Dealiasing Doppler Spectra in Meteorological Radars
A method of dealiasing of Doppler spectra is proposed. The method is based on the fact that overlaid spectral components of the received signals are statistically independent, and that there is no overlay in range. For implementation of the method, the radar is supposed to transmit trains of pulse...
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irk-123456789-982532016-09-29T19:01:43Z Dealiasing Doppler Spectra in Meteorological Radars Sosnytskiy, S.V. Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования A method of dealiasing of Doppler spectra is proposed. The method is based on the fact that overlaid spectral components of the received signals are statistically independent, and that there is no overlay in range. For implementation of the method, the radar is supposed to transmit trains of pulses with two or more pulse repetition frequencies. In contrast to usual dual-PRF techniques, it does not require the Doppler spectrum to be narrow. The method is validated on experimental data from a meteorological radar, the dealiased spectra are compared with those measured directly at higher pulse repetition frequency Предложен метод восстановления допплеровских спектров. Метод основан на том, что наложенные спектральные компонентыпринятых сигналов статистическинезависимы, а также на отсутствии наложений сигнала по дальности. Для реализации метода локатор должен излучать пачки импульсов на двух или более частотах повторения импульсов. В отличие от обычных методов, основанных на двух частотах повторения импульсов, он не требует узости допплеровского спектра. Метод проверен на экспериментальных данных, записанныхметеорологическимлокатором; восстановленные спектрысравниваются со спектрами, измеренными на большей частоте повторения импульсов. Запропоновано метод відновлення допплерівських спектрів. Метод базується на тому, що накладені спектральні компоненти прийнятого сигналу є статистично незалежними, а також на відсутності накладання сигналу за відстанню. Для реалізації методу локатор має випромінювати пакети імпульсів на двох або більше частотах повтору імпульсів. На відміну від звичайних методів, що використовують дві частоти повтору імпульсів, він не вимагаємалоїширини допплерівського спектру. Метод перевірено на експериментальних даних, записаних метеорологічним локатором; відновлені спектри порівнюються зі спектрами, отриманими на більшій частоті повтору імпульсів. 2012 Article Dealiasing Doppler Spectra in Meteorological Radars / S.V. Sosnytskiy // Радиофизика и радиоастрономия. — 2012. — Т. 17, № 1. — С. 89–94. — Бібліогр.: 7 назв. — англ. 1027-9636 http://dspace.nbuv.gov.ua/handle/123456789/98253 en Радиофизика и радиоастрономия Радіоастрономічний інститут НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования |
| spellingShingle |
Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования Sosnytskiy, S.V. Dealiasing Doppler Spectra in Meteorological Radars Радиофизика и радиоастрономия |
| description |
A method of dealiasing of Doppler spectra is proposed. The method is based on the fact that overlaid spectral components of the
received signals are statistically independent, and that there is no overlay in range. For implementation of the method, the radar
is supposed to transmit trains of pulses with two or more pulse repetition frequencies. In contrast to usual dual-PRF techniques,
it does not require the Doppler spectrum to be narrow. The method is validated on experimental data from a meteorological
radar, the dealiased spectra are compared with those measured directly at higher pulse repetition frequency |
| format |
Article |
| author |
Sosnytskiy, S.V. |
| author_facet |
Sosnytskiy, S.V. |
| author_sort |
Sosnytskiy, S.V. |
| title |
Dealiasing Doppler Spectra in Meteorological Radars |
| title_short |
Dealiasing Doppler Spectra in Meteorological Radars |
| title_full |
Dealiasing Doppler Spectra in Meteorological Radars |
| title_fullStr |
Dealiasing Doppler Spectra in Meteorological Radars |
| title_full_unstemmed |
