2. Substantiation of the Proposed Approach for Frequency Estimation
Analysis was carried out for continuous time dependence of an analyzed signal. It
was done on the basis of the inner product determined according to (1) for an analyzed signal and auxiliary sine function with zero initial phase and arbitrarily
selected frequency:
where A$_{2}$ denotes the amplitude of the analyzed signal, ${\varphi}$$_{2}$ and
${\omega}$$_{2}$ correspondingly denote the values of the initial phase and angular
frequency of the analyzed signal, b denotes the duration of the analyzed signal, and
${\omega}$$_{1}$ denotes the value of the angular frequency of the employed auxiliary
sine function. All three parameters of the analyzed signal, A$_{2}$, ${\omega}$$_{2}$
and ${\varphi}$$_{2}$, are considered to be unknown. The value of the initial phase
${\varphi}$$_{2}$ is considered to stay within the range of values from 0$^{\circ}$
to 90$^{\circ}$. The inner product (1) represents the sine Fourier transform of the continuous analyzed signal determined
over some time interval from 0 to b. According to one study [11], for the case of ${\omega}$$_{1}$ ${\neq}$ ${\omega}$$_{2}$, the results of the calculation
of the inner product (1) can be expressed as:
where all terms s$_{1}$..s$_{5}$ can be determined according to:
The inner product determined according to (1) of the analyzed signal and auxiliary sine function has not been complemented by the
corresponding inner product of the analyzed signal and auxiliary cosine function.
This is due to not applying the complete system of the orthogonal functions, which
cannot be directly used for frequency estimation. Therefore, as the maximum value
of (1) is attained for the case when ${\omega}$$_{1}$ ${\neq}$ ${\omega}$$_{2}$, the frequency
estimator is biased. Increasing the duration of the analyzed signal b partially allows
us to mitigate this error of the frequency estimation. But practically applying the
inner product (1) for the purposes of frequency estimation is complicated. In order to attain an admissible
level of accuracy, various values of the initial phases will require various durations
of the analyzed signal. Besides, the duration of the analyzed signal for practical
applications is limited for the vast majority of cases, and this limitation of the
analyzed signals causes additional difficulties in the practice of frequency estimation.
For analyzed signals with different duration, different values of the initial phase
${\varphi}$$_{2}$ obviously will lead to different levels of accuracy.
2.1 The Affect of the Selected Frequencies of the Auxiliary Sine Functions on Sidelobe
Properties
Eq. (2) can be substantially simplified by means specific selection of the frequencies of
the auxiliary sine functions. This simplified form of the inner product can be efficiently
used for the purposes of frequency estimation. The simplification of (2) can be attained by the selection of the frequencies used in (1) for auxiliary sine functions according to the following relations:
where f$_{s}$$_{1}$ denotes the frequency used in (1) for the auxiliary sine function selected in order to simplify (2) by eliminating terms s$_{2}$ and s$_{3}$. f$_{s}$$_{2}$ denotes the frequency used
in (1) in the auxiliary sine function to simplify (2) by eliminating terms s$_{4}$ and s$_{5}$, k = 1..n. It also denotes the number of
a particular auxiliary sine function from the total amount of auxiliary sine functions
n.
As it is easy to infer from the analysis of (2), for the case when the values of the frequencies used in auxiliary sine functions
fit (8), the value determined according to (2) of the inner product S will contain only terms s$_{1}$, s$_{4}$, and s$_{5}$. Due
to the presence of the dependence on the ${\omega}$$_{1}$ multiplier in cos(b${\omega}$$_{1}$),
the dependence of the terms s$_{2}$ and s$_{3}$ in (2) on ${\omega}$$_{1}$ is represented by the cosine functions. The selection of the
frequencies of the auxiliary sine functions according to (8) allows us to eliminate the impact of the terms s$_{2}$ and s$_{3}$on the frequency
dependence of the results of the inner product calculation S.
