CC BY
3
Visnyk N'l'UU KP1 Seriia Radiolekhnika tiadioaparat.obuduuannia, "2022, Iss. 88, pp. 42—49
UDC 621.39
Iterative Method for Noise Power Estimating at Unknown Spectrum Occupancy
Buhaiov M. V.
S. P. Korolov Military institute, Zhytomyr, Ukraine E-mai 1: karunen&ukr. ne I.
Noise power estimating is the core of modern radio monitoring systems for solving tasks of spectrum occupancy calculation, detecting and estimating signal parameters. The growth of electronic devices number leads to an increase in overall noise level and its fast fluctuations. These devices often emit pulses or separate carriers. Since radio monitoring equipment must operate under these conditions, it may not be possible to exclude these components from radio noise measurements. It was shown that in some cases an increase in the noise power by 20% of the expected value leads to an increase in the false alarm rate by an order. The aim of this work is to develop and explore an iterative method for estimating the noise power with an unknown occupancy of the analysis frequency band, which will have low computational complexity and estimates independent of spectrum occupancy. The essence of the proposed method consists in two-threshold division of frequency samples into signal and noise by a statistical criterion using the coefficient of variation of spectral estimates. Thresholds are selected for a given false alarm rate. When threshold value of the coefficient of variation is exceeded, it is considered that there are occupied frequency channels in the spectrum, and each frequency sample is compared with the second threshold. Those samples that have exceeded the threshold are considered signal, and the rest noise. The described procedure is then repeated for noise samples until all signal samples have been discarded. Also was developed method for calculating the noise power in time domain using the obtained noise power in frequency domain. Algorithm evaluation has shown that it remains robust for spectrum occupancy up to 60%. In this case, the relative error in estimating the noise power does not exceed 5%, and the average number of iterations of the algorithm grows with increasing occupancy and does not exceed 10.
Keywords: spectrum occupancy: iterative method: coefficient of variation: periodogram: radio monitoring: noise power
DOI: 10.20535/RADAP. 2022.88.42-49
Introduction
In modern radio monitoring systems noise power estimating problem is essentially important for solving tasks of spectrum occupancy calculation, detecting and estimating signal parameters. Knowing noise level can improve processes of signal recognition and demodulation. Uncertainty of the noise power is mainly due to variations in the gain of the low-noise amplifier, calibration errors and the presence of interference. If the first two factors lead to slow changes in the noise level and can be easily taken into account, the latter leads to its fast changes [1]. Interference is mainly due to an increase of electronic systems number, in particular those that nse low-bandwidth signals with low power density, and leads to an increase in the overall noise level and its oscillations. Therefore, considerable attention is paid to measurement of snch kind of noise [2]. If there is no predominance of single radio sources at the measurement site, noise characteristic has a normal amplitude distribution and can be considered as white Gaussian noise. However, in the conditions
of high density of electronic devices, which are often found in large cities and residential areas, it is virtually impossible to find a place where at least temporarily not dominated by noise or radiation generated by a single source. These sources often emit pnlses or separate carriers. As radio monitoring equipment must operate in snch conditions, it may not be possible to exclude these components from radio noise measurements. In addition, in some cases, the signals may be weaker than the background noise. As a result, it is difficult to detect and locate a weak signal with low power density, nsing existing radio control systems with low sensitivity. To extract signals from background noise the newest processing methods should be nsed in future spectrum control systems [3].
The resulting estimate of frequency band occupancy depends on the valne of the threshold, which is determined by the noise power [4]. To detect as ninch radio emissions as possible, regardless of their power, it is advisable to nse a dynamic threshold, which is calculated regarding on the current noise
power. Therefore, the development of methods for estimating noise power in conditions of dynamic changes in the electronic environment will provide a reliable estimates of spectrum occupancy, as well as detection of low power density signals by measuring changes in background noise level.
1 Related works
The methods of estimating noise power proposed in the literature are mainly based on statistical analysis of signal spectra, autocorrelation function and calculation of eigenvalues of the covariance matrix.
