CN110855246B - 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 general method of digital synthesis of gaussian white noise is to generate white noise with uniform distribution, and then transform the uniform distribution into gaussian distribution to obtain gaussian white noise. Document "implementation of a gaussian white noise generator in an FPGA" (Huang Benxiong, etc.) first adopts a Tausworthe algorithm to generate uniformly distributed white noise, and then realizes conversion from the uniformly distributed white noise to the gaussian white noise 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 rapid method for generating a white gaussian noise sequence by using an FPGA (Guan Yu, 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 module for generating m sequence, if the selected number of stages of the linear feedback shift register is n, the repetition period of the m sequence can be 2 ^ｎ -1. The data generated by the m sequence is regarded as unsigned integer, and the value range of the data is 1~2 ^ｎ -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 a feedback coefficient of the i-th bit register. When c is going to _i When =0, no feedback is indicated; when c is going to _i If =1, feedback is indicated. In this structure c ₀ =c _r =1，c ₀ Not 0,c ₀ A value of 0 does not constitute a periodic sequence, since c ₀ =0 means no feedback, being a static shift register. c. C _r And also cannot be 0, i.e. the r-th bit register must participate in feedback, otherwise, the feedback shift register of the r stages will be reduced to the feedback shift register of the r-1 stage or lower. Different feedback logic, i.e. c _i (i =1,2,3, …, r) take different values, which will result 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 the rising edge (or falling edge) of each clock to derive a new pseudo code word 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 to a 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 by one clock cycle; the phase of S3 lags S2 by one clock cycle; phase lag S3 of S4 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 required, and this 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 the 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 S1 state is:

the feedback codeword corresponding to the S2 state is:

a is needed to be calculated when the feedback code word corresponding to the S2 state is calculated _n+3 The feedback coefficient is brought into a formula to participate in calculation, and the formula can be simplified according to the actual feedback coefficient. Similarly, the feedback code words corresponding to the states of 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), its 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), the above formula can be reorganized as follows:

order to

Then the above formula can be written as:

in summary, although this is a common 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 the white gaussian noise is not controllable, but 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 noise with arbitrary variance, which comprises filtering the output of an m-sequence generator through an FIR filter, combining the filtered sequences, converting the combined 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 arbitrary variance gaussian noise mapping module comprises the following steps:

s1, obtaining probability statistics P according to input signals, namely sequences output by FIR filter ₁ ；

S2, generating a Gaussian white noise density function according to the user-specified variance;

s3, obtaining Gaussian white noise probability statistics P according to the Gaussian white noise density function ₂ ；

S4. From P ₁ =P ₂ Obtaining the input signal mapA mapping curve of white gaussian noise to a user specified variance;

and S5, obtaining an output signal after the input signal is mapped according to the mapping curve.

Further, step S1 includes the following substeps:

s11, normalizing the input signal to make the input signal obey Gaussian distribution between [ -1,1 ];

s12. Suppose for [ -1]Random noise betweenx ₁ Obeying Gaussian distribution, the simplest probability statistical mode is as follows:

。

further, step S2 includes the following substeps:

s21, defining the variance as

White gaussian noise of (1);

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

wherein- ∞ <y＜+∞。

Further, in step S2, the noise is in [ -4 ]

，4/>

]The probability of the distribution of the interval outside is small, so that only the interval of [ -4 ^ is considered when converting to white Gaussian noise>

，4/>

]The noise point within.

Further, the method can be used for preparing a novel materialStep S3 specifically includes: any one obeying a Gaussian distribution variance of

White noise ofx ₂ The corresponding probability statistics are as follows:

。

further, step S4 includes the following substeps:

s41, order P ₁ =P ₂ Then for any one is [ -1,1]Noise points obeying a certain Gaussian distributionx ₁ The Gaussian white noise point of the variance specified by the corresponding user isx ₂ And is andx ₂ satisfies P ₂ (x≤x ₂ )=P ₁ (x≤x ₁ )；

S42, taking the axis of abscissa to represent any random noisex ₁ The ordinate represents the user-defined variance corresponding thereto as

White gaussian noise ofx ₂ Obtaining a mapping curve k;

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 output signal after mapping the input signal:y=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 is a time domain and frequency domain signal diagram of the m sequence;

FIG. 8 is a chart of statistical probability density for 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 statistical signal probability density after filtering;

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

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

FIG. 14 gives variance

The Gaussian signal probability statistical characteristic diagram;

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

FIG. 16 maps a curve fit plot;

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 2 ³² 1 (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) Variance according to user's requirement

= 0.4. Synthesize the required Gaussian probability density function, from which a given variance ≦ is derived as shown in FIG. 13>

The statistical characteristics of the gaussian signal probability of (a), as shown in fig. 14;

(7) Here P is the same according to the statistical properties of the probabilities ₁ =P ₂ According 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 signal by using a fitted curve, wherein the time domain and frequency domain signals are as 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:

(1) Signal generator, gaussian noise source, etc. any system that needs 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 (1)

1. A method for generating Gaussian noise with arbitrary variance, which filters the output of m sequence generator through FIR filter, after merging the filtered sequences, converts them into analog signals through D/A converter, then linear amplification and filtering to get broadband noise signal, characterized in that arbitrary variance Gaussian noise mapping module is added 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 filter ₁ ；

S2, generating a Gaussian white noise density function according to the variance specified by the user;

s3, obtaining Gaussian white noise probability statistics P according to the Gaussian white noise density function ₂ ；

S4. From P ₁ =P ₂ Obtaining a mapping curve of the input signal to Gaussian white noise of a user-specified variance;

s5, obtaining an output signal after the input signal is mapped according to the mapping curve;

step S1 includes the following substeps:

s11, normalizing the input signal to make the input signal obey Gaussian distribution between [ -1,1 ];

s12. Suppose for [ -1]In betweenRandom noisex ₁ Obeying to the gaussian distribution, the simplest probability statistical method is as follows:

；

step S2 includes the following substeps:

s21, defining the variance as

White gaussian noise of (1);

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

wherein- ∞ <y＜+∞；

In step S2, the noise is [ -4 ]

，4/>

]The probability of the distribution of the interval outside is small, so that only the interval of [ -4 ^ is considered when converting to white Gaussian noise>

，4/>

]A noise point within;

the step S3 specifically comprises the following steps: any one obeys a Gaussian distribution variance of

White noise ofx ₂ The corresponding probability statistics are as follows:

；

step S4 includes the following substeps:

s41, let P ₁ =P ₂ Then for either one is [ -1,1]Noise points obeying a certain gaussian distributionx ₁ The Gaussian white noise point of the variance specified by the corresponding user isx ₂ And is andx ₂ satisfy P ₂ (x≤x ₂ )=P ₁ (x≤x ₁ )；

S42, taking the axis of abscissa to represent any random noisex ₁ The ordinate represents the user-defined variance corresponding thereto as

White gaussian noise ofx ₂ Obtaining a mapping curve k;

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

in step S5, the output signal after mapping the input signal:y=k×x。

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