The elimination of the impact of the terms s$_{2}$ and s$_{3}$ also leads to a simultaneous
increase of the impact of the terms s$_{4}$ and s$_{5}$on the frequency dependence
of this inner product. The selection of the frequencies of the auxiliary frequencies
according to (9) leads to an antithetical result and allows us to eliminate the impact of the terms
s$_{4}$ and s$_{5}$ with a simultaneous increase of the impact of s$_{2}$ and s$_{3}$
on the dependence of the inner product S determined according to (1) on the frequencies of the auxiliary sine functions. Therefore, the selection of the
frequencies of the auxiliary sine functions according to (8) and (9) allows us to segregate the input of the terms s$_{2}$ and s$_{3}$ from the input
of the terms s$_{4}$ and s$_{5}$ to the ultimate value of the inner product S. Fig. 1 presents typical dependencies of the inner product (2) on frequencies calculated according to (8) used in auxiliary sine functions. All calculations have been carried out for the
case of f$_{2}$ = 46 Hz, ${\varphi}$$_{2}$ = 60$^{\circ}$, A$_{2}$= 1, and b = 0.2.
Fig. 2 presents the same frequency dependence calculated for the case when the frequencies
of the sine functions fit (9).
Fig. 1. Typical frequency dependence of inner product S calculated according to(1)determined for the case when the frequencies of auxiliary sine functions fit the values determined according to(8).
For the case where the frequencies of the auxiliary sine functions have been selected
according to (9), frequency estimation can be carried out on the basis of the observation of the particular
points of the sidelobes in the frequency dependence of the inner product S. The results
of simulations have shown that for the case when some particular frequency coincides
with the frequency of the analyzed signal, the sum of the specifically selected values
of the frequency dependence of the inner product S is equal to zero, regardless of
the value of the initial phase of the analyzed signal. This behavior of the sidelobes
is illustrated in Fig. 3, which presents the fragment of the frequency dependence of the inner product (1) for the case of b = 3 and, f$_{2}$ = 40 Hz. A previously mentioned area of interest
which contains characteristic elements of the sidelobes with a sum equal to zero is
marked with an elliptical curve.
In the case of calculated frequencies according to (9) of the auxiliary sine functions for the selected duration of the analyzed signal
with b = 3 and k = 240, the value of f$_{\mathrm{s2}}$ will be equal to 40 Hz. Therefore,
this frequency will exactly coincide with the frequency of the analyzed signal. Fig. 4 presents the frequency dependence of the inner product (1) calculated for the case of f$_{2}$ = 40 Hz and b = 2 s.
For all results of the calculations presented in Figs. 1-4, the values of the inner
product (1) have been calculated numerically by applying the trapezium formula. The results of
the calculations presented in Fig. 4 show that the existence of some difference between the frequency of the analyzed
signal and auxiliary sine function used in (1) leads to the gradual increase of the previously distinguished characteristic points
of the sidelobes. Therefore, the sum of these elements is not equal to zero.
The described behavior of the sidelobes which allows us to remove the bias of the
frequency estimation caused by the initial phase of the analyzed signal is determined
by the previously mentioned removal of the terms s$_{4}$ and s$_{5}$ in (2). For the case when the initial phase of the analyzed signal coincides with the initial
phases of the auxiliary sine functions (i.e., for the case of ${\varphi}$$_{2}$ =
0 in (1)), the frequency dependence of S will be determined only by terms s$_{2}$ and s$_{4}$.
Since the selection of the frequencies according to (9) allows us to remove only the terms s$_{4}$ and s$_{5}$ and does not allow us to eliminate
the term s$_{1}$, such a selection of the frequencies disrupts a possible approach
for frequency estimation based on analysis of the interaction of the terms s$_{2}$
and s$_{4}$. However, it provides another option suitable for frequency estimation.
Provided s$_{4}$= s$_{5}$ = 0, which is attained by the selection of the frequencies
of the auxiliary sine functions according to (9), and for the case of the duration of the analyzed signal fitting the condition sin(b${\omega}$$_{2}$)
= 0, the value of s$_{2}$ = 0 and, consequently, the value of inner product S will
be determined by a single alternating term s$_{3}$. The residual of the independent
on b term s$_{1}$ and the remaining alternating term s$_{3}$determines the presented
in Fig. 3 frequency dependence of marked with an elliptic curve characteristic points. The
described interaction of the terms s$_{1}$, s$_{2}$, and s$_{3}$ and the resulting
frequency dependence of the inner product S are illustrated in Fig. 5.