In [5,G], a consecutive mean excision (CME) algorithm for detecting signal samples is proposed, which can be nsed to estimate the noise power. Algorithms snch as CME are slow, as they take over a large number of frequency samples, and some modifications need sort operation. A similar approach is also proposed in [7]. In [8]. the noise power is estimated from the free adjacent frequency channel. The problems of estimating the noise level for cases of its slow change in time, frequency dependence and in the presence of signals in the analyzed frequency band are considered in [9]. Initially, the Forward CME (FCME) algorithm was nsed to estimate the noise power. After that, another algorithm constantly monitors significant changes in the noise level in the frequency domain, and in case of its occurrence, the FCME algorithm is nsed again. It is shown that the proposed approach has less computational complexity compared to CME. However, the algorithm for detecting changes in the noise level requires a complex procedure for calculating hyperparameters. which values depend on the signal-to-noise ratio (SNR). In [10] an iterative method of simultaneous estimation of channel state and noise power using EM-algorithrii is proposed, and in [11] noise level is measured using maximum likelihood estimates at free time intervals. If there are OFDM signals with cyclic prefix in the analyzed frequency band, it is proposed to estimate the noise power in [12] by-analyzing the autocorrelation function. In [13]. the noise variance is estimated based on the assumption that signal can be described by a system of Yule-Walker equations with known coefficients. In [14] noise power is estimated by spectrogram processing. It is assumed that the probability of a signal occurrence in a given frequency-time domain is 0.5. When deviating from this valne, as well as at high SNR values, the probability of false alarm rate deviates from the required. In [15]. noise and signal levels are proposed to be estimated in the frequency domain nsing the method of maximum likelihood by iterative approximation. The most reliable way to prove whether a frequency band contains only white Ganssian noise is to nse the mathematical concept of singular valne decomposition. This is the most practical way to choose frequency band for measuring the noise level [1]. In [4] to detect the maxi-
mum number of signals, it is recommended to define the threshold as a noise level pins 3-5 dB. The noise level is measured at an unused frequency or calculated as an average of 20% of the frequency samples with a minimum values. However, this approach will result in reduced sensitivity to signal detection at low SNR values, which is an actual today problem [16].
A common drawback of these works is absence of accurate estimates of the measured noise power for different levels of spectrum occupancy, which will significantly affect the performance of signal processing algorithms.
2 Problem statement
The aim of the work is to develop and explore an iterative method of estimating the noise power at unknown occupancy of analyzed frequency band, which will have low computational complexity and independent of spectrum occupancy estimates.
3 Dependence of false alarm probability on noise power
Before describing the method of estimating the noise power, we will establish how the probability of false alarm depends on the deviation of the actual valne of the noise level from the expected one. This will further make it possible to formulate the requirements for the required accuracy of noise power estimation with the allowable valne of the false alarm rate error.
To separate received samples into signal and noise one the Neyman-Pearson criterion is most often nsed [ ]. In this approach, the false alarm rate Pf is fixed at some level, and decisive function is found by maximizing the probability of detection. In any case, the test is to compare the valne of a function with threshold. Threshold valne is chosen based on the valne of false alarm rate. Most signal processing is performed in time or frequency domains. Therefore we will consider these cases.
Noise power determines the false alarm rate at a fixed threshold valne. At Fig. 1 is shown dependence of the ratio of the actual probability of false alarm rate Pf to its Med value Pf0 with increasing standard deviation (SD) of zero-mean Gaussian noise Threshold for PFo was calculated for a^0. From this figure it is seen that when the required valne of the probability of false alarm rate Pf0 decreases, its deviation from the actual value of Pf increases for a fixed noise SD
Thus, when the deviation of the expected value of noise SD from the actual is only 20% the value of Pf increases by about 1.4 times for PFo = 0.1 and almost an order for Pp0 = 0.0001. Moreover, the Pf deviation rate will be the maximum for small values of a^/a%0. For ^ to the error in the value of the false alarm rate will be Pf/Pf0 ^ 0.5.
