13th International Conference on DEVELOPMENT AND APPLICATION SYSTEMS, Suceava, Romania, May 19-21, 2016
High frequency resolution harmonic analysis
Bujor Mircea
PhD student, IEEE Student Member
Polytechnic University of Timisoara
Timisoara, Romania
Abstract—The acquisition time of a real life signal limits the
resolution of its spectrum, because it directly affects the
orthogonality of its spectrum components. There is no spectral
analysis yet to distinguish frequencies that are not orthogonal. A
spectral analysis for distinguishing non orthogonal frequencies is
proposed here. A system of linear equations is solved for the
sampled points, producing a result equivalent to FFT of a longer
acquisition time. There is an extra requirement for such an
algorithm: to cover frequencies up to a value a couple of times
higher than the maximum frequency of the signal. The simulations
show it is possible to distinguish between non orthogonal
frequencies. The spectral analysis consists of only a matrix
multiplication, once it is computed, but building that matrix
requires a high computational cost. Beside spectral analysis, the
proposed approach can be used for extrapolation.
Keywords-harmonic analysis; frequency spectrum; matrix;
I.
INTRODUCTION
The lest obvious limitation in spectral analysis is the
impossibility to resolve two close frequencies if the signal has a
short acquisition time. In contexts such as digital modulation, or
real time spectrum analysis, it becomes obvious. For example in
OFDM modulation, the subcarrier spacing is strictly determined
by the duration of the symbol. Any closer frequency spacing
leads to loss of orthogonality between subcarriers.
There are a number of spectral analysis techniques [1], [2],
like MUSIC or ESPRIT, claiming to achieve higher frequency
resolution. The term higher frequency resolution is misleading,
because it doesn't mean they can distinguish frequencies that are
not orthogonal. They produce values for frequencies that are not
orthogonal, but those values can't help distinguish non
orthogonal frequencies. Had any such spectral analysis existed,
it would have been used in OFDM or other digital modulation.
This is the reason why the Discrete Fourier Transform is
considered state of the art and absolute reference for this work.
Just by zero padding the signal, any "high frequency resolution"
can be achieved, just like the Chirp-Z Transform or Goertzel
algorithm to name a few.
The proposed spectral analysis attempts to achieve the same
result as the Discrete Fourier Transform, but with a shorter
acquisition time. This implies distinguishing frequencies beyond
the orthogonal ones. Such a thing is possible because sinusoids
of different frequencies are linearly independent, i.e. there is no
linear combination of non-null sinusoids of different
frequencies, that produces a null sinusoid. A system of equations
for example can distinguish the unknowns based on linear
independence. There is no spectral analysis yet to distinguish
spectral components based on linear independence.
The Discrete Fourier Transform is fed a finite number of
samples and produces a finite number of spectral components,
but requires the samples to extend over a certain amount of time.
A system of equations can do the same, but there is no
requirement for the length of the acquisition time. This is
important as in practice acquisition time is often limited.
II.
SPECTRAL ANALYSIS BASED ON LINEAR INDEPENDENCE
Over the whole paper, the following presumptions are
considered valid, regarding the properties of a real life signal. A
real life signal is supposed to have a continuous spectrum, not a
discrete one. The consequence of having a continuous spectrum
is that it is aperiodic, or its period is infinite. Also real life signals
are supposed to be frequency limited, as most of sources and
propagation mediums are frequency limited.
A. The deterministic character of DFT
The spectrum of a periodic and frequency limited signal is
clearly a discrete set of sinusoids. Those sinusoids can be
computed by means of Fourier Series. The alternative is to build
a system of equations, and get exactly the same result. The
difference is Fourier Series requires integration over a whole
period while a system of equations can deal with a much shorter
acquisition time.
Real life signals as defined above are not periodic, or their
period is infinite, since they have a continuous spectrum.
Consequently the Fourier Series are to be replaced by the Fourier
Transform, which turns into Discrete Fourier Transform(DFT)
for signals described by samples instead of mathematical
formula.
The DFT has a less obvious deterministic character. The
DFT output contains a discrete set of frequencies from
f 0 = 1 / T , where f 0 is the fundamental frequency in Hz, and
T is the duration of the acquisition time in seconds, up to
f max = f s / 2 , where f s is the sampling frequency. The
number of samples is M = 2 * N , where N = f max / f 0 .
