CN110855246A - Method for generating Gaussian white noise with arbitrary variance - Google Patents

Abstract

The invention discloses a method for generating Gaussian white noise with arbitrary variance, which comprises the steps of filtering the output of an m sequence generator through an FIR filter, combining the output sequences, converting the combined output sequences into analog signals through a digital-to-analog converter, and performing linear amplification and filtering to obtain broadband noise signals, wherein an arbitrary variance Gaussian noise mapping module is added between the FIR filter and the digital-to-analog converter, and the output noise source signals are mapped into Gaussian white noise signals with arbitrary adjustable variance by utilizing a probability statistical method and a curve mapping method. The invention adds a new function without paying too much cost, the noise source can adjust the variance according to the requirement of a user, different waveforms are generated, the invention is suitable for the generation of noise under different signal-to-noise ratios under the same power, the test scenes and the test cases are effectively increased, and the use scenes of the noise source are expanded.

Description

Method for generating Gaussian white noise with arbitrary variance

Technical Field

The invention relates to the technical field of noise generators, in particular to a method for generating arbitrary variance Gaussian white noise.

Background

The Gaussian white noise source has wide application in the engineering debugging stage, and can be used for simulating noise between satellites and between grounds and noise brought by receiver channel equipment, so that the performance index of the receiver is tested.

Currently, noise generators are divided into physical noise generators and digital noise generators. The physical noise generator has high precision, but the realization circuit is complex. In order to obtain broadband white gaussian noise quickly and conveniently, a noise generation method based on an FPGA is discussed below, and a corresponding hardware platform is combined to quantize and output a digital signal.

The current general method for digital synthesis of white gaussian noise is to generate white gaussian noise with uniform distribution, and then transform the uniform distribution into white gaussian noise to obtain white gaussian noise. In the literature, "implementation of a gaussian white noise generator in an FPGA" (yellow-benxite, etc.), firstly, a Tausworthe algorithm is adopted to generate uniformly distributed white noise, and then, the conversion from the uniformly distributed white noise to the gaussian white noise is realized by a table look-up method. In the document "design and implementation of high performance programmable white gaussian noise" (jiang le, etc.), uniformly distributed random sequences are generated by using a lagged-Fibonacci algorithm, and then the white gaussian noise is generated by a formula method. In the document, "a fast method for generating a white gaussian noise sequence by using an FPGA" (manage and universe, etc.), an M sequence generator is used to generate uniformly distributed random numbers, and the white gaussian noise sequence is realized by a 15-segment polygonal line approximation method.

The principle of generation of gaussian white noise is described:

the system architecture of the wideband noise generator is shown in fig. 1, and the core of the design is an m-sequence generator (longest linear feedback shift register) and an FIR filter (finite long single-bit impulse response filter). The designed m sequence generator has 4 paths of output, filtering is carried out through an FIR (finite impulse response) polyphase filter, the output sequences are combined into 2 paths of signals and then converted into analog signals through a DAC (digital-to-analog converter), and linear amplification and filtering are carried out to obtain broadband noise signals.

(1) m sequence generation

The pseudo-random sequence is a definite sequence with some random characteristics, and has random statistical characteristics and can be repeatedly generated, so that the pseudo-random sequence has wide application. The m sequence is a commonly used pseudo random sequence, and is a short for the longest linear feedback shift register sequence. The m sequence is widely researched since Shannon information theory is produced, and the theory is mature and widely applied at present. The linear feedback shift register is the main functional block for generating m-sequence if the linear feedback shift register selectsIs n, the repetition period of the m-sequence may be 2ⁿ-1. Considering data generated by the m sequence as unsigned integers, the value range of the data is 1-2ⁿ-1, and each unsigned integer occurs only once in a repetition period. The data generated with m-sequences are random numbers subject to uniform distribution.

The m-sequence is the longest linear shift register and is formed by adding feedback to the shift register. The structure is shown in fig. 1. In the figure a_n-i(i ═ 1,2,3, …, r) is the state of each bit register in the shift register; c. C_i(i ═ 1,2,3, …, r) is the feedback coefficient of the ith bit register. When c is going to_iWhen 0, no feedback is indicated; when c is going to_iWhen 1, feedback is indicated. In this structure c₀＝c_r＝1，c₀Cannot be 0, c₀A value of 0 does not constitute a periodic sequence, since c ₀0 means no feedback and is a static shift register. c. C_rIt cannot be 0, i.e. the r-th bit register must participate in feedback, otherwise, the feedback shift register of r stages will be reduced to the feedback shift register of r-1 stage or lower. Different feedback logic, i.e. c_i(i-1, 2,3, …, r) take different values, resulting in different shift registers. The formula expression is as follows:

when the structure shown in fig. 2 is implemented in an FPGA, the shift register performs a right shift operation on each rising edge (or falling edge) of the clock to derive a new pseudo code codeword a_n-r(ii) a At the same time due to the shift operation, a_n-i(i-1, 2,3, …, r) state is changed, before the next clock edge comes, the corresponding state is subjected to exclusive-or calculation to obtain a new feedback value, and a is sent in under the driving of the next clock_n-1. This structure can only output one pseudo code at a time, taking parallel four ways as an example, and the improved structure is shown in fig. 3.

