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CN111525976A - Hidden communication method based on normal random process mean parameter modulation - Google Patents

Hidden communication method based on normal random process mean parameter modulation Download PDF

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CN111525976A
CN111525976A CN202010138833.XA CN202010138833A CN111525976A CN 111525976 A CN111525976 A CN 111525976A CN 202010138833 A CN202010138833 A CN 202010138833A CN 111525976 A CN111525976 A CN 111525976A
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CN111525976B (en
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王亢
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Zhejiang Police College
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04K—SECRET COMMUNICATION; JAMMING OF COMMUNICATION
    • H04K1/00—Secret communication
    • H04K1/003—Secret communication by varying carrier frequency at or within predetermined or random intervals
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W12/00—Security arrangements; Authentication; Protecting privacy or anonymity
    • H04W12/02—Protecting privacy or anonymity, e.g. protecting personally identifiable information [PII]
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

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Abstract

A hidden communication method based on normal random process mean parameter modulation takes a normal random sequence with a non-0 mean value as a carrier, the mean polarity of the normal random sequence is modulated by hidden bits transmitted by a legal user, and then the carrier is multiplied by a bipolar pseudo-random code, so that the hidden signal in the whole bit period presents the characteristic of zero mean value; the concealed signal transmitted by this method is very similar to the gaussian noise that is necessarily present in the receiver, so that the presence of the concealed communication is not easily perceived by an eavesdropper. The invention provides a structure diagram of a covert communication system, a probability density function of a covert signal, an average value estimator, and a bit error rate formula under an additive white Gaussian noise channel, and indicates the relationship between the bit error rate and the number of samples, the signal-to-noise ratio and the size of a carrier average value in a bit period. The invention is easy to realize, has low requirement on the calculation performance of equipment and has better imperceptibility.

Description

Hidden communication method based on normal random process mean parameter modulation
Technical Field
The invention relates to the technical field of wireless communication and information security, in particular to a covert communication method based on normal random process mean value parameter modulation.
Background
In the data interaction process between the current and future wireless communication devices and legal users, the data security problem is more and more emphasized. One solution is to encrypt the network connection between the transport layer and the application layer by means of a standard protocol SSL/TLS or the like or a proprietary protocol. However, another problem is introduced during encryption, and encryption technology encrypts plaintext into a bunch of messy codes and reminds an eavesdropper that important information exists in transmitted data. An eavesdropper may not be able to decipher the communication content between the legitimate users, but may transmit an interference signal, so that the legitimate users cannot communicate normally.
The invention adopts a covert communication method, and the transmitted covert signal has the random statistical characteristic similar to the environmental interference, so that an eavesdropper can not distinguish the source of the received signal (from the covert signal transmitted by a legal user or the environmental interference under multiple users). By the aid of the hidden communication method with the low interception probability, an eavesdropper is likely to fail to detect the existence of hidden transmission between legal users, and no mention is made of cracking of transmitted information, so that the purpose that important information of the users is not intercepted is achieved.
The invention provides a concealed communication method hidden in environmental interference by using a random sequence normally distributed with a non-0 mean value as a carrier and combining a bipolar pseudo-random code, wherein an additive white Gaussian noise channel is a classic channel in communication.
Disclosure of Invention
In order to realize the safe transmission of the hidden data by a legal user, the invention provides a hidden communication method based on the mean value parameter modulation in the normal distribution random process, the hidden signal emitted by the method presents the normal distribution of 0 mean value, and is similar to the statistical characteristic of the Gaussian noise inevitably existing in a receiver, so that an eavesdropper cannot distinguish whether the received signal is from environmental interference or hidden signal, and the hidden communication method has strong concealment. In addition, the structure of the covert communication system and the synchronization of the pseudo-random codes are unknown to the eavesdropper, which increases the difficulty of the eavesdropper in cracking.