Thus a DFT result consists only of discrete orthogonal
components by definition. In order to compute more
components the signal has to be expanded in time, even if only
zeroes are used for that.
This is the deterministic character of DFT: M input values
produce M unique output values, like in a system of equations.
A system of equations can be thought of as a matrix
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multiplication. Both the DFT and matrix multiplication are
linear transforms producing unique results. That means they
reach the same amplitude for each frequency and the same
phase, provided they start from the same set of frequencies and
the same samples. This is not common practice because matrix
multiplication is more expensive than the FFT algorithm.
III.
B. Proposed spectral analysis
Let a signal of maximum frequency f max be sampled over T
seconds. Let the needed frequency resolution correspond to a
DFT output of an acquisition time k times larger.
The proposed spectral analysis consists of the next steps:
• a virtual maximum frequency vf max
depending on the maximum frequency
original signal and acceptable output error
is estimated
f max of the
• the spectrum is split into sinusoids of the fundamental
frequency f 0 = 1 / kT and its harmonics up to vf max .
• the acquisition time is placed in the center of the virtual
acquisition time kT. The time reference of the first
sample is thus t = ( k − 1)T / 2 instead of zero.
• a system of equations is formed and the necessary
number of samples are measured within T seconds,
evenly spaced. As the system of equations has to be
linear, its coefficients are a sine and a cosine for each
frequency, and for each sampling time, like this:
e jωt = cos ωt + j sin ωt
• the solution of the system of equations represents the
output of the proposed spectral analysis
The step of choosing a virtual maximum frequency is the
trickiest part, because using the maximum frequency of the
actual signal produces errors.
Because actual sampling is faster than it is for conventional
DFT, the higher frequencies start overlapping at the Nyquist rate
of the actual sampling only. Due to windowing, the frequency
spectrum spreads beyond the maximum frequency of the signal,
up to infinite. The frequencies between the maximum frequency
of the signal and the Nyquist rate of actual sampling are not null
and they don't overlap with the lower ones. The significant ones
have to be included in the system of equations, or the linear
independence will be affected.
The part of the spectrum beyond the maximum frequency of
the signal will have values very close to zero most of the times,
but without them the result is erred. The virtual maximum
frequency is not a fixed value. Depending on the acceptable level
of errors it can be moved up or down. They are related to the
window of period kT, the virtual one, not to the window of
period T. If the virtual window is different than the rectangular
window, the needed maximum frequency is significantly lower.
A virtual window is applied just like a real one. The only
requirement is that the time must have the same reference as in
the system of equations.
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The optimum choice of the virtual maximum frequency is a
subject left for future research, because it depends on the specific
application where the algorithm is used. This paper focuses only
on the proof of concept for the possibility of distinguishing non
orthogonal frequencies.
NUMERICAL SIMULATION
Using this algorithm for increased resolution leads to illconditioned systems of equations. Numerical algorithms
available today support matrices up to 12000x12000 size and
precision of 32768 bits (approximately 10000 decimal digits)
[3]. Also there is Rump's algorithm [4] that works for matrices
2t
with condition number up to B if the precision of the software
t
is B .
The maximum precision(precision is not the same with
accuracy) needed was 1000 digits only for the given examples.
So for practical implementations there are already good enough
tools to support the required complexity of numerical
computations.
A. Proof of concept examples
For the proof of concept two examples are given. The virtual
maximum frequency was about six to seven times larger than
maximum frequency of the signal. The windows were
rectangular. Two examples are given to illustrate the fact that the
frequency resolution can be improved by faster sampling leading
to larger systems of equations.
The first example is for a signal shown in Fig. 1. It lasts 5.16
milliseconds, and the minimum frequency difference between
two orthogonal frequencies is 193.7 Hz (the frequency spacing
between two orthogonal frequencies is the inverse of the
acquisition time, except for the particular case where they are in
phase and the frequency spacing halves). Any spectral analysis
algorithm, like FFT, can distinguish frequencies spaced 193.7
Hz apart. No spectral analysis exists yet to distinguish closer
frequencies if the acquisition time is that short, because they are
no longer orthogonal. The signal contains 75 Hz cosine and 95
Hz sine functions (sine and cosine so they are not in phase).