44
Buhaiov M. V.
10:
10'
10
10'
10
10
1 1 1 1
- PF0= 0.1 - PF0= 0.01 ......... Pf0= 0.001 ----- PFq= 0.0001
—
/ / /
/ / f i .-■ / /
/ ft-/
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Fig. 1. Dependence of the false alarm rate increasing Fig. 2. Dependence of false alarm rate increasing via
via noise SD
noise SD for Nfft = 1024, N = 4NFFT and R = 0.5Nfft
Similar curves can be constructed for the case of signal processing in the frequency domain using the Welch periodogram. The length of the analysis sequence is N samples, the length of the fast Fourier transform ^pp-x) is NFFF and the overlap between the windows is R samples. At Fig. is shown the dependence of false alarm rate increasing via noise SD increasing for NFFF — 1024, N = 4NFFF, R = 0.5NFFF and Hamming window function. If the value of noise SD deviates from the expected at 20%, the obtained value of the probability of false alarm rate will be greater than expected by almost 4 times for PFo = 0.1 and 65 times for PFo = 0.0001. With increasing ppx length depicted at Fig. 2 dependences practically do not change. As the overlap between the windows R decreases, the curves will be Hatter, and as N increases, the curves will be steeper. This is due to the dependence of noise SD in the frequency domain on the parameters of the periodogram: with decreasing R noise SD will increase, and with increasing N -decrease. For the frequency domain for a^ ^ m the error of false alarm rate will be PF/PFo ^ 1.
4 Iterative method for noise power estimating in frequency domain
Iterative algorithms are often used in signal processing due to their adaptability and recursive representation capabilities [17]. They provide an approximate result and have less computational complexity compared to analytical methods.
In time domain, noise and signal can be contained in all samples to be analyzed. In this case, it is difficult to estimate the noise power. In the frequency domain, signal samples are superimposed on noise only in occupied frequency channels. Therefore, it is advisable to estimate noise power in frequency domain.
In [18, 19] was proposed an iterative method of detecting occupied spectrum bands. Its essence is the two-threshold separation of frequency samples into signal and noise according to the statistical criterion using the coefficient of variation of spectral estimates. First, the coefficient of variation is calculated for power spectral density (PSD) samples and compared with the threshold value. The threshold is selected for a given probability of erroneous assignment of noise samples to signal.
If the threshold value of the coefficient of variation is exceeded, it is assumed that there are occupied frequency channels in the spectrum and each frequency sample is compared with the second threshold. This threshold is chosen similarly to the previous one. Those samples that exceed the threshold are considered signal, and the rest - noise. The described procedure is then repeated for noise samples until all signal samples will be detected.
Using the above ideas in Fig. 3 shows the algorithm for estimating the noise power in frequency domain. We will use the Welch periodogram to calculate the PSD of the received signal.
In block 1 we enter such parameters: x - vector of received signal samples; M - number of averaged spectra; w - vector of window function samples; Qtr -vector of threshold values for coefficient of variation for a given probability of false alarm rate; PF - required false alarm rate in frequency domain; ns - number of detected signal samples.
M
px (k)=YJ \xj (k)f
which is then normalized to energy tS px^)
Npft — 1
E Px^)
k=0
(3)
In block 3. the valne of the PSD coefficient of variation Q is calculated, which in block 4 is compared with the threshold value for a given number of detected signal samples Qtr(ns). The probability density function of noise samples has central chi-sqnare distribution with M degrees of freedom. Then, for the separation of signal and noise samples in the frequency domain, the threshold value is calculated in block 5 by the following expression [18]:
T =-
1
Nfft - ns
fl- —
y 8.64M V 8.64M J
where
1.24 + 0.85H0'657 1 + 0.0001H—3 + ''
H = — ln
(l -Pf) .
(4)
(5)
Using this threshold in block 6 vector Px is divided into the vectors of signal Ps and noise P^ samples. If at the current iteration of algorithm signal samples is not detected (block 7), the noise level L^ (block 11) is estimated by the following expression:
L* = wx pi (*).
