CN113595943B - Maximum likelihood-based MPSK signal-to-noise ratio estimation method - Google Patents
Maximum likelihood-based MPSK signal-to-noise ratio estimation method Download PDFInfo
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Abstract
The application discloses a MPSK signal-to-noise ratio estimation method based on maximum likelihood, which comprises the following steps of S1, receiving complex baseband signals r (N), wherein n=0, 1, … and N-1; and S2, estimating the signal-to-noise ratio SNR of the complex baseband signals r (N), n=0, 1, … and N-1 by adopting a maximum likelihood estimation method. The method is based on a maximum likelihood method, a statistical histogram of phase angle distribution is obtained from a received complex baseband signal, then the histogram is matched with probability density functions of phase angle distribution in different signal-to-noise ratios under the modulation mode, and the signal-to-noise ratio corresponding to the probability density function with the highest matching degree is used as an estimated value of the signal-to-noise ratio of an input signal.
Description
Technical Field
The application belongs to the technical field of signal-to-noise ratio calculation, and particularly relates to a maximum likelihood-based MPSK signal-to-noise ratio estimation method.
Background
The SIGNAL-to-NOISE RATIO, called SNR or S/N (Signal-NOISE RATIO), is also called SNR. Refers to the ratio of signal to noise in an electronic device or electronic system. The signal refers to an electronic signal from outside the device that needs to be processed by the device, the noise refers to an irregular additional signal (or information) that does not exist in the original signal generated after the signal passes through the device, and the signal does not change with the change of the original signal.
The signal-to-noise ratio is defined as the ratio of the average power of the useful signal component in the received signal to the average power of the noise, which is an important parameter of the received signal. The existing signal-to-noise ratio estimation method mainly comprises an estimation method based on a baseband signal symbol moment and an estimation method based on a subspace, wherein the symbol moment-based method needs a larger estimation sample number due to the fact that the moment of a random variable is utilized, and the calculation amount of the subspace-based method is larger.
Disclosure of Invention
The present application aims to solve or improve the above-mentioned problems by providing a MPSK signal-to-noise ratio estimation method based on maximum likelihood.
In order to achieve the above purpose, the application adopts the following technical scheme:
a MPSK signal to noise ratio estimation method based on maximum likelihood comprises the following steps:
s1, receiving a complex baseband signal r (N), n=0, 1, …, N-1;
s2, estimating the complex baseband signal r (N) by using a maximum likelihood estimation method, where n=0, 1, …, and the signal-to-noise ratio SNR of N-1:
wherein ,SNRmin and SNRmax Respectively a lower limit and an upper limit of a signal-to-noise ratio possible value of the signal to be estimated; phase angleIs the argument of the complex number r (n) in the range of [0,2 pi ]; m is the modulation order of the MPSK modulation signal;when the signal-to-noise ratio is SNR, the M-element phase keying modulated complex baseband signal has a carrier phase of theta c Probability density function of phase angle distribution at that time.
Further, a probability density function p (phi; M, theta) is calculated c ,SNR)：
Wherein l is the number of M constellation points of the MPSK signal constellation diagram.
Further, the signal-to-noise ratio in step S2 is approximately calculated:
further, when searching the signal-to-noise ratio estimation value, the following formula is adopted to approximate the calculation of the signal-to-noise ratio:
wherein L is about θ c In interval [0 ] , 2 pi) the total number of searches at which the maximum is searched, m is the number of all L searches, SNR step Is a step factor, L is a value related to θ c Number of searches when searching for maximum value over interval [0,2 pi ], SNR step Is a step factor;
definition i est The method comprises the following steps:
definition i max The method comprises the following steps:
wherein ,is a round-up operation.
Further, approximation calculation i est The method comprises the following steps:
wherein the phase angle isIs->Is a discretization result of (2);
for i obtained by approximation calculation est Is calculated in a simplified manner:
wherein ,nk Is thatMiddle equals->By counting the phase angle +.>The distribution histogram is calculated by a method that: n is n k The value of k=0, 1, …, L-1 is equal to the phase angle +.>The number of times falling within the interval [ 2pi.k/L, 2pi (k+1)/L;
will beSeen as two lengths L Sequence n of (2) k Andand is implemented using a fast Fourier transform:
the LogLF (k; i) is:
wherein: k is more than or equal to 0 and less than or equal to L-1, i is more than or equal to 0 and less than or equal to i max ；
The MaxLLF (i) is:
then i est The method comprises the following steps:
further, i is obtained based on solving est Calculating to obtain the estimated SNR of the signal to noise ratio of the complex baseband signal r (N), n=0, 1, …, N-1 est ：
SNR est ≈SNR min +i est ·SNR step 。
The MPSK signal to noise ratio estimation method based on the maximum likelihood has the following beneficial effects:
the method is based on a maximum likelihood method, a statistical histogram of phase angle distribution is obtained from a received complex baseband signal, then the histogram is matched with probability density functions of phase angle distribution in different signal-to-noise ratios under the modulation mode, and the signal-to-noise ratio corresponding to the probability density function with the highest matching degree is used as an estimated value of the signal-to-noise ratio of an input signal.