Further analysis will be carried out based on the following premises:
1. The exact value of frequency of the analyzed signal stays within the frequency
interval limited by the maximum value of frequency dependence of the inner product
(1) and the closest minimum value of this frequency dependence which follows the closest
zero crossing of the main lobe. These particular points of the dependence of the inner
product (1) on frequencies of the auxiliary sine functions allow us to distinguish the array
of the frequencies, which is supposed to contain the exact frequency of the analyzed
signal, and are correspondingly marked by the points A, B, and C in Figs. 1 and 2.
2. For the case of a multifrequency signal, the duration of the analyzed signal and
the distance between its frequency components are supposed to have values which allow
us to consider that the main lobe and its closest sidelobes of a particular frequency
component do not affect the location and size of the sidelobes, which are determined
by other frequency components. This stipulation in fact implies that the case of a
multifrequency signal can be reduced to a separate analysis of the contribution of
each frequency component to the overall frequency dependence of the inner product
S.
3. The exact value of frequency of the analyzed signal should not necessarily correspond
to the maximum value of inner product (1), but belongs to the previously distinguished frequency interval limited by the maximum
value of frequency dependence of the inner product (1) and the closest minimum value of this frequency dependence.
4. Frequency estimation is carried out on the basis of the consecutive calculation
of the series of the inner products of the analyzed signal and auxiliary sine functions
with various frequencies. For each inner product calculation, the analyzed signal
is limited by its various durations, which allow us to determine various arrays of
the frequencies calculated according to (9) for the auxiliary sine functions.
5. According to the results of the simulations presented in Fig. 3, the most accurate frequency estimation is attained for the case when the sum of
the results marked by the elliptical curve in Fig. 3 of the inner product calculation is closest to zero. Thus, the criterion for the
assessment of the proximity of the estimated value of frequency to the exact frequency
of the analyzed signal should be based on calculation of the sum of these particular
values of S. This criterion will be accomplished for the case when for a certain duration
of the analyzed signal among the frequencies determined according to (9) for the auxiliary sine functions some particular frequency will have its value as
close possible to the exact frequency of the analyzed signal.
6. For the case of conventional algorithms for frequency estimation, based on applying
the discrete Fourier transform, the frequencies used in auxiliary sine and cosine
functions usually are selected to be a multiple of the sampling frequency. For this
case, their circular frequencies can be calculated according to the following relation
[12]:
Fig. 2. Typical frequency dependence of the inner product S calculated according to(1)determined for the case when the frequencies of auxiliary sine functions fit the values determined according to(9).
Fig. 3. The affect of the calculated according to(9)frequencies of the auxiliary sine functions on sidelobes of the frequency dependence of the result of the inner product calculation for the case when a certain frequency of the auxiliary sine function coincides with the frequency of the analyzed signal.
Fig. 4. Typical frequency dependence of inner product S calculated according to(1)determined for the case when the frequencies used in auxiliary sine functions fit the values determined according to(9)and do not coincide with the frequency of the analyzed signal.
where N denotes the total amount of signal samples, and k denotes the number of a
particular component of the Fourier transform. For the considered in frequency estimator
instead of applying (10) for the circular frequencies of the auxiliary trigonometric functions, the values
of frequencies will be selected according to (9). Also, it should be noted that calculation of the frequencies of the auxiliary sine
functions according to (9) usually allows us to attain much lower frequency resolution in comparison with the
conventional approach based on applying (10). In the case of applying (9) for the selection of the frequencies, a high level of frequency resolution can be
attained only for analyzed signals with a sufficiently large duration. For analyzed
signals with the same duration determined according to (9), frequency resolution is considerably lower than for the case where the frequencies
of the auxiliary sine functions are selected according to (10).