The structure shown in fig. 3, with the rising edge of each clock cycle, pushes out the last 4 codewords of the serial shift register, while all values of the register are shifted to the right by 4 positions. In the serial structure, only the feedback code word corresponding to the current state needs to be calculated each time. In a parallel architecture, each state requires the computation of a feedback word, since four states are generated at a time.

When the program is initialized, the four serial shift registers are respectively initialized, and the difference between two adjacent groups of shift registers is a pseudo code phase. As shown in FIG. 4, the phase of S2 lags by S1 one clock cycle; the phase of S3 lags by S2 one clock cycle; the phase of S4 lags by S3 one clock cycle. In order to calculate the feedback codeword corresponding to the current state of S1, the states of S2, S3 and S4 are needed, and the feedback word represents the feedback codeword lagging S1 by four clock cycles. The calculation of the feedback codeword corresponding to the state of S2 requires the use of a feedback word corresponding to the state of S1 in addition to the states corresponding to the two sets of registers of S3 and S4. The calculation formula of these four feedback codewords is as follows.

The feedback codeword corresponding to the state of S1 is:

the feedback codeword corresponding to the state of S2 is:

a is needed to be calculated when the feedback code word corresponding to the S2 state is calculated_n+3The feedback coefficient is brought into a formula to participate in calculation, and the formula can be simplified according to the actual feedback coefficient. Similarly, feedback codewords corresponding to the states of calculation S3 and S4 can be written. Since the parallel structure does not change the structure of the feedback calculation, the generated pseudo code is completely consistent compared with the pseudo code generated by the serial shift register.

(2) Multiple term filtering process

The power spectrum of random numbers generated by multiple M sequences in parallel is basically horizontal, digital filtering is required to be carried out on the generated random numbers to generate a digital noise sequence with adjustable bandwidth, and a Finite Impulse Response (FIR) digital filter is adopted in the method. According to the Central Limit Theorem (CLT), the sum of a large number of independent and identically distributed random variables must be a normal random variable. Therefore, the random numbers which are uniformly and randomly distributed are converted into random numbers which are approximately normally distributed after passing through the FIR digital filter, and the distribution characteristic of the output noise is closer to Gaussian distribution along with the increase of the order of the filter. The FIR digital filter not only realizes the function of adjusting the noise bandwidth, but also completes the conversion from uniform noise to white Gaussian noise.

The basic principle of polyphase filtering is as follows: assuming that the impulse response of the digital filter is h (n), the z-transform H (z) is defined as:

let N ═ mD + k (M ═ 0,1,2, … M-1; k ═ 0,1,2, …, D-1; N ═ mD), then the above formula can be reorganized as follows:

order to

Then the above formula can be written as:

in summary, although this is a commonly used white gaussian noise generation method, since one very important feature of white gaussian noise is variance, which determines the shape of probability density of white gaussian noise, the variance of white gaussian noise is not controllable, and is determined by the tap coefficient of the filter, rather than being artificially controllable.

Disclosure of Invention

In order to solve the above problems, the present invention provides a method for generating gaussian white noise with arbitrary variance, which comprises filtering the output of an m-sequence generator through an FIR filter, combining the output sequences, converting the combined output sequences into analog signals through a digital-to-analog converter, and performing linear amplification and filtering to obtain wideband noise signals, wherein an arbitrary variance gaussian noise mapping module is added between the FIR filter and the digital-to-analog converter, and the output noise source signals are mapped into gaussian white noise signals with arbitrary adjustable variance by using a probability statistical method and a curve mapping method, wherein the arbitrary variance gaussian noise mapping module comprises the following steps:

s1, obtaining probability statistics P according to input signals, namely sequences output by FIR filter₁；

S2, generating a Gaussian white noise density function corresponding to the variance according to user requirements;

s3, obtaining the Gaussian white noise probability P according to the Gaussian noise density function₂；

S4. from P₁＝P₂Obtaining a mapping curve of the input signal mapped to variance Gaussian white noise;

and S5, according to the mapping curve, outputting the input signal after mapping.