Compared with the prior art, the estimation method of the application does not have the phenomenon, and adopts the maximum likelihood estimation method to estimate the signal-to-noise ratio of the MPSK modulation signal at the baseband, has simple principle, does not involve complex operation and is suitable for engineering application.
Drawings
Fig. 1 is a flowchart of a MPSK signal-to-noise ratio estimation method based on maximum likelihood.
Fig. 2 shows the root mean square error of the snr estimation obtained by simulating the MPSK signal snr estimation method based on the maximum likelihood when the number of the complex baseband symbols used for the estimation is 2048 symbols.
Fig. 3 shows the root mean square error of the snr estimation obtained by simulating the MPSK signal snr estimation method based on the maximum likelihood when the number of complex baseband symbols used for the estimation is 8192 symbols.
Detailed Description
The following description of the embodiments of the present application is provided to facilitate understanding of the present application by those skilled in the art, but it should be understood that the present application is not limited to the scope of the embodiments, and all the applications which make use of the inventive concept are protected by the spirit and scope of the present application as defined and defined in the appended claims to those skilled in the art.
According to embodiment one of the application
For M-element phase keying modulation signals, receiving complex baseband signals:
wherein ,A to send the modulus of the complex baseband signal, A>0 is an unknown quantity;for power normalized MPSK modulated complex baseband signal transmitted at time n, i.e. with +.>Wherein E [. Cndot.]Representing a mathematical expectation operation; θ c Representing the carrier phase, which is an unknown constant; w (n) is power +.>Is added to the white gaussian noise (complex additive white Gaussian noise, complex AWGN).
The signal-to-noise ratio snr is defined as the ratio of the average power of the useful signal component to the average power of the noise component in the received signal, i.e.:
order theThe signal to noise ratio snr can be reduced to:
in practice, the signal-to-noise ratio is often expressed in decibels, namely:
the objective of this embodiment is to estimate the db SNR after receiving the complex baseband signal r (N), n=0, 1, …, N-1, and calculate by using the maximum likelihood estimation method:
wherein ,SNRmin and SNRmax Respectively a lower limit and an upper limit of a signal-to-noise ratio possible value of the signal to be estimated; phase angleIs the argument of the complex number r (n) in the range of [0,2 pi ];
function ofRepresenting that the signal-to-noise ratio is SNR, the M-ary phase keying modulated complex baseband signal has a carrier phase of theta c A probability density function of the phase angle distribution, the probability density function being:
from formula (6):
meanwhile, it can be seen from the formula (6): probability density function p (phi; M, theta) c SNR), and p (phi; m,0, SNR) with respect to phase angleWith 2 pi as the period. Therefore, the formula (7) can be further written as:
to achieve maximum likelihood signal-to-noise ratio estimation in equation (5), the following process is performed on equation (5):
when searching for the maximum value in the expression (9), the following approximation calculation method is adopted:
wherein M is the modulation order of the MPSK modulation signal; l is a positive integer representing θ c The greater the number of searches for a maximum value over the interval [0,2 pi ]), the greater the approximation of the value to the right of the second about equal sign in the formula (10) to the value to the left of the about equal sign, and conversely the lower the approximation; SNR of step Is a step factor;
i est the definition is as follows:
i in formula (11) max The definition is as follows:
therein, whereinIs a round-up operation.
Considering the equation relationship of equation (8), equation (11) can be written as:
phase angleDiscretization into->The discretization method comprises the following steps:
therein, whereinRepresenting a rounding down operation.
By discretizingReplace->Approximation of equation (13), namely:
recording deviceMiddle equals->The number of times of n k The following equation can be verified to hold:
in practice, n k The values of k=0, 1, …, L-1 do not need to be aligned first according to the above stepsDiscretizing to obtain->Reckoning->Obtained directly by means of the distribution of the statistical phase angle +.>The method of the histogram of the distribution is that: n is n k The value of k=0, 1, …, L-1 is equal to the phase angle +.>Times falling within the interval [ 2pi.k/L, 2pi (k+1)/L). In actual implementation, the embodiment calculates n k The above method is also used for the value of (2).
Substituting the formula (16) into the formula (15) can obtain:
the last equal sign in equation (17) exploits the monotonicity of the exponential function exp (.