Frequency estimation is carried out based on the assumption that the exact value of
frequency of the analyzed signal belongs to the range of frequencies limited by the
points A and B in Figs. 1 and 2. The existence of undesirable noise components in
analyzed signals, however, might complicate visual recognition of this frequency interval.
Therefore, for practical applications, it is desirable to develop complementary methods
which would provide additional tools which could be used in order to identify this
frequency interval. In the current study, such a tool was developed on the basis of
applying the antiderivative function determined for the dependence of the result of
the inner product calculation (2) on values of angular frequencies ${\omega}$$_{1}$of auxiliary sine functions.
2.2 Estimation of the Frequency Interval which Contains the Frequency of the Analyzed
Signal
The necessity of employing the distinguished frequency interval is determined by the
selected approach for frequency estimation, which implies that the exact frequency
of the analyzed signal does not necessarily correspond to the maximum value of the
determined according to (1) inner product. Since the value of frequency cannot be attributed to the maximum value
of the inner product, it should be associated with some other specific properties
of the dependence of the inner product on frequencies of the auxiliary sine functions.
According to the previously made assumption the exact frequency is associated with
the frequency interval limited by the maximum value of S and the closest minimum value
of this dependence (points A and B in Fig. 1). The contamination of the analyzed signal by noise components in analyzed signals,
however, distorts the dependence of the inner product (1) on frequencies f$_{1}$ used in auxiliary sine functions and consequently complicates
recognition of this frequency interval. Nevertheless, the mentioned problem of the
frequency interval recognition partially can be mitigated on the basis of the analysis
of the properties of the antiderivative function determined for the dependence of
the inner product (1) on values of angular frequencies of the auxiliary sine functions S = f(${\omega}$$_{1}$).
The selection of this particular function for the purpose of the identification of
the necessary frequency interval was motivated by the intention to smooth the function
S = f(${\omega}$$_{1}$) and conceal its random oscillations caused by the contamination
with some undesirable noise component. In a simplified form, assuming zero integration
constant, this antiderivative function can be expressed as:
where all terms can be determined according to:
The results of the calculations presented in Fig. 6 illustrate typical frequency dependencies of the antiderivative function according
to (11).
The results of the calculations presented in Fig. 7 illustrate the impact of the duration of the analyzed signal on frequency dependence
of the expressed antiderivative function according to (11).
The previously mentioned impact of the undesirable noise components on S(f$_{1}$)
is illustrated in Figs. 8 and 9. Fig. 9 presents this frequency dependence for an analyzed signal without any undesirable
noise component, while Fig. 9 presents the same frequency dependence calculated for a signal contaminated with
noise. Fig. 10 presents the frequency dependence of the antiderivative function for the cases of
the absence and presence of noise components in analyzed signals.
For all cases, noise contamination has been represented by normally distributed white
noise. The results of the calculations illustrate the distortions in the calculated
frequency dependence of the inner product determined according to (1) caused by the presence of the noise components in analyzed signals. In Fig. 8, the frequency of the auxiliary sine function approaching the exact frequency of
the analyzed signal leads to gradual increase of the sidelobes. According to the results
in Fig. 9, the presence of the noise components disrupts the gradual increase and can lead
to a false conclusion about the existence of a certain frequency component in the
analyzed signal. Also, it may cause difficulties in the identification of the previously
distinguished frequency interval, which contains the exact frequency of the analyzed
signal.
The results of the calculations presented in Fig. 10 indicate that the presence of the noise components also distorts the frequency dependence
of the antiderivative function. However, in the vicinity of the frequency interval
limited by the maximum value of B(f$_{1}$) and its closest local minimum value, such
a disruptive impact becomes less evident. The maximum value of frequency dependence
presented in Fig. 8 still corresponds to the main lobe frequency equal to 1002 Hz. However, from the
prospective of the impact of the noise components on the antiderivative function according
to the results presented in Fig. 10, it is possible to distinguish a certain frequency region in the vicinity of the
exact frequency of the analyzed signal for which the impact of the noise components
does not lead to such drastic distortions. This frequency interval is distinguished
by the frequencies of 998.1 Hz and 1009 Hz in Fig. 10. Therefore, this impact becomes mitigated for the range of frequencies in the vicinity
of the exact frequency of the analyzed signal.