Further, the step S1 includes the following sub-steps:

s11, normalizing the input signal to enable the input signal to be subjected to Gaussian distribution between [ -1,1 ];

s12, suppose for [ -11]Random noise x in between₁Obeying to the gaussian distribution, the simplest probability statistical method is as follows:

P₁(x₁)＝p(x<x₁)。

further, the step S2 includes the following sub-steps:

s21, defining Gaussian white noise with variance sigma by a user;

s22, according to a formula, the probability density of generating the Gaussian white noise is as follows:

wherein- ∞ < y < + > infinity.

Further, in step S2, since the probability of the interval distribution of noise outside the range of [ -4 σ, 4 σ ] is small, only noise points within the range of [ -4 σ, 4 σ ] are considered when converting to white gaussian noise.

Further, the step S3 is specifically: any white noise x obeying a Gaussian distribution with variance of σ₂The corresponding probability is:

further, the step S4 includes the following sub-steps:

s41, order P₁＝P₂Then for either one is [ -1,1 [)]Noise points x obeying a certain gaussian distribution therebetween₁The white Gaussian noise point is x₂And x is₂Satisfy P₂(x≤x₂)＝P₁(x≤x₁)；

S42, taking the axis of abscissa to represent any random noise x₁The ordinate represents the corresponding gaussian noise x with a user-defined variance σ₂And obtaining a mapping curve k.

Further, the mapping curve k is directly obtained through calculation, or is realized by piecewise fitting on the premise of not introducing large loss.

Further, in step S5, the input signal mapped output signal: y is k × x.

The invention has the beneficial effects that:

(1) the method adds a Gaussian noise mapping module with any variance, and utilizes a probability statistical method and a curve mapping method to map an output noise source signal into a Gaussian white noise signal with the variance capable of being adjusted at will;

(2) the method is simple, is realized through an algorithm, does not need to consume excessive computing units, and has high realizability;

(3) for any waveform emitter or noise source, the Gaussian noise source with variable variance is designed according to the requirements of customers, and is a product advanced function.

Drawings

FIG. 1 is a schematic diagram of a Gaussian white noise generator;

FIG. 2 is a schematic diagram of a serial shift register architecture;

FIG. 3 is a schematic diagram of a parallel shift register structure;

FIG. 4 is a parallel shift register state diagram;

FIG. 5 is a schematic diagram of a novel Gaussian white noise generator;

FIG. 6 is a flow chart of a novel Gaussian white noise generating device;

FIG. 7 m sequence time and frequency domain signal diagrams;

FIG. 8 is a graph of statistical probability density characteristics of the m-sequence;

FIG. 9 is a graph of filter characteristics;

FIG. 10 is a time and frequency domain plot of the filtered signal;

FIG. 11 is a graph of the probability density statistics of the filtered signal;

FIG. 12 is a graph of the probability statistics of filtered Gaussian signals;

FIG. 13 is a graph of Gaussian signal probability density characteristics given variance;

FIG. 14 is a graph of Gaussian signal probability statistics given a variance σ;

FIG. 15 is a graph of a mapping of an arbitrary variance Gaussian noise signal;

FIG. 16 is a graph of a mapping curve fit;

FIG. 17 is a time and frequency domain plot of the mapped signal;

FIG. 18 is a graph of statistical signal probability density after mapping.

Detailed Description

In order to more clearly understand the technical features, objects, and effects of the present invention, embodiments of the present invention will now be described with reference to the accompanying drawings. It should be understood that the detailed description and specific examples, while indicating the preferred embodiment of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. The components of embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the present invention, presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.

Example 1

The embodiment provides a method for generating arbitrary variance gaussian white noise, which comprises the following steps:

(1) first, a set of m-sequence signals is generated, the order of the signals is 32, and the generation length is 2³²1 (cut-off 1000 points), and the resulting m-sequence is shown in fig. 7, from which it can be seen that the power spectrum of the random number is substantially horizontal in the whole frequency domain;

(2) fig. 8 is a statistical characteristic of the probability density of the m-sequence, and it can be found from statistics that, except for the left and right sides, the data distribution in the middle value interval is basically a straight line, which indicates that the signal corresponding to the pseudo code sequence is uniformly distributed in the interval, and meets the requirement of white noise;

(3) the m-sequence then enters a filter module, the amplitude-frequency characteristic of the filter is shown in FIG. 9, and the filter is 51-order;

(4) the time domain and frequency domain characteristics of the filtered signal are shown in fig. 10, and it can be seen that the signal bandwidth is consistent with the filter bandwidth; the statistical properties of the probability density of the filtered signal are shown in fig. 11, in which it can be seen that the current signal meets the requirement of gaussian white noise, but the variance of the probability density map cannot be determined;