The inability to distinguish non orthogonal frequencies is
exemplified in Fig.2. The FFT for the same sampled points as in
Fig. 1 is represented. The signal was zero padded so that more
points are covered (the conventional "high frequency
resolution"), but no spectral analysis can distinguish frequencies
that are not orthogonal. Only the bandwidth from 0Hz to 640Hz
is displayed, for consistency with the other figures.
The proposed algorithm clearly identifies the two peaks as
distinct. In Fig. 3 is presented the result of proposed spectral
analysis in 64+1 points, corresponding to frequencies multiples
of 10 from 0Hz to 640 Hz. The test signal frequencies were
chosen for as much as possible virtual "scalloping loss". As the
frequencies were pushed to the maximum frequency too, their
shape is not the best.
Fig. 1.
Time domain signal for a signal sum of 75Hz cosine and 95Hz sine
functions
Fig. 4.
Example of a spectrum in 128 points of 82.5Hz cosine and 92.5
sine sampled over 5.14 ms
B. Example of low virtual maximum frequency and
extrapolation capabilities
It can be seen in this example what happens if the virtual
maximum frequency is not placed high enough. The desired
spectrum is buried in low frequency spectrum, even if nothing
of the kind was fed to the input. The linearity of the system of
equations was lost because it neglected significant components
of high frequency. The spectrum in Fig. 5 corresponds to 145Hz
cosine and 165 Hz sine signal sampled over 2.58 ms. The
spectral analysis was performed for the same number of
points(64+1) as the one in Fig. 3, and the same bandwidth(0640Hz).
Fig. 2.
Fig. 3.
FFT spectrum for the signal shown in Fig.1
While the two peaks are still distinguishable, the low
frequency part of the spectrum is a clear corruption. If the
sinusoids from the computed spectrum are summed over 100
milliseconds, which is a period for a spectrum in steps of 10 Hz,
a reconstructed signal can be obtained as in Fig. 6.
The 64 points spectrum of the signal in Fig.1 computed with
proposed algorithm
Another example is given, for a signal of approximately the
same duration, 5.14 ms. The two frequencies are a 82.5 Hz
cosine and 92.5 Hz sine. This time the same spectrum is split in
128+1 bins covering the same bandwidth from 0Hz to 640 Hz..
The two frequencies are clearly distinguishable in Fig.4.
Fig. 5.
The spectrum computed for 145Hz cosine and 165Hz sine signal
sampled oer 2.58 ms.
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Fig. 6.
Recomposed signal from spectrum shown in Fig. 5.
It can be seen that the irregularities cause high disturbances
at the ends of the virtual period of the signal. Had the spectrum
been divided in 20 Hz steps, the disturbances would have been
located at ends of 0.5 seconds interval.
This example was given to show that the extrapolation
capabilities of this algorithm are better than its capabilities as a
spectral analysis tool. The extrapolation is correct for more than
ten times the original time span to both left and right, even if the
spectral analysis is compromised.
C. Using different windows
Next a signal lasting 5.16 ms, consisting of a sum of 165 Hz
cosine and 205 Hz sine function is given. The spectrum is
computed in 64+1 points, multiple of 10Hz. This time the
Nuttall window coefficients shown in Fig. 7 are used.
The spectrum obtained after that windowing is given in Fig.
8. Again the covered bandwidth is from 0Hz to 640 Hz. With
this window the spectral components can go up to about twice
the previous maximum frequency.
Fig. 8.
Spectrum for 165Hz cosine and 205Hz sine functions sampled
over 5.16 ms to which Nuttall coefficients in Fig. 7 was applied.
IV.
CONCLUSION
An algorithm with frequency resolution so high it can
distinguish non orthogonal frequency components was
presented. The challenge is to pick a virtual maxim frequency
high enough to preserve the linearity of the system but low
enough to still get as many as possible meaningful spectral
components.
The algorithm requires a heavy computational cost for the
inverse matrix of the system. Once it is computed the spectral
algorithm consists of a mere multiplication with the inverse
matrix. It must be mentioned here for reference, that FFT
algorithm has a lower computational cost than a matrix
multiplication.
The proposed algorithm is to be used especially in
telecommunications, where the acquisition time of a symbol is
limited from the start. It can also be used as a numeric calculus
tool for extrapolation, with applications in statistics. This last
domain is left for future research, as well as the optimum choice
of the virtual maximum frequency.
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Fig. 7.
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