Although the presence of the noise components can lead to some shifting of the previously
distinguished frequency interval, it still covers the exact frequency of the analyzed
signal. Therefore, for the case of the analyzed signals with noise components, the
existence of a certain frequency component can be established by calculating the antiderivative
function (11), at least in some range of signal-to-noise ratios. The selection of the frequencies
of the auxiliary sine functions according to (9) changes the dependence of the antiderivative function on values of frequencies of
the auxiliary sine functions. For the case of frequencies selected according to (9), this dependence can be expressed as:
where D(${\omega}$$_{1}$) denotes the antiderivative function determined for inner
product (1) and calculated for the case when the frequencies of the auxiliary sine functions
fit the values determined according to (9), and the expressions for D$_{1}$..D$_{8}$ can be determined according to:
Fig. 11 illustrates the dependencies of the antiderivative function on frequency of the auxiliary
sine functions.
The results of the calculations presented in Fig. 11 indicate that despite the presence of the noise contamination, the location of the
minimum value of the antiderivative function that has not been affected by the noise
component is close to the location of the minimum of the antiderivative function that
has not been distorted by noise contamination. In the vicinity of this minimum value,
two antiderivative functions also display higher similarity than in distant regions
of the considered frequency dependence.
For the case of frequencies selected according to (9) for the auxiliary sine functions mentioned, the similarity of the antiderivative
functions in the vicinity of their minimum value becomes more evident than for the
case of results of the calculations in Fig. 10, which were carried out for arbitrarily selected frequencies of the auxiliary sine
functions. Such similarity of the antiderivative functions of the contaminated and
uncontaminated signals confirms that the existence of the certain frequency component
in noise can be determined by identifying the minimum value of the antiderivative
function calculated for the dependence of the inner product (2) on frequencies used for auxiliary sine functions.
For the purposes of simplicity, all antiderivative functions in Figs. 10 and 11 have been represented by cumulative sums of the corresponding elements of the frequency
dependence S(f$_{1}$). Consequently, the proposed approach for frequency estimation
for a single tone signal which does not contain any noise component consists of the
following procedures:
1. Limitation of the whole array of values of the preliminary sampled signal by some
intermediate value which will correspond to some intermediate signal duration b.
2. Calculation of the array of frequencies of the auxiliary sine functions with zero
initial phases according to (9).
3. Calculation of the values used in auxiliary sine functions in particular moments
of time which are determined by the value of sampling frequency used in order to obtain
the array of values of the analyzed signal.
4. Calculation of the series of the inner products of the samples of the analyzed
signal and auxiliary sine functions.
5. Calculation of the sum of the results marked with an ellipse in Figs. 3-5 for the inner product calculation. The first element of this sequence directly follows
the maximum calculated value of the inner product (point A in Figs. 1 and 2). All other elements of this sequence alternate with non-zero results of the inner
product calculation, as illustrated by the results of the calculations presented in
Figs. 3-5.
6. The selection of the other intermediate value among the array of samples of the
analyzed signals which will correspond to other intermediate signal duration b and
reiteration of procedures 2-5.
7. The selection of the most accurately estimated value of frequency. The value of
frequency which corresponds to the maximum value (point A in Figs. 1 and 2) of the dependence of the inner product on frequencies of the auxiliary sine functions
indicates the upper boundary of the frequency interval which contains the exact frequency
of the analyzed signal. Frequency associated with the minimum value of this frequency
dependence (point B in Figs. 1 and 2) indicates the lower boundary of the frequency interval with the exact value of frequency.
The value of frequency which corresponds to the sum closest to zero of the elements
marked with ellipse in Figs. 3-5, which stays between the upper and lower boundaries and represents the best available
frequency estimation.
Fig. 5. Interaction of the terms s$_{1}$, s$_{2}$, and s$_{3}$ which result in equality to zero of the points of the frequency dependence of the inner product S marked with elliptical curve particular for the case of A$_{1}$ = A$_{2}$ = 1, b = 3 s, and f$_{2}$ = 46 Hz.