N
? t—
(6)
k=1
Fig. 3. Block diagram of noise power estimation algorithm
In block 2, the PSD of the signal is calculated using the Welch peroiodogram. First, the spectrum of the window-weighted j-th signal segment is calculated by the following expression:
Npft — 1
Xi (k)= ^ x(n) w(n)e—j2'n. (1)
n=0
The accumulation of M energy spectra will give estimate of PSD:
where N^ = NFFT — ns - number of noise samples.
In block 8 for every iteration the number of signal samples ns is updated. In block 9, the value of the coefficient of variation Q is calculated for the energy-normalized vector for noise samples PIn block 10, the previous vector of frequency samples Px is replaced by the vector of noise samples P^ . After that, among the noise samples we again search for signals. Evaluation of the algorithm continues until all signal samples will be detected and rejected.
For samples of TctW dcttct, it is recommended to perform one measurement with duration at least 0.5 s every 10-30 s [1]. The proposed algorithm will give an adequate estimate of noise if its level is approximately the same at all analyzed frequencies.
5 Noise power calculation in time domain
To determine the Stcirt and end time of ct SI gnal in a frequency channel, it is necessary to know the noise SD to set the desired threshold value. Noise SD in time domain will be calculated using Parseval's theorem [17] according to such expression:
N —1
(2)
=1
\xi (n)\2 = '
(7)
n=0
x
3
U =
46
Buhaiov M. V.
where N = (M — 1)R + NppT - number of samples for periodogram calculation; g - normalization coefficient, which depends on the parameters of the periodogram and the type of window function.
If noise is zero-mean, then its variance can be calculated by this expression:
1
N
N— 1
E
n=0
Tt Y. \x« (r
(8)
The value of g coefficient is determined based on the fact that N samples of the signal in the time domain must contain the same amount of energy as M windowed segments of Nppp length. Then the value of g can be determined from the following expression:
N
9 =
NfFT — 1
m j2 H«0
n=0
(9)
Taking into account expressions (7-9) we can calculate noise SD with such equation:
L.
«
NfFT — 1
m K«0
n=0
(10)
Having estimate and assuming that the noise has normal distribution, we can calculate the valne of the threshold and for the required false alarm rate in the time domain using expression 5.
Also we can calculate SNR in j-th frequency-channel:
~2
SNR
2 2 of«- VÎ
4
(ID
where ajs - noise variance in j-th frequency channel with width A/j; a^ - noise variance in given frequency-channel.
If the noise SD a^ was calculated for bandwidth An, than for channel with bandwidth Afi noise power can be calculated according such expression:
= M'
= Âïï) •
(12)
Similar calculations can be performed in the frequency domain:
E p* w
SNRi
ki m
(ki
0 L«
- 1,
(13)
where ki max mid ki min - maximum and minimum value of frequency sample index in j-th channel.
Calculated values of SNR can be used to predict the quality of signals detection or demodulation when designing electronic systems for operation in sophisticated electromagnetic environments.
6 Simulations and numerical results
In this paper bandwidth occupancy 'q will be calculated as the ratio of the sum of bandwidths of all occupied frequency channels to the analyzed band-
An
spectrnm shape cases. One for signals with almost rectangular spectrum envelope (OFDM, filtered PSK) and another for the rest spectrum shape envelopes. The occupancies of frequency band was chosen such that developed algorithm will remain stable.
For the first case analyzed frequency band contains one OFDM signal, which occupies from 3% to 60% of
An
An
rials with different powers and bandwidths. Spectrum occupancy will vary from 1% to 30%. Fig. 4a shows result of OFDM signal processing with a spectrum width of 0.58An with 10 dB SNR and the following parameters of the periodogram: M = 7 NppT = 1024, R = 512, w = hamming. False alarm rate Pp was chosen at 0.01. In Fig. b is shown the case of narrowband signals processing. The noise level is calculated according to expression 6 clt SI gnal-free frequencies.