The invention provides the following technical scheme:
a hidden communication method based on normal random process mean parameter modulation comprises information hiding processing of a hidden communication system transmitter and information recovery processing of a hidden communication system receiver;
the information hiding process of the covert communication system transmitter comprises the following steps:
1-1: the random sequence generated by a normally distributed noise generator, the mean of which is modulated by a masked bit b, is also generatedThat is, when b is '0', a normal distribution noise generator as a carrier generates a normal distribution noise generator having a length n, a mean μ, and a variance σ in one bit period2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(μ，σ2) (ii) a For b of '1', a normal distribution noise generator generates a mean of- μ and a variance of σ2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(-μ，σ2)；
1-2: a pseudo-random code generator for generating an orthogonal or quasi-orthogonal pseudo-random code sequence with length n in one bit period, a single-polarity and double-polarity conversion for converting unipolar '0' and '1' into bipolar '+ 1' and '1', respectively, and a pseudo-random sequence after the single-polarity and double-polarity conversion is marked as { mk，k＝1，...，n}；
1-3: multiplying bipolar pseudo-random code sequence by normal distribution sequence modulated by hidden bit to obtain hidden signal sequence { sk＝mkxkK is 1,.. multidot.n }, the concealment signal exhibits the characteristic of zero-mean normal distribution, i.e. the mean value of each bit concealment signal sequence is 0;
the information recovery process of the covert communication system receiver comprises the following steps:
2-1: in case of synchronization, the receiver generates a pseudo-random sequence that is identical to the transmitter; the pseudo-random sequence is converted into unipolar '0' and unipolar '1' through single-polarity and bipolar '1' and bipolar pseudo-random code { mk，k＝1，...，n}；
2-2: n random sequences r received by the receiver in one bit periodk＝skwk, k1, n, where wk, k1, n is an additive white gaussian noise sequence with a mean of 0 and a variance of 0
Figure BDA0002398293100000031
Then { rkK 1.. multidata, n is synchronized with the pseudo-random sequence of the synchronous bipolarity { m }kMultiplying k by 1.. multidot.n to obtain { u }k＝rkmkThe sequence k 1.. multidot.n is fed to the mean estimator, and then the length n is used as the length n of the sequence ykObtaining the estimated value of the parameter of mean value in random sequence
Figure BDA0002398293100000032
The estimator is as follows:
Figure BDA0002398293100000033
2-3: obtaining estimates of concealed bits using hard decisions
Figure BDA0002398293100000034
The hard decision rule is:
Figure BDA0002398293100000035
in covert communication systems, a pair of synchronized pseudo random code generators are required, where the pseudo random codes function: firstly, the transmitted concealed signal presents the characteristic of zero mean value; secondly, the method is used for distinguishing different users, and the receiver can recover the non-0-mean normal random sequence modulated by the concealed bit from the received signal only if the receiver is consistent with the pseudo-random code of the transmitter; thirdly, the eavesdropper must master the structure of the covert communication system and the synchronous pseudo-random code to break the covert communication.
Further, the pseudo-random codes pick approximately orthogonal Gold codes.
The pseudo-random code sequence, if used to distinguish users, is the same at each concealed bit.
Still further, as the non-0-mean normally distributed random noise of the carrier, when the ratio of the mean value mu to the standard deviation sigma is less than 0.4, that is, mu is less than or equal to k sigma (k is more than 0 and less than or equal to 0.4), the hidden signal obtained by multiplying the bipolar pseudo random code sequence is the normally distributed random noise with the approximate zero mean value, and the variance is the normally distributed random noise with the zero mean value
Figure BDA0002398293100000041
Specifically, the random variables X to N (μ, σ)2) The distribution column P (M ═ 1) ═ 1/2, M and X are independent from each other, and when S is MX, the probability density function of S is
Figure BDA0002398293100000042
When mu is not more than k sigma (k is more than 0 and not more than 0.4), fSG for(s)S(s) approximation
Figure BDA0002398293100000043
In the formula
Figure BDA0002398293100000044
Further, a mean estimator
Figure BDA0002398293100000045
Is composed of
Figure BDA0002398293100000046
Accordingly, when the pseudorandom code sequence of the receiver is synchronized with the pseudorandom code sequence of the transmitter, the estimator has an average value of
Figure BDA0002398293100000051
The variance of the estimator is
Figure BDA0002398293100000052
In the formula
Figure BDA0002398293100000053
Is the variance of additive white gaussian noise.