Dealiasing Doppler Spectra in Meteorological Radars |
| title_sort |
dealiasing doppler spectra in meteorological radars |
| publisher |
Радіоастрономічний інститут НАН України |
| publishDate |
2012 |
| topic_facet |
Радиофизические аспекты радиолокации, радионавигации, связи и дистанционного зондирования |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/98253 |
| citation_txt |
Dealiasing Doppler Spectra in Meteorological Radars / S.V. Sosnytskiy // Радиофизика и радиоастрономия. — 2012. — Т. 17, № 1. — С. 89–94. — Бібліогр.: 7 назв. — англ. |
| series |
Радиофизика и радиоастрономия |
| work_keys_str_mv |
AT sosnytskiysv dealiasingdopplerspectrainmeteorologicalradars |
| first_indexed |
2025-07-07T06:15:10Z |
| last_indexed |
2025-07-07T06:15:10Z |
| _version_ |
1836967695448276992 |
| fulltext |
ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012 89
Радиофизика и радиоастрономия. 2012, Т. 17, № 1, c. 89–94
ÐÀÄÈÎÔÈÇÈ×ÅÑÊÈÅ ÀÑÏÅÊÒÛ
ÐÀÄÈÎËÎÊÀÖÈÈ, ÐÀÄÈÎÍÀÂÈÃÀÖÈÈ,
ÑÂßÇÈ È ÄÈÑÒÀÍÖÈÎÍÍÎÃÎ
ÇÎÍÄÈÐÎÂÀÍÈß
S. V. SOSNYTSKIY
Institute of Radio Astronomy of the National Academy of Sciences of Ukraine,
4, Chervonopraporna st., Kharkiv, 61002, Ukraine
E–mail: sergey@ri.kharkov.ua
DEALIASING DOPPLER SPECTRA IN METEOROLOGICAL RADARS
A method of dealiasing of Doppler spectra is proposed. The method is based on the fact that overlaid spectral components of the
received signals are statistically independent, and that there is no overlay in range. For implementation of the method, the radar
is supposed to transmit trains of pulses with two or more pulse repetition frequencies. In contrast to usual dual-PRF techniques,
it does not require the Doppler spectrum to be narrow. The method is validated on experimental data from a meteorological
radar, the dealiased spectra are compared with those measured directly at higher pulse repetition frequency.
Key words: meteorological radar, Doppler spectra, aliasing
© S. V. Sosnytskiy, 2012
1. Introduction
Any designer of a pulsed radar system faces the prob-
lem of choosing a pulse repetition frequency (PRF)
value. Higher PRF values provide a wider range of
unambiguous Doppler velocity measurement, while
lower values give a wider range of unambiguous mea-
surement of slant distance. Generally, when no addi-
tional measures are taken, the maximal unambiguous
slant distance maxR and maximum unambiguous ve-
locity maxv are limited by the following relation
max max 8,R v c= λ (1)
where c is the light speed, and λ is the wavelength
used.
To overcome this limit, different methods have been
proposed. The most widely used ones are based on
the staggered pulse repetition time [1, 2], or on the
dual pulse repetition frequency [3–5]. These tech-
niques allow estimating the average radial speed
for essentially larger maxv values than is predicted
by (1). The accuracy of such estimation, however,
depends on the spectrum width, effectively implying
that the spectrum width should be essentially less
than the PRF value.
The large number of existing methods reflects the
complexity of the problem and, probably, impossibility
of a universal solution applicable to any radar design.
This paper gives yet another method of dealiasing
Doppler spectra, which is based on some other as-
sumptions than those mentioned above. The most
important assumption in the proposed method is that
different components of Doppler spectrum are sta-
tistically independent. No assumptions regarding the
shape of the power spectrum or its continuity are
used. Similarly to dual-PRF techniques, this method
is based on transmitting trains of pulses at different
pulse repetition frequencies with all of them provi-
ding the needed value of max .R
The paper has the following structure. Description
of the proposed method is given in Section 2. In
Section 3, validation of the method is given. Some
problems peculiar to the method and options for their
solutions are discussed in Section 4. Section 5 dis-
cusses possibility of using the method with more than
two PRFs values, and gives example of dealiasing
of three spectra measured at three different PRFs.
