RU2611102C1 - Method for spectral analysis of polyharmonic signals - Google Patents

Method for spectral analysis of polyharmonic signals Download PDF

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RU2611102C1
RU2611102C1 RU2015153710A RU2015153710A RU2611102C1 RU 2611102 C1 RU2611102 C1 RU 2611102C1 RU 2015153710 A RU2015153710 A RU 2015153710A RU 2015153710 A RU2015153710 A RU 2015153710A RU 2611102 C1 RU2611102 C1 RU 2611102C1
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signal
spectrum
spectral analysis
samples
zero
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Иван Иванович Канатов
Дмитрий Владимирович Миненков
Вячеслав Викторович Гульванский
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Федеральное государственное автономное образовательное учреждение высшего образования "Санкт-Петербургский государственный электротехнический университет "ЛЭТИ" им. В.И. Ульянова (Ленина)"
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01RMEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R23/00Arrangements for measuring frequencies; Arrangements for analysing frequency spectra
    • G01R23/16Spectrum analysis; Fourier analysis

Abstract

FIELD: physics, measurement technology.
SUBSTANCE: method relates to digital signal processing, particularly to spectral analysis of signals in a Fourier basis, and can be used in radar, radio communication and measurement technology. The disclosed method comprises supplementing a sample of the analysed signal with zeros, performing Fourier transform, multiplying real and imaginary parts of spectrum readings with like parts of an adjacent reading, multiplying the sum vector with minus one and zeroing all readings less than zero.
EFFECT: invention enables to reduce the level of side lobes without deterioration of resolution of spectral analysis, and increase signal-to-noise ratio.
7 dwg, 1 tbl

Description

The invention relates to digital signal processing, in particular to spectral analysis of signals in the Fourier basis, and can be used in radar, radio communications and measuring equipment.

A known method of spectral analysis using weighted windows of Hanning, Kaiser, Dolph-Chebyshev and others (L. Rabiner, B. Gold. Theory and application of digital signal processing. M., 1978)

Figure 00000001

where n = 0, ... N-1, is the number of time reference,

m = 0, ... M-1, is the number of the spectral component, M = N,

Figure 00000002

Figure 00000003

Signal weighting is used to reduce side lobe levels. The side lobes not only affect the spreading of the spectrum, in which the monoharmonic signal is reproduced by a whole group of spectral components; their level means the transmission coefficient in the suppression band, and the gain in the signal-to-noise ratio obtained by spectral transformation of the signal depends on its magnitude.

The best of the "classical" window functions, such as Kaiser, Hanning, Dolph-Chebyshev, atomic functions of Kravchenko, etc., provide suppression of the "tails" of the frequency response to levels (-80 ÷ -100) dB. However, suppression of the side lobes is achieved at the cost of a twofold decrease in resolution due to the expansion of the main lobe of the frequency response of the spectrum analyzer. This is a common drawback for all window functions: whatever the shape of the weight window, it narrows the already short time interval of the signal, and narrowing the interval in the time domain inevitably leads to the expansion of the spectrum.

Twofold expansion of the main lobe not only degrades the resolution, but also doubles the noise level. The efficiency of isolating a narrow-band signal from noise by a window function is usually estimated by an equivalent noise band (ESR) equal to the bandwidth of a conventional rectangular frequency filter, whose area under the amplitude frequency response curve is equal to the corresponding area of the window function, and the amplitude is equal to the effective value of the transmission coefficient of its main lobe :

Figure 00000004

ESR is measured in bins - the number of steps in frequency. For an ideal (rectangular) frequency response ESR = 1. For a rectangular time window ESR = 4.1, for a window of Hanning ESR = 4.01. Such a slight difference shows the inefficiency of using window functions to suppress noise. For comparison, the proposed method gives ESR within 1.1 ÷ 1.5.

In FIG. Figure 1 shows the frequency characteristics of two adjacent channels of the fast Fourier transform (FFT) - a spectrum analyzer. The bold line shows the desired (ideal) characteristics. In the ideal case, the squareness of the characteristic ensures the independence of the transmission coefficient from the frequency and guarantees the absence of a signal skip at the boundary between the channels in the absence of their overlap. The more gentle the borders, the more overlap is necessary for reproducing the “boundary” frequencies. For the FFT method, the real characteristic of the first zeros is twice as wide as the ideal one. When using the Hanning window, the characteristic becomes twice as wider. Not much smaller expansion of the main petal is provided by the windows of Dolph-Chebyshev and Kravchenko-Gauss.

It is also known that in the case where resolution is the main parameter, you have to abandon the method of direct Fourier transform of the original signal, and use such "superresolution" methods as the Pisarenko, Proni, and Kapon methods (Modern methods of spectral analysis: Review. S.M. Kay, S.L. Marple, TIIER, vol. 69, No. 11, November 1981). However, these methods are characterized by the appearance of false frequencies and a change in amplitude from the number of analyzed signals.