(5) in order to map the gaussian noise signal to a gaussian noise signal source with an arbitrarily defined variance, the signal is input into a gaussian white noise mapping module with an arbitrary variance, and the signal is normalized first, so that the probability statistical characteristic of the filtered signal is shown in fig. 12;

(6) then synthesizing a required gaussian probability density function according to the variance σ required by the user being 0.4, as shown in fig. 13, and obtaining a gaussian signal probability statistical characteristic of a given variance σ according to the synthesized probability density function, as shown in fig. 14;

(7) here P is the same according to the statistical properties of the probabilities₁＝P₂According to the principle, the original input signal is mapped to obtain a mapping curve as shown in fig. 15, the curve can be found to be symmetrical about the 0 point, and then the curve is fitted to obtain fig. 16;

(8) mapping and outputting the filtered signals by using a fitted curve, wherein time domain and frequency domain signals are shown in FIG. 17, and the frequency domain of the signals in the graph is not influenced; the probability density is shown in fig. 18, in which a black line frame represents the probability density of the filtered signal, and a black strip is the probability density of the mapped gaussian noise signal, it can be seen that the signal values spread to both sides and are not concentrated near 0, and the amplitude value probability increases at both sides, which indicates that the variance increases, and the former gaussian noise signal is changed into the gaussian noise signal with the variance of 0.4 required by the user.

It should be noted that the application environment of the method for generating gaussian white noise with arbitrary variance provided in this embodiment is as follows:

(1) a signal generator, a Gaussian noise source and the like, and any system needing to generate Gaussian white noise can be used;

(2) a filter may be added to change the generated noise signal to the bandwidth required by the user, and if not, other signals (e.g., PRBS, white noise signal) may also be changed to gaussian noise signal with arbitrary variance.

The foregoing is illustrative of the preferred embodiments of this invention, and it is to be understood that the invention is not limited to the precise form disclosed herein and that various other combinations, modifications, and environments may be resorted to, falling within the scope of the concept as disclosed herein, either as described above or as apparent to those skilled in the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (8)

1. A method for generating Gaussian white noise with arbitrary variance includes filtering output of m sequence generator by FIR filter, merging output sequences, converting output sequences into analog signal by D/A converter, linear amplifying and filtering to obtain broadband noise signal, adding arbitrary variance Gaussian noise mapping module between FIR filter and D/A converter, said mapping module includes following steps:

s1, obtaining probability statistics P according to input signals, namely sequences output by FIR filter₁；

S2, generating a Gaussian white noise density function corresponding to the variance according to user requirements;

s3, obtaining the Gaussian white noise probability P according to the Gaussian noise density function₂；

S4. from P₁＝P₂Obtaining a mapping curve of the input signal mapped to variance Gaussian white noise;

and S5, according to the mapping curve, outputting the input signal after mapping.

2. The method for generating arbitrary variance white gaussian noise according to claim 1, wherein said step S1 comprises the sub-steps of:

s11, normalizing the input signal to enable the input signal to be subjected to Gaussian distribution between [ -1,1 ];

s12, suppose for [ -11]Random noise x in between₁Obeying to the gaussian distribution, the simplest probability statistical method is as follows:

P₁(x₁)＝p(x<x₁)。

3. the method for generating arbitrary variance white gaussian noise according to claim 2, wherein said step S2 comprises the sub-steps of:

s21, defining Gaussian white noise with variance sigma by a user;

s22, according to a formula, the probability density of generating the Gaussian white noise is as follows:

wherein- ∞ < y < + > infinity.

4. The method of claim 3, wherein in step S2, only noise points within the interval of [ -4 σ, 4 σ ] are considered when converting to white Gaussian noise because the probability of distribution of the interval of noise outside of [ -4 σ, 4 σ ] is small.

5. The method of claim 3, wherein the step S3 specifically comprises: any white noise x obeying a Gaussian distribution with variance of σ₂The corresponding probability is:

6. the method for generating arbitrary variance white gaussian noise according to claim 5, wherein said step S4 comprises the following sub-steps:

s41, order P₁＝P₂Then for either one is [ -1,1 [)]Noise points x obeying a certain gaussian distribution therebetween₁The white Gaussian noise point is x₂And x is₂Satisfy P₂(x≤x₂)＝P₁(x≤x₁)；

S42, taking the axis of abscissa to represent any random noise x₁The ordinate represents the corresponding gaussian noise x with a user-defined variance σ₂And obtaining a mapping curve k.

7. The method of claim 6, wherein the mapping curve k is obtained directly by calculation or by piecewise fitting without introducing large loss.

8. The method of claim 6, wherein in step S5, the input signal is mapped to output signal: y is k × x.

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