Finally, conclusions are given in Section 6.
90 ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012
S. V. Sosnytskiy
2. Description of the Method
Measuring Doppler spectra by analyzing reflections
in pulsed radars is like analyzing a sampled complex
signal with the sampling frequency being equal to
the pulse repetition frequency. Let us consider what
happens with a continuous complex signal during
sampling when the sampling frequency Sf is less
than the signal bandwidth. If a signal component
has frequency f outside the unambiguous bandwidth
( )2, 2 ,S Sf f− after sampling its frequency appears
to be within the mentioned range, modified as
,Sf f mf′ = + (2)
where m is integer. If we calculate the spectrum of
the sampled signal, the complex amplitude at the fre-
quency f ′ is equal to sum of the complex amplitudes
at all frequencies f satisfying (2).
In the case of weather radars, the reflected sig-
nal from a single range bin can be assumed random.
And, actually, the point of interest of meteorologists
is not the signal itself but the average Doppler
spectrum at a given range bin. In this paper, it is
assumed that signal components with frequencies
1f and 2f separated by an integer number of PRF
1 2( )PRFf f mf= + are statistically independent. In this
case, the average value of power in the aliased spec-
trum at a frequency of f is a sum of average powers
of the original spectrum at several frequencies:
( ) ( ).ALIASED ORIGINAL PRF
m
s f s f mf= +∑ (3)
This relation can be conveniently described in a ma-
trix form. In practice, the Doppler spectrum is usually
calculated as a discrete Fourier transform of the re-
flected signal. Let us calculate the power of each
Fourier component and use the obtained values
to compose a vector 1s which is referred below as
a measurement vector. Then, let us sample power of
the original spectrum of the signal with a frequency
interval of 1 1 ,PRFf f NΔ = which is the same as
in the Doppler spectrogram. Here, 1N is the number
of samples in DFT. These samples are also combined
in a vector .s Now, the effect of sampling on the
spectrum can be described as
1 1 .= ⋅s M s (4)
Here, 1M is a matrix which in what follows is re-
ferred to as a measurement matrix. The number of
rows in this matrix is equal to the number of samples
used in the discrete Fourier transform 1( ).N The num-
ber of columns reflects the width of the full spectrum.
The matrix is of block-diagonal type. Most of its com-
ponents are zeros, and only some are units. Here
is an example of a measurement matrix, which de-
scribes sampling of a signal when the signal band-
width can be twice as large as the sampling frequen-
cy, and 4-samples DFT is used:
1
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
.
1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0
⎛ ⎞
⎜ ⎟
⎜ ⎟=
⎜ ⎟
⎜ ⎟
⎝ ⎠
M
As is seen, in this case two elements in each row are
units. This reflects the fact that due to aliasing, each
component of the spectrogram is a sum of two spec-
trum components, as was expressed in (3). In terms
of relation (3), the units in the middle part of the ma-
trix correspond to 0,m = while the units in the top
right and bottom left corners correspond to 1m = −
and 1,m = respectively.
In practice, the vector 1s in equation (4) is what
we have measured and the vector s is what we
need to obtain. It would be convenient to solve the
equation by finding the inverse of the matrix 1 :M
1
1 1.
−= ⋅s M s
However, the rank of the matrix 1M is smaller than
the number of its columns and, therefore, a unique
solution cannot be found. One of the ways to have
a unique solution is to decrease the number of co-
lumns to the rank of the measurement matrix, but this
would mean that the bandwidth of the original spec-
trum be less than the sampling frequency. In order
to provide a single solution when the original spec-
trum is wider than the measured one, the rank of the
matrix can be increased by adding more rows. In this
paper, this is made by measuring one more vector 2s
using another sampling frequency 2.PRFf For con-
venience, the second vector should be calculated
with another number of samples 2N in order to have
the same frequency interval fΔ as in the vectors 1s
and .s This can be satisfied provided that
2 2 1 1 .PRF PRFf N f N= (5)
The two measurement vectors can be combined in
one vector ,M
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
1
2
s
s
s
and the measurement matrix
ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012 91
Dealiasing Doppler Spectra in Meteorological Radars
for this vector will be 1
2
.