There are also known methods for eliminating the spreading effect of the spectrum by adjusting the sampling frequency of the signal to the fundamental frequency when analyzing polyharmonic signals. So, in the method of spectral analysis of polyharmonic signals and a device for its implementation (RU 2363005), such adjustment is carried out by multiplying the basic Fourier functions by the calculated coefficient. This method is applicable only for large signal-to-noise ratios, when the fundamental frequency can be measured without resorting to spectral processing.

There is also a known method of spectral analysis of polyharmonic signals (Middleton V. Introduction to the statistical theory of communication. T. 2. - M .: Sov. Radio, 1962), in which for the analysis of polyharmonic signals in the frequency domain, the non-amplitude spectrum is often used - the complex samples module obtained by the FFT procedure, and the instantaneous power spectral density (SPM):

Figure 00000005

Given the complex nature of S (m), the operations of calculating the PSD are represented as the sum of the products of vectors of the real and imaginary parts (we multiply the complex and imaginary parts of the signal spectra elementwise, respectively):

Figure 00000006

Multiplying the material and imaginary parts of the samples by their copies does not eliminate the side lobes of the frequency response, however, in terms of the combination of signal conversion operations, it is the closest analogue to the prototype of the proposed method.

The technical result of the proposed method is to reduce the level of the side lobes without compromising the resolution of the spectral analysis, which entails an increase in the signal-to-noise ratio.

The indicated technical result is achieved by the fact that the Fourier transform is performed on the sample of the analyzed signal, the obtained real and imaginary parts of the complex spectrum samples are multiplied by their copies and the products are summed, while the original signal sample is supplemented with zero intervals in front and behind, and the real and imaginary parts of the spectrum samples multiplied by the same parts of the neighboring sample and after adding the product vectors, the total vector is multiplied by minus one and all the samples are reset to zero.

The essence of the claimed technical solution is illustrated in FIG. 2-8.

In FIG. 2, the solid line shows the frequency response of two adjacent channels obtained by the proposed method, and the dotted line shows the frequency response of the FFT channel. Black color shows the overlap zone, in which the spectrum of the monoharmonic signal will contain two nonzero samples instead of one. The width of the main lobe is almost twice as narrow as the FFT lobe.

In FIG. 3 shows the frequency characteristics of one FFT channel (dotted line) and obtained by the proposed method. The width of the main lobe can be adjusted by changing the number of zero samples.

In FIG. 4a shows the spectrum of a sinusoidal signal obtained by the proposed method (Fig. 4a) before the operation of zeroing negative samples of the spectrum, and in FIG. 4b shows the spectrum obtained by the FFT method.

In FIG. Figure 5 shows the phase change in the spectrum of a monoharmonic signal of frequency 11.5 when noise is applied (FFT with a rectangular window) - even in noise, the phase difference between 11 and 12 samples remains equal to π.

In FIG. 6a shows the amplitude spectrum of a noisy signal; FIG. 6b is a spectrum obtained by the proposed method from the same FFT samples.

In FIG. 7 shows the frequency characteristics of the Hanning weighing window and PS.

The sequence of signal conversion operations of the proposed method is presented in paragraphs 1-4:

1. The signal sample x (n), n = 1, 2, ... N, is supplemented on both sides by k zeros:

Figure 00000007

2. The Fourier transform (FFT) is performed:

Figure 00000008

3. The real and imaginary parts of each m-th spectral sample are multiplied by the same parts of the next (m + 1) -th sample, and the products are summed:

S (m) = a (m) + jb (m);

P (m) = a (m) ⋅ a (m + 1) + b (m) ⋅ b (m + 1).

4. The sign of the vector P is reversed, after which the samples with a negative sign are assigned a zero value or the value of the selected small value in the case of recalculation of P in decibels:

Figure 00000009

Comparing the result with relation (4), one can write the operations listed in the form:

Figure 00000010

We call the obtained characteristic "the product of neighboring samples of the spectrum (PS)."

The proposed method is based on the following regularity of the phase-frequency characteristic of the main lobe of any channel of the Fourier spectrum: on the entire frequency interval of the main lobe, the phase varies linearly from -π to + π. A monoharmonic signal that has got into two adjacent channels of a spectrum analyzer due to overlap will form two spectral coefficients, which can be represented as vectors on the complex plane. The linearity of the phase-frequency characteristic provides the opposite direction of these vectors - a constant phase shift of 180 degrees. The most efficient way to isolate such vectors is their scalar product, which is formed by the proposed method.

Regardless of the phase and frequency of the monoharmonic signal, both the real and imaginary parts of the complex coefficients of adjacent spectrum channels will have opposite signs. Therefore, the product of their material parts, as well as the product of their imaginary parts, will be negative. Their sum will also be negative. To bring the result to its usual form, we multiply the sum by -1. Now all samples with a minus sign can be discarded as not belonging to the main lobe.

This method can be implemented using computing devices.

The main advantages of the proposed method are:

1. The width of the main lobe is the smallest of all known window functions and even 1.42 times narrower than a rectangular window. This minimizes spreading of the spectrum.

2. Low level of side lobes - most of them are zeros, and the total weight of all side lobes is 0.16% of the main lobe.

3. The rectangularity of the main lobe (the ratio of the width of the lobe at 0.5 to the width at the level of zeros) is significantly higher than that of existing windows (Table 1).