When the hidden bit '0' and '1' are synchronous and the pseudo-random code sequences of the transmitting and receiving parties are synchronous, the bit error rate rho under the additive white Gaussian channel is
Figure BDA0002398293100000054
The complementary error function in the formula is defined as
Figure BDA0002398293100000055
The formula indicates the relationship between the bit error rate and the number of samples n, the ratio r of the mean value mu and the standard deviation of the concealed signal and the signal-to-noise ratio r, when the pseudo-random code sequences of the two parties of the transmitter and the receiver are not synchronous, the bit error rate is different
Figure BDA0002398293100000056
This means that an eavesdropper cannot acquire any valid information if it cannot synchronize with the pseudorandom code of the transmitter.
As the signal-to-noise ratio increases, the bit error rate decreases, but there is an error bit rate platform (error floor), when the signal-to-noise ratio tends to infinity, the bit error rate platform value is the bit error rate of the ideal channel, and is equal to the bit error rate of the ideal channel
Figure BDA0002398293100000057
In the invention, a covert communication system comprises a covert transmitter and a receiver, wherein in the transmitter, a covert bit modulates a mean value parameter of normally distributed random carriers; estimating the mean value at the receiver, thereby recovering the concealed bits;
the transmitter in the covert communication system takes a covert bit b as '0' and serves as a normal distribution noise generator of a carrier wave in a bit period to generate a signal with the length of n, the mean value of mu and the variance of sigma2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(μ，σ2) (ii) a For a masked bit b of '1', a normal distribution noise generator generates a mean of- μ and a variance of σ2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(-μ，σ2). Then, the normal distribution random sequence with the mean value of n being not 0, and the bipolarPseudo-random code sequence of equal probability { m }kMultiplying k by 1.. multidot.n to obtain a concealment signal sequence { s }k＝mkxkAnd k is 1. The concealment signal exhibits a zero-mean normal distribution characteristic, i.e., the mean of each bit concealment signal sequence is 0, which is similar to additive white gaussian noise in wireless communication.
The receiver in the covert communication system multiplies the received sequence by the synchronous pseudo-random code sequence to recover a normal random sequence with a non-0 mean value; obtaining the estimated value of the mean value by a mean value estimator
Figure BDA0002398293100000061
And obtaining the estimation of the hidden bit through hard decision.
In covert communication systems, a pair of synchronized pseudo-random code generators are required. Approximately orthogonal Gold codes are chosen as pseudo-random codes. The function of the pseudo-random code is: firstly, the transmitted concealed signal presents the characteristic of zero mean value; secondly, the method is used for distinguishing different users; thirdly, the difficulty of the eavesdropper in cracking is increased.
The conception of the invention is as follows: the non-0-mean normal distribution random sequence is used as a carrier, the mean polarity of the carrier is modulated by transmitted hidden bits, and then the carrier is multiplied by a bipolar pseudo-random code, so that a hidden signal presents the characteristic of zero-mean normal distribution; this is very similar to the gaussian noise that the receiver must have, and eavesdroppers are insensitive to this, thus achieving the goal of covert communication.
The invention has the beneficial effects that: the hidden communication system has low calculation complexity and is suitable for equipment with weak calculation power, such as the Internet of things and the like; the probability density function formula of the concealed signal and the performance analysis of the estimator are given, and the bit error rate formula of the concealed communication system under the additive white Gaussian noise channel is deduced.
Drawings
Fig. 1 is a block diagram of a covert communication system.
Fig. 2 is a graph of relative error when the probability density function of the mixed random process is approximately normal distribution when μ is 0.4 σ and σ is 1.
Fig. 3 is a normal distribution probability chart of the estimator when n is 1000 and the channel snr is-10 dB.
Fig. 4 shows the bit error rate of the concealment system in the additive white gaussian noise channel when μ is 0.4 σ and n is 100.
Detailed Description
The present invention will now be described in detail with reference to the accompanying drawings.