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
M
M
M
If 1N and 2N
do not have common multipliers, the rank of the
combined measurement matrix will be equal to
1 2 1.N N+ − This allows the number of columns
to be greater than 1N and 2 ,N which means that the
bandwidth of the original spectrum can be greater
than any of the two pulse repetition frequencies.
It will be observed that the measurement matrix is
not square and, therefore, a pseudoinverse of the
measurement matrix should be used. Provided that
the length of the original spectrum is equal to the rank
of the measurement matrix, the original spectrum
can be calculated as
1( ) ,−= ⋅ ⋅ ⋅* *
Ms M M M s
where asterisk denotes conjugate transpose of the
matrix.
For implementation in radars, it is convenient to
have the matrices multiplied before the radar starts
its operation:
1( ) .−= ⋅ ⋅* *A M M M
Then, during the radar operation, dealiasing of the
measured spectra can be done through a single ma-
trix-vector multiplication:
.= ⋅ Ms A s (6)
Summing up, the practical implementation should
be as follows. The transmitted signal should consist
of trains of pulses with two different PRFs. That is,
1N pulses with pulse repetition frequency of 1PRFf
should be followed by 2N pulses with PRF of 2.PRFf
For each range bin, radar returns measured during
the first train of pulses are used to calculate the
vector 1s and the ones measured during the second
train are used to calculate the vector 2.s The com-
plete measurement vector M
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
1
2
s
s
s
is then used
to calculate the dealiased spectrum in accordance
with (6).
3. Validation of the Method
In order to verify the validity of the assumptions made
when deriving the method of spectrum dealiasing, raw
radar data were used. The data were recorded with
a 36 GHz Doppler meteorological radar during its nor-
mal operation. Namely, it was the MIRA-36 radar
manufactured by the Institute of Radio Astronomy,
Kharkiv, Ukraine and METEK GmbH, Elmshorn,
Germany [6, 7]. During the radar operation, the pulse
repetition frequency was 9 kHz. Signal for the me-
thod validation was taken from a single range bin.
Its central Doppler frequency was about 2.5 kHz and
its bandwidth about 900 Hz. Before using the data
for validation, they have been multiplied by a comp-
lex harmonic signal with frequency of –2.5 kHz in
order to shift the whole spectrum to low frequencies.
Then, measurement with lower values of the pulse
repetition frequency was modeled by decimating
the original signal by some decimation factor DECN
(an integer value). This was equivalent to having the
signal recorded with a PRF value of 9 kHz.DECN
The decimated signals were then analyzed as described
in Section 2. Thus, three spectra were obtained: two
aliased spectra corresponding to the two PRFs, and
a dealiased spectrum calculated from them.
In order to obtain a benchmark spectrum for com-
parison, the original signal was decimated choosing
such value of ,DECN that the total bandwidth were
the same as in the dealiased spectrum.
In Fig. 1, one can see two aliased spectra obtai-
ned for PRF values of 500.0 and 529.4 Hz (deci-
mation factors used were 18 and 17, respectively).
The numbers of pulses per train were 170 and 180,
respectively. In Fig. 2, a comparison between the
dealiased and benchmark spectra is given. The bench-
mark spectrum was calculated using decimation fac-
tor of 9 with 340 samples DFT.
Fig. 1. Aliased spectra corresponding to PRF values
of 500.0 and 529.4 Hz (decimation factors 18 and 17,
respectively). The numbers of pulses per train are 170 and 180,
respectively. Both spectra are results of averaging of 171
spectrograms
92 ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012
S. V. Sosnytskiy
As is seen from Fig. 2, the two spectra are very
close that proves applicability of the proposed me-
thod to meteorological Doppler measurements. Some
difference observed between the two spectra has its
origin in random fluctuations of the spectrum power
around its average value, which will be discussed
below.