Figure 00000011

4. The location of the maximum of the main lobe corresponds to the middle of the interval between the samples of the spectrum, which is most convenient for the combined FFT-PS processing, which guarantees a twofold increase in resolution.

5. The width of the main lobe can be adjusted by the number of zeros added, gradually increasing it to ΔF = 2 (Fig. 5). Since the addition of zeros leads to an increase in the time base of the analysis, at the same time, the resolution can be doubled.

6. This is the most significant property of the proposed method: it gives an increase in signal-to-noise ratio by 10-20 dB ;.

This result is due to the combined influence of the following factors:

- narrowing the main lobe almost twice and a sharp decrease in the side lobes gives a total gain, which can be estimated by the ratio of the ESR of the rectangular window and the PS:

Figure 00000012

- the basis of the method of PS formation is the scalar product of two vectors a * b = | a || b | cosϕ, which allows you to sharply increase the amplitude of the in-phase (antiphase) factors and throw out (zero) about 50% of the noise due to the use of the difference in the shapes of the laws of distribution of difference phases of adjacent samples of the noise spectrum and the monoharmonic signal;

- like any other quadratic characteristic (dispersion, PSD, correlation), the PS contrasts even the subtle difference in amplitude values.

The efficiency of signal noise cleaning allows us to estimate the spectra in FIG. 6a and 6b. In FIG. 6a shows the amplitude spectrum of a noisy signal

Figure 00000013
in FIG. 6b is a spectrum obtained by the proposed method from the same FFT samples.

Even more striking is the difference in the spectra on a logarithmic scale. In FIG. 7 shows the superposition of the amplitude spectrum of the monoharmonic signal weighted by the Hanning window and the PS spectrum.

Claims (1)

  1. The method of spectral analysis of polyharmonic signals, namely, that the Fourier transform is performed on the sample of the analyzed signal, the obtained real and imaginary parts of the complex spectrum samples are multiplied by their copies and the products are summarized, characterized in that the original signal sample is supplemented with zero intervals in front and behind, and the real and imaginary parts of the spectrum samples are multiplied by the same parts of the neighboring sample, and after adding the product vectors, the total vector is multiplied by minus one in and zero all samples less than zero.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
RU2685972C1 (en) * 2017-12-01 2019-04-23 Акционерное общество "Всероссийский научно-исследовательский институт радиотехники" Method and device for filtering frequency-modulated signals
RU2702917C1 (en) * 2019-03-11 2019-10-14 Федеральное государственное бюджетное учреждение науки Институт проблем морских технологий Дальневосточного отделения Российской академии наук (ИПМТ ДВО РАН) Method of detecting noisy objects in shallow and deep sea

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SU1256044A1 (en) * 1985-02-19 1986-09-07 Рязанский Радиотехнический Институт Digital spectrum analyzer
SU1691852A1 (en) * 1986-12-05 1991-11-15 Предприятие П/Я В-8251 The spectrum analyzer
US6882947B2 (en) * 2001-12-31 2005-04-19 Teradyne, Inc. Discrete fourier transform (DFT) leakage removal
RU2363005C1 (en) * 2008-01-25 2009-07-27 Открытое акционерное общество ордена Трудового Красного Знамени Всесоюзный научно-исследовательский проектно-конструкторский институт "Тяжпромэлектропроект" имени Ф.Б. Якубовского Method of spectral analysis of polyharmonic signals and device to this end
US8321491B2 (en) * 2007-06-26 2012-11-27 The United States Of America As Represented By The Secretary Of The Army System and method for detecting a weak signal in a noisy environment

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
SU1256044A1 (en) * 1985-02-19 1986-09-07 Рязанский Радиотехнический Институт Digital spectrum analyzer
SU1691852A1 (en) * 1986-12-05 1991-11-15 Предприятие П/Я В-8251 The spectrum analyzer
US6882947B2 (en) * 2001-12-31 2005-04-19 Teradyne, Inc. Discrete fourier transform (DFT) leakage removal
US8321491B2 (en) * 2007-06-26 2012-11-27 The United States Of America As Represented By The Secretary Of The Army System and method for detecting a weak signal in a noisy environment
RU2363005C1 (en) * 2008-01-25 2009-07-27 Открытое акционерное общество ордена Трудового Красного Знамени Всесоюзный научно-исследовательский проектно-конструкторский институт "Тяжпромэлектропроект" имени Ф.Б. Якубовского Method of spectral analysis of polyharmonic signals and device to this end

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
RU2685972C1 (en) * 2017-12-01 2019-04-23 Акционерное общество "Всероссийский научно-исследовательский институт радиотехники" Method and device for filtering frequency-modulated signals
RU2702917C1 (en) * 2019-03-11 2019-10-14 Федеральное государственное бюджетное учреждение науки Институт проблем морских технологий Дальневосточного отделения Российской академии наук (ИПМТ ДВО РАН) Method of detecting noisy objects in shallow and deep sea

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