Fig. 5a demonstrates dependence of the relative error of noise power estimate S via occupancy. For one OFDM signal S does not exceed 1%. Numerical results have shown that for the OFDM signal, the algorithm with the above parameters remains stable for occupancies up to 60%. At higher occupancies, the algorithm gives inflated estimates of the noise power. For direct sequence spread spectrum signals proposed algorithm remains stable for signal bandwidth up to
0.4An
smooth PSD by accumulating more spectrum realizations M.
The dependence of the average number of iterations of the algorithm required to estimate the noise power is shown at Fig. 5b. When the band occupancy is about 20% of narrowband signals with different spectrum widths and powers, the average number of iterations is about 4, and the error is about 3%. For comparison, in case of presence in analyzed frequency band one OFDM signal with a spectrum width of 0.2An under constant preconditions, the average number of iterations did not exceed 3, and the estimation error was less than 0.5%.
The influence of the window function on the valne of the error in estimating the noise power and the average number of iterations of the algorithm is insignificant and for practical applications it is advisable to choose the window for the required level of sidelobes. Only the rectangular window compared to other windows for the OFDM signal has a relative error of estimating noise power by an average of 2% while for other windows it does not exceed in average 0.5%.
2
2
2
a
«
2
k
1тератишшй метод оцшювашш ршпя шуму ири цсиадомШ заипятосп смуги частот aiia;ii:sy
47
Frequency, MHz
(a)
Frequency,
(b)
Fig. 4. Spectrum after processing for first (a) and second (b) cases of signal environment
J_1_1_1_l_ ---1 OFDM signal - Multiple narrowband signals
j
/
0 10 20 30 40 50 60 q,%
(a)
0 10 20 30 40 50 60
n,%
(b)
Fig. 5. Relative error of noise power estimate (a) and mean number of iterations (b) via occupancy
If the signals dynamic range exceeds 30-40 dB or in case of very high SNR, which leads to ont-of-band radiation, the use of window functions with low si-delobes will provide less error for noise power estimates. It is recommended to nse nnttall and blackman-harris windows to obtain stable estimates of noise power.
As the number of analyzed signal segments M increases, the values of the relative error and the average number of iterations to obtain an estimate of the noise power decrease. Increasing the overlap between adjacent windows R from 0.5 to 0.75 has little effect on the error valne. Increasing FFT length reduces the error valne with a slight increase in the number of algorithm iterations.
complexity. The result is achieved by two-threshold iterative separation of frequency samples into signal and noise according to the statistical criterion nsing the coefficient of variation of spectral estimates. The algorithm remains stable for bandwidth occupancy up to 60%. The relative error in noise power estimating does not exceed 5%. and the average number of iterations of the algorithm is not more than 10. The developed algorithm should be nsed to improve the performance of radio monitoring systems.
Prospects for further research in this area should foens on improving the proposed method to process spectrum bands with higher than 60% occupancy levels.
Conclusions
The scientific novelty of the obtained result is the development of an iterative method for estimating the noise power at unknown occupancy of analyzed frequency band, which has low computational
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1теративний метод оцшювання р!вня шуму при невщомш зайнятост! смуги частот анал!зу
Бугайов М. В.