Referring to fig. 1 to 4, a hidden communication method based on normal distribution random process mean parameter modulation, which conceals information of a transmitter of a hidden communication system and recovers information of a receiver of the hidden communication system;
the information hiding process of the covert communication system transmitter comprises the following steps:
1-1: generation of obedient mean value (-1) using a normally distributed noise generatorbMu, variance of sigma2The random distribution sequence of (1) is marked as N ((-1)bμ，σ2) Where b is a binary masked bit, b ∈ {0, 1}, μ and σ are real numbers greater than 0, and the bit period is TbAt a time TbHas n random sequences, marked as x, which are independently and identically distributedkN, specifically, if the concealment bit b is 0, N (μ, σ) N are generated2) A random sequence of distributions; if the concealment bit b is 1, N N (-mu, sigma) bits are generated2) A random sequence of distributions;
1-2: the pseudo-random code generator generates pseudo-random codes with equal probability of '0' and '1', and converts unipolar '0' and '1' into bipolar '+ 1' and '-1' respectively through single-polarity and double-polarity conversion. Marking the bipolar pseudo-random code sequence as mk，k＝1，...，n}；
1-3: the random sequence output by the normally distributed noise generator is multiplied by the bipolar pseudo-random sequence, which is denoted as sk＝xkmkK ═ 1,..., n }; after the multiplication of '1' of single and double polarity transformation, the polarity of the random sequence mean value is changed; multiplication with '+ 1', without changing the polarity of the random sequence mean; containing covert informationSequence skTransmitting the signal into a channel;
setting M as a discrete random variable, and arranging P (M-1) or 1/2; let X be a continuous random variable and the probability density function be fX(x) (ii) a M and X are independent of each other; conditional probability density function f of X when M is MX|M(x | m); let S be MX, it can be shown that S is a continuous random variable and the probability density function is fS(s) is
Figure BDA0002398293100000081
Subsequent figures 2 and 3 demonstrate that under certain conditions, the random variable S can be approximated as a normal distribution, further, as can be derived from equation (1), fS(-s)＝fS(s), i.e. transmitted { skIn addition, the distribution of the k-1, n-n sequence is symmetrical about s-0, so that in a concealment bit period, although the random sequence output by the noise generator is a normal distribution with a non-0-mean value, the random sequence is approximately a normal distribution with a 0-mean value after being multiplied by the bipolar pseudo random code, and thus, a transmission signal containing concealment information follows a normal distribution with a mean value of 0 as the noise of the gaussian channel, and is difficult for an eavesdropper to distinguish, so that the transmission signal has good concealment.
The information recovery process of the covert communication system receiver comprises the following steps:
2-1: in case of synchronization, the receiver generates a pseudo-random sequence that is identical to the transmitter; the pseudo-random sequence is converted into unipolar '0' and unipolar '1' through single-polarity and bipolar '1' and bipolar pseudo-random code { mk，k＝1，...，n}；
2-2: the sequence after passing through the additive white Gaussian noise channel is rk＝sk+nkK1, …, n; then multiplied by a synchronous bipolar pseudorandom sequence mk, k1, n, to obtain { r }kmkThe sequence k 1.. multidot.n is fed to a mean estimator, which estimates the mean value as:
Figure BDA0002398293100000091
2-3: obtaining estimates of concealed bits using hard decisions
Figure BDA0002398293100000092
The hard decision rule is:
Figure BDA0002398293100000093
referring to fig. 2, a hidden communication method based on normal distribution random process mean parameter modulation, where a hidden signal containing hidden bits is approximately normally distributed according to 0 mean, is specifically described as follows:
equation (1) gives the probability density function of the concealment signal, where the probability density function of the random variable X is:
Figure BDA0002398293100000094
thus, the probability density function of the concealment signal is:
Figure BDA0002398293100000095
when mu is not more than k sigma (k is more than 0 and not more than 0.4), fS(s) may be used as gS(s) to approximate the difference between the first and second values,
Figure BDA0002398293100000096
wherein
Figure BDA0002398293100000097
Fig. 2 shows the relative error when σ is 1 and k is 0.4, and equation (5) is approximated to equation (6), which is defined as
Figure BDA0002398293100000098
The value range in FIG. 2 is [ -3 σ ]S，3σS]According to the "3 σ" criterion of normal distribution, 99.7% probability falls within this range. In addition, the curve shapes of the graphs are identical with different sigma values. As can be seen from the figure, the maximum relative error is within 6%, which is acceptable. In particular, the approximation is good with a smaller k value. For example, k is 0.3, with a maximum relative error within 2%; k is 0.2, and the maximum relative error is within 0.4%.