4. Implementation Problems
and Possible Solutions
The proposed method has some problems which need
to be addressed when implementing the method in
practical systems. First, there is a problem of Dis-
crete Fourier Transform. The most common imple-
mentations of DFT are Fast Fourier Transform algo-
rithms which require the number of samples to be a
power of 2. These algorithms cannot be used with the
proposed method because if both of the measure-
ment sub-vectors 1s and 2s have lengths equal to some
powers of 2, the rank of the measurement matrix would
be equal to the length of the larger sub-vector, that
is there would be no gain in spectrum bandwidth.
At least three solutions to this problem are possible.
1) Fast Fourier Transform algorithms for numbers
of samples other than powers of 2 are possible, though
less common.
2) With the rapid increase of digital processing
boards capabilities, it is possible that slower DFT
algorithms would be acceptable.
3) Though in Section 2 it was insisted that the
frequency resolution of all spectra should be the same
(resulting in requirement (5) which implies restriction
on the choice of 1N and 2 ),N actually it is possib-
le to generalize the method for the case of diffe-
rent frequency resolutions of the measured spectra.
In this paper, the restriction on frequency resolution
(5) is made rather for the sake of convenience of
explanation of the method, and in order to avoid in-
terpolations of spectra when deriving a correct
measurement matrix.
The second problem of method implementation
is increase of noise in the dealiased spectrum as
compared to the original spectra. The source of this
problem is the fact that the method was derived for
average spectral powers. In practice, we always have
some fluctuations around the average values, and
these fluctuations increase during dealiasing. So far,
the only solution to this problem is averaging of the
measured spectra. Averaging can be done either in
time (by accumulating the spectra during several
repeats of pulse trains), or in frequency (smoothing
the spectra with some kind of filtering technique),
or both. The results in Section 3 are obtained using
averaging both in time and in frequency. In time, 171
pulse trains of each PRF were analyzed, so both
vectors were obtained by averaging 171 spectra.
5. Using More than Two Pulse
Repetition Frequencies
In Section 2, it has been described how to dealias spec-
tra measured at two different PRFs. When two PRFs
are used, the bandwidth of the dealiased spectrum
is usually two times larger than the lesser of the two
PRFs. Compared to the larger PRF, the bandwidth
is only 1.6–1.9 times larger. It is possible to enhance
this value by using more than two PRFs. For this,
measurement vector and measurement matrix are
composed as
1
2 ,
...
⎛ ⎞
⎜ ⎟
⎜ ⎟= ⎜ ⎟
⎜ ⎟
⎝ ⎠
M
N
s
s
s
s
1
2 .
...
⎛ ⎞
⎜ ⎟
⎜ ⎟= ⎜ ⎟
⎜ ⎟
⎝ ⎠N
M
M
M
M
It will be observed, however, that using many PRFs
with close values makes the dealiased spectrum even
more distorted with fluctuations than in the case of
two PRFs. Averaging spectra in time also becomes
less efficient: the maximum time during which spec-
tra can be averaged is limited by the dynamics of at-
Fig. 2. Comparison of the dealiased spectrum with the benchmark.
The benchmark spectrum is calculated using decimation factor
of 9 and the DFT length is 340 samples
ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012 93
Dealiasing Doppler Spectra in Meteorological Radars
mosphere, and the larger is the number of PRFs
we use, the more time it takes to transmit and receive
all trains of pulses. Also, the values of PRF should
be chosen carefully so that all columns of the final
measurement matrix would not be linearly dependent.
Figs. 3 and 4 show an example of using pulse trains
with three different PRFs. This example was calcula-
ted using the same experimental data as in Section 3.