Оцшюваппя потужпост! шуму е ключовим елемеп-том сучаспих систем радюмошторипгу для вгцяшеппя завдапь визпачеппя зайпятост! смуги частот, виявлеппя i оцшюваппя параметр!в сигпал1в. Зростаппя кглькост! радюелектрошшх пристрош призводить до зросташ1я загалыюго р!впя шуму i його р!зких коливапь. Щ при-стро! часто випромипоють 1мпульси або окрем! песучь Осшльки обладпаппя радюмошторипгу мае працювати в таких умовах, виключити ц! складов! з вим1рювапь радюшуму може виявитися поможливим. У робот! показано, що в деяких випадках збглынеппя р!впя шуму в!д очшувапого па 20% призводить до зростаппя ймов!рпо-ст! хибпо! тривоги па порядок. Метою роботи е розро-блоппя та досл1джеппя 1теративпого методу оцшюваппя р!впя шуму при пев!домш зайпятост! смуги частот апа-л!зу, що матиме певисоку обчислювальпу складшсть та пезалежш в!д заваптажепост! оцшки. Сутшсть запро-поповапого методу полягае в двопороговому роздшепш частотпих в!дл1к1в па сигпальш та шумов! за статисти-чпим критер!ем 1з використаппям коефщ!епта вар!ацп спектральпих оцшок. Пороги обираються для задано! fiMOBipnocTi хибио! тривоги. У раз! перевищешш порогового значения коефщ!епта вар!ацГ! вважаеться, що у спектр! е зайпят! частотш капали i кожей частотпий в!дл1к пор1вшоеться з другим порогом. Ti в!дл1ки, що перевищили nopir вважаються сигпалышми, а решта шумовими. Шсля цього описана процедура повторюе-ться для шумових в!дл1к1в до тих nip, доки по буде в1дкипуто yci сигпальш в!дл1ки. Розроблепо методику розрахупку середпьоквадратичпого в1дхилеппя шуму в часовш облает! 1з використашшм отримапого р!впя шуму в частотшй облает!. Досл1джеппя алгоритму показали, що вш залишаеться стшким для заваптажепост! смуги частот апал!зу до 60%. При цьому в1дпоспа помил-ка оцшюваппя р!впя шуму по перевищуе 5%, а середпя шльшеть 1терацш алгоритму зростае 3i збглынеппям заваптажепост! i складае по бглыне 10.
Клюноог слова: зайпятасть смуги частот: иератив-пий метод: коефщ!епт вар1ацп: перюдограма: радюмош-торипг: радючастотпий спектр: р!вепь шуму
Гтеративний метод оцшювання р!вня шуму при невщомш зайнятост! смуги частот анашзу
49
Итеративный метод оценивания уровня шума при неизвестной занятости полосы частот анализа
Бугаев Н. В.
Оценка мощности шума является ключевым элементом современных систем радиомониторинга для решения задач определения занятости полосы частот, обнаружения и оценки параметров сигналов. Рост количества радиоэлектронных устройств ведет к росту общего уровня шума и его резким колебаниям. Эти устройства часто излучают импульсы или отдельные несущие. Поскольку оборудование радиомониторинга должно работать в таких условиях, исключить эти составляющие из измерений радиошума может оказаться невозможным. В работе показано, что в некоторых случаях увеличение уровня шума на 20% от ожидаемого ведет к возрастанию вероятности ложной тревоги на порядок. Целью работы является разработка и исследование итеративного метода оценки уровня шума при неизвестной занятости полосы частот анализа, что будет иметь невысокую вычислительную сложность и независимые от загруженности оценки. Сущность предлагаемого метода состоит в двухпороговом разделении частотных
отсчетов на сигнальные и шумовые по статистическому критерию с использованием коэффициента вариации спектральных оценок. Пороги выбираются для заданной вероятности ошибочной тревоги. При превышении порогового значения коэффициента вариации считается, что в спектре имеются занятые частотные каналы, и каждый частотный отсчет сравнивается со вторым порогом. Те отсчеты, что превысили порог, считаются сигнальными, а остальные — шумовыми. После этого описанная процедура повторяется для шумовых отсчетов до тех пор, пока не будут отброшены все сигнальные отсчеты. Разработана методика расчета среднеквадра-тического отклонения шума во временной области с использованием полученного уровня шума в частотной области. Исследования алгоритма показали, что он остается устойчивым для загруженности полосы частот анализа до 60%. При этом относительная ошибка оценки уровня шума не превышает 5%, а среднее количество итераций алгоритма растет с увеличением загруженности и составляет не более 10.
Ключевые слова: занятость полосы частот; итеративный метод; коэффициент вариации; периодограмма; радиомониторинг; радиочастотный спектр; уровень шума