The term in formula (17)
Considered as two sequences n of length L k Andis calculated by means of a fast Fourier transform (Fast Fourier Transform, FFT), i.e. the amount of computation of equation (17) can be reduced by means of FFT computation.
The LogLF (k; i) is:
wherein: k is more than or equal to 0 and less than or equal to L-1; i is more than or equal to 0 and less than or equal to i max
The MaxLLF (i) is:
then equation (17) can be written as:
obtaining i est Then the estimated SNR of the signal to noise ratio can be obtained by (10) est ：
SNR est ≈SNR min +i est ·SNR step (21)。
According to a second embodiment of the present application, the MPSK modulated signal-to-noise ratio estimation method of the present application is simulated. The simulation parameters were set as follows: the signal modulation and modulation modes comprise BPSK, QPSK and 8PSK; the SNR value range of the signal to noise ratio is 4-10 dB; the number N of complex baseband symbols is 2048 or 8192; the phase angle quantization level L is 512; SNR of min 0dB; SNR of max 15dB; SNR of step 0.05dB; phase angle theta c Obeying uniform distribution at [0,2 pi); monte Carlo simulations were 500 times.
The root mean square error (Root Mean Square Error, RMSE) defining the signal-to-noise ratio estimate is:
the simulation results in the root mean square error of the signal to noise ratio estimation when the number of complex baseband symbols used for the estimation is 2048 symbols and 8192 symbols, respectively, as shown in fig. 2 and 3, respectively.
The method is based on a maximum likelihood method, a statistical histogram of phase angle distribution is obtained from a received complex baseband signal, then the histogram is matched with probability density functions of phase angle distribution in different signal-to-noise ratios under the modulation mode, and the signal-to-noise ratio corresponding to the probability density function with the highest matching degree is used as an estimated value of the signal-to-noise ratio of an input signal.
Compared with the prior art, the estimation method of the application does not have the phenomenon, and adopts the maximum likelihood estimation method to estimate the signal-to-noise ratio of the MPSK modulation signal at the baseband, has simple principle, does not involve complex operation and is suitable for engineering application.
Although specific embodiments of the application have been described in detail with reference to the accompanying drawings, it should not be construed as limiting the scope of protection of the present patent. Various modifications and variations which may be made by those skilled in the art without the creative effort are within the scope of the patent described in the claims.
Claims (1)
1. The MPSK signal to noise ratio estimation method based on the maximum likelihood is characterized by comprising the following steps:
s1, receiving a complex baseband signal r (N), n=0, 1, …, N-1;
s2, estimating the complex baseband signal r (N) by using a maximum likelihood estimation method, where n=0, 1, …, and the signal-to-noise ratio SNR of N-1:
wherein ,SNRmin and SNRmax Respectively a lower limit and an upper limit of a signal-to-noise ratio possible value of the signal to be estimated; phase angleIs the argument of the complex number r (n) in the range of [0,2 pi ]; m is the modulation order of the MPSK modulation signal; />When the signal-to-noise ratio is SNR, the M-element phase keying modulated complex baseband signal has a carrier phase of theta c Probability density function of phase angle distribution at time;
calculating probability density function p (phi; M, theta) c ,SNR)：
Wherein l is the number of M constellation points of the MPSK signal constellation diagram;
the signal-to-noise ratio in the approximation calculation step S2:
when searching the signal-to-noise ratio estimation value, the following formula is adopted to approximate the calculation of the signal-to-noise ratio:
wherein L is about θ c The total number of searches when searching for the maximum value over the interval [0,2 pi ], m is the number of all L searches, SNR step Is a step factor;
definition i est The method comprises the following steps:
definition i max The method comprises the following steps:
wherein ,is an upward rounding operation;
approximation calculation i est The method comprises the following steps:
wherein the phase angle isIs->Is a discretization result of (2);
for i obtained by approximation calculation est Is calculated in a simplified manner:
wherein ,nk Is thatMiddle equals->By counting the phase angle +.>The distribution histogram is calculated by a method that: n is n k The value of k=0, 1, …, L-1 is equal to the phase angle +.>The number of times falling within the interval [ 2pi.k/L, 2pi (k+1)/L;
will beConsidered as two sequences n of length L k Andand is implemented using a fast Fourier transform:
the LogLF (k; i) is:
wherein: k is more than or equal to 0 and less than or equal to L-1, i is more than or equal to 0 and less than or equal to i max ；
The MaxLLF (i) is:
then i est The method comprises the following steps:
based on solvingObtained i est Calculating to obtain the estimated SNR of the signal to noise ratio of the complex baseband signal r (N), n=0, 1, …, N-1 est ：
SNR est ≈SNR min +i est ·SNR step 。
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