Fig. 3 shows the measured spectra. The PRF va-
lues used for this calculation are 333.3, 375.0, and
428.6 Hz. The corresponding decimation factors are
27, 24, and 21. The numbers of pulses in the trains
are 168, 189, and 216. To reduce the effect of spec-
trum fluctuations, 77 pulse trains of each PRF were
analyzed.
Fig. 4 shows comparison of the dealiased spec-
trum with the benchmark one. The benchmark spec-
trum was calculated using the decimation factor of 9
with 504 samples DFT. The shape of the spectrum
is reconstructed correctly, even though the measured
spectra had more severe aliasing than in the previous
example. One may notice that spectra in Fig. 4 are
less detailed than those in Fig. 2. This is a result of
stronger averaging in frequency, which was needed
to attenuate the effect of fluctuations.
6. Conclusions
A method of dealiasing Doppler spectra measured by
meteorological radars is proposed. It is based on trans-
mitting trains of pulses with different values of pulse
repetition frequency. The difference between the spec-
tra of received signals allows restoring (dealiasing)
of the original spectrum. The method uses no assump-
tions regarding the shape of the power spectrum ex-
cept that it fits within the dealiased spectrum band-
width. The theoretical upper limit of relation between
the spectral width and PRF values is equal to the num-
ber of PRF values used.
The proposed method has been validated using
raw data from a meteorological radar. Some prob-
lems of method implementation have been discussed
and solutions to these problems proposed.
The proposed method can be used for dealiasing
of spectra not only in meteorological radars, but also
in any system where the analyzed signal is random
and only the average spectrum is wanted.
Acknowledgments
The author is thankful to METEK GmbH company
for providing the data used in method validation.
REFERENCES
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spectrum. The benchmark spectrum is calculated using
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94 ISSN 1027-9636. Радиофизика и радиоастрономия. Т. 17, № 1, 2012
S. V. Sosnytskiy
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С. В. Сосницкий
Радиоастрономический институт НАН Украины,
ул. Краснознаменная, 4, г. Харьков, 61002, Украина
ВОССТАНОВЛЕНИЕ ДОППЛЕРОВСКИХ СПЕКТРОВ
В МЕТЕОРОЛОГИЧЕСКИХ РАДИОЛОКАТОРАХ
Предложен метод восстановления допплеровских спектров.
Метод основан на том, что наложенные спектральные ком-
поненты принятых сигналов статистически независимы, а так-
же на отсутствии наложений сигнала по дальности. Для реа-
лизации метода локатор должен излучать пачки импульсов
на двух или более частотах повторения импульсов. В отли-
чие от обычных методов, основанных на двух частотах по-
вторения импульсов, он не требует узости допплеровского
спектра. Метод проверен на экспериментальных данных,
записанных метеорологическим локатором; восстановленные
спектры сравниваются со спектрами, измеренными на боль-
шей частоте повторения импульсов.
С. В. Сосницький
Радіоастрономічний інститут НАН України,
вул. Червонопрапорна, 4, м. Харків, 61002, Україна
ВІДНОВЛЕННЯ ДОППЛЕРІВСЬКИХ СПЕКТРІВ
У МЕТЕОРОЛОГІЧНИХ РАДІОЛОКАТОРАХ
Запропоновано метод відновлення допплерівських спектрів.
Метод базується на тому, що накладені спектральні компо-
ненти прийнятого сигналу є статистично незалежними,
а також на відсутності накладання сигналу за відстанню.
Для реалізації методу локатор має випромінювати пакети
імпульсів на двох або більше частотах повтору імпульсів.
На відміну від звичайних методів, що використовують дві
частоти повтору імпульсів, він не вимагає малої ширини доп-
плерівського спектру. Метод перевірено на експерименталь-
них даних, записаних метеорологічним локатором; віднов-
лені спектри порівнюються зі спектрами, отриманими
на більшій частоті повтору імпульсів.
Received 25.12.2011
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