Fig. 6. The dependencies of the antiderivative function on frequencies of the auxiliary sine functions for the case of A$_{2}$ = 1 and b = 1.
Fig. 7. The dependencies of the antiderivative function for the case of ${\varphi}$ = 45$^{\circ}$, A$_{2}$ = 1.
Fig. 8. Typical frequency dependence of the inner product S calculated according to(1)determined for arbitrarily selected frequencies of the auxiliary functions.
Fig. 9. Typical frequency dependence of the inner product S calculated according to(1)determined for arbitrarily selected frequencies of auxiliary sine functions and signal-to-noise ratio of 0.026.
Fig. 10. B(f) for the case of arbitrarily selected frequencies of the auxiliary sine functions: 1 - for the case of the analyzed signal without any noise component, 2 - for signal-to-noise ratio equal to 0.026.
Fig. 11. The dependence of the antiderivative function D(f$_{1}$) for the case of the frequencies determined according to(9)of the auxiliary sine functions: 1- for the case of the analyzed signal without any noise component, 2 - for the case of presence of the additive normally distributive white noise.
3. The Results and Discussion of the Simulations
All simulations have been carried out for the case of a simple sine signal with a
preliminary selected amplitude, frequency, and initial phase. Simulations also have
been carried out without taking into consideration the possible presence of noise
components in analyzed signals, which may arise from diverse sources and have their
own specific spectral properties. The criterion for the most appropriate duration
of the analyzed signal was based on the results of the calculation of the finite differences
between the necessary for frequency estimation elements of the frequency dependence
marked with an ellipse in Fig. 3. For the purposes of the simplification instead of applying the relation (26), which directly stems from the trapezium rule, all simulations have been carried
out on the basis of the calculation of the auxiliary value P which was determined
according to (28):
where h denotes the time duration between the adjacent samples of a signal, and all
Q$_{i}$ for i = 1..n can be calculated as:
All simulations have been carried out for an analyzed sine signal with the values
of A$_{2}$ = 1, f$_{2}$ = 50 Hz, and ${\varphi}$$_{2}$ = 60$^{\circ}$. The results
of the simulations are presented in Fig. 12
Figs. 13 and 14 present frequency dependencies of the inner product (1) for different durations of the analyzed signal.
According to the results of the simulations presented in Fig. 12, it can be concluded that the sum closest to zero of the finite differences of the
sequence determined according to (26) exhibits periodical alternations. The results presented in Fig. 13 illustrate proper selection of the duration of the analyzed signal, which corresponds
to one of the local values closest to zero of P in Fig. 12 and allows us to use (9) for the array of the frequencies which contains the exact frequency of the analyzed
signal. For this case, the maximum value of S corresponds to the upper boundary of
the frequency interval which contains the exact frequency of the analyzed signal (f$_{1}$
= 50.17 Hz). The minimum value of S corresponds to the lower boundary of this frequency
interval (f$_{1}$ = 49.83 Hz), whereas the intermediate value (f$_{1}$ = 50 Hz) represents
the exact frequency of the analyzed signal, as it corresponds to the value of b which
matches with one of the local values closest to zero of P in Fig. 12.
The results presented in Fig. 14 illustrate the unsuitable selection of the duration of the analyzed signal b, which
corresponds to one of the local minimum values of P in Fig. 12. For this case, the intermediate value of S which stays between the estimated upper
(50.08 Hz) and lower (47.75 Hz) boundaries of the frequency interval does not allow
us to carry out accurate frequency estimation. In further analysis, special attention
should be paid to the analysis of the multifrequency signals and the effect of the
noise contamination, which might distort the frequency dependence of the inner product.
Consequently, the criteria based on calculation of the finite differences or on calculation
of the sum used for frequency estimation characteristic points of the sidelobes could
be inapplicable.
Fig. 12. The dependence of P calculated according to(28)on the duration of the analyzed signal b.
Fig. 13. S(f) for the case of b = 2.97 s.
Fig. 14. S(f) for the case of b = 2.955 s.