expression
named
( ) ( )
expression , , cos 2
mm
f t ft
ϕ πϕ
= −
.
And, similarly, taking into account (29), (30), and (51),
we can write
( ) (
) ( )
( )
( )
( ) cos 2 cos
cos 2 expression , ,
rr
m r rm
xt A t c
ft A f t
ϕν ϕ
πϕ
=⋅ =⋅⋅
⋅=⋅
, (52)
where
2
r
Ac
=
denotes an amplitude associated
with another expression called
(
)
expression , ,
rm
ft
ϕ
=
( ) ( )
cos cos 2
m
ft
ϕπ
=
.
Obviously, the expressions
( )
expression , ,
m
ft
ϕ
and
( )
expression , ,
rm
ft
ϕ
differ from each other.
Thereby,
( )
xt
and
(
)
r
xt
differ from each other,
too. However, we want to have
12
r
AA c= = =
. (53)
From (53), we get
12c =
, what applied in (50)
gives the expected result. Finally, this ends the proof
of the lemma.
♣
To complete the topic of this section, let us show
also that both the choices
0c =
and
1c
=
mentioned before lead to results which are worse than
the one achieved for
12c =
. So, consider first
0c =
.
Substituting this value in (50) gives
( )
0t
ν
≡
, which
applied finally in (29) leads to
( )
0
r
xt≡
.
Let us now interpret the above reconstructed
signal
( )
r
xt
using the terminology of approximation
theory. In this convention,
( )
r
xt
will be simply
viewed as an approximation of the original signal
( )
xt
. But, note that the dc component being
identically zero is rather a very poor approximation of
any possible function of a continuous time variable t
that can be inscribed into the series of the signal
samples given by (26).
Consider next the case of
1c =
. Substituting this
value in (50), similarly as before, gives
( ) ( )
2 cost tT
νπ
= ⋅
. And, the latter applied in (29)
leads to
( ) ( ) ( )
2cos cos
r
x t tT
ϕπ
=
.
The latter result seems to be a better
approximation of
( )
xt
than the previous one. Now,
the approximation consists of two components of the
Fourier series of the periodic function
(
)
xt
given by
(23). The dc component is perfectly determined
because it equals identically zero in (23) as well as in
(
) (
)
(
)
2cos cos
r
x t tT
ϕπ
=
. The Fourier series
coefficient multiplying
( )
cos tT
π
in (23) is equal to
( )
cos
ϕ
, but here
(
)
2cos
ϕ
. So, in terms of the
approximation theory, it is overestimated. Further,
the Fourier series coefficient multiplying
( )
sin
tT
π
in (23) equals
( )
sin
ϕ
, but in our approximation
( ) ( )
( )
2cos cos
r
x t tT
ϕπ
=
is identically equal to
zero. Thus, we can say that it is evidently
underestimated.
Comparison of the three cases regarding possible
choice of the coefficient c, which were discussed
above, shows that the best of them is the first one with
12c =
. Why? Because this choice assures a correct
calculation (reconstruction) of two from a total
number of three Fourier series coefficients of the
periodic signal given by (6).
4 CONCLUSIONS
It is well known that a critical sampling of an analog
signal can lead to ambiguous results in the sense that
the reconstructed signal is not unique. Such is the case
of sampling of a cosinusoidal signal of any phase
considered in very detail in this paper.
The non-unique results obtained for this case as
well as the reasons of a lack of uniqueness are
thoroughly explained here and in an accompanying
paper (Borys A., Korohoda P. 2020). Furthermore, it is
also shown that manipulating values of the transfer
function of an ideal rectangular reconstruction filter at
the transition edges does not eliminate the ambiguity
incorporated in the result of signal reconstruction
achieved.
REFERENCES
Marks II R. J. 1991. Introduction to Shannon Sampling and
Interpolation Theory, Springer-Verlag, New York.
Vetterli M., Kovacevic J., Goyal V. K. 2014. Foundations of
Signal Processing, Cambridge University Press, Cambridge.
Landau H. J. 1967. Sampling, data transmission, and the
Nyquist rate, Proceedings of the IEEE, vol. 55, no. 10,
1701 – 1706.
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.
736
Borys A., Korohoda P. 2020. Impossibility of perfect
recovering cosinusoidal signal of any phase sampled
with Nyquist rate, TransNav, the International Journal on
Marine Navigation and Safety of Sea Transportation,
submitted for publication.
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.
Brigola R. 2013. Fourier-Analysis und Distributionen,
edition swk, Hamburg.