Referring to fig. 3, a normal distribution probability map for the estimator when n is 1000 and the channel snr is-10 dB. The transmitted concealment signal S ═ MX, whose probability density function is equation (1), can be approximated as having a mean of 0 and a variance of 0, given by equation (6), when μ ≦ k σ (0 < k ≦ 0.4) is selected
Figure BDA0002398293100000101
The normal distribution of (1) is a random process. The output after passing through the additive white Gaussian channel is R ═ S + W ═ MX + W (8)
Wherein W is the mean value of 0 and the variance of
Figure BDA0002398293100000102
White Gaussian noise (SNR), defined as the ratio of the power of the concealment signal to the power of the additive white Gaussian noise (i.e. the ratio of the power of the concealment signal to the power of the additive white Gaussian noise)
Figure BDA0002398293100000103
If the pseudo-random code generator M 'of the receiver is synchronized with the transmitter, and M' is equal to M, the input average estimator is
Figure BDA0002398293100000104
When the pseudo-random code sequence of the receiver is synchronous with the pseudo-random code sequence of the transmitter, namely M 'is M, M' is 1; when the pseudo random code sequences at the transmitting and receiving ends are not synchronous,
Figure BDA0002398293100000107
in addition, M' W obeys a mean of 0 and a variance of
Figure BDA0002398293100000105
Is normally distributed. When the pseudo-random code sequences at the transmitting end and the receiving end are synchronous, X and M' W are independent of each other due to different sources, and then Y obeys mean value mu and variance
Figure BDA0002398293100000106
Is normally distributed. The estimator of the mean value μ is, as shown in equation (2)
Figure BDA0002398293100000111
Accordingly, the mean of the estimator is
Figure BDA0002398293100000112
The variance of the estimator is
Figure BDA0002398293100000113
It can be verified by simulation that the estimator follows normal distribution, as shown in fig. 3, where the simulation conditions are n 1000, k 0.4, and r-10 dB.
Obviously, when the pseudo random code sequences at the transmitting end and the receiving end are not synchronous, Y ═ M' W does not contain the hidden bit modulation signal X, and the hidden bit cannot be demodulated, that is, the bit error rate is 0.5.
Referring to fig. 4, a bit error rate performance curve for blind communications over an additive white gaussian noise channel. The simulation conditions are n-100, k-0.4, and r-10, 10 dB. As can be seen from the simulation diagram, the simulation is very close to the theoretical derivation. The theoretical bit error rate is derived as follows:
in conjunction with equations (12) and (13), the probability density function of the estimator is
Figure BDA0002398293100000114
And combining the hard decision rule given by the formula (3), the bit error rate rho is
Figure BDA0002398293100000115
The complementary error function in the formula is defined as
Figure BDA0002398293100000116
It can be seen from the formula that increasing the number of bit samples n, the mean of the concealment signal, and the signal-to-noise ratio can reduce the bit error rate. However, increasing the number n decreases the transmission rate; increasing the mean value mu increases the transmit power of the concealment signal, reducing concealment. It can be seen from the figure that, under the condition of low signal-to-noise ratio, the bit error rate is relatively high, that is, the influence of gaussian noise on the estimator is relatively large; on the other hand, when the signal-to-noise ratio is large, there is an error floor (error floor), that is, no matter how high the signal-to-noise ratio is, the bit error rate can not be reduced any more. This is because the limit at this time is the ideal channel, so the bit error rate platform value is the bit error rate at the ideal channel, which is equal to the bit error rate at the ideal channel
Figure BDA0002398293100000121

Claims (7)

1. A hidden communication method based on normal random process mean parameter modulation is characterized in that: the covert communication method comprises information hiding processing of a covert communication system transmitter and information recovery processing of a covert communication system receiver;
the information hiding process of the covert communication system transmitter comprises the following steps:
1-1: the random sequence generated by the normal distribution noise generator has its mean value modulated by the hidden bits, i.e. when b is '0', the normal distribution noise generator as the carrier generates the random sequence with length n, mean value mu and variance sigma in one bit period2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(μ，σ2) (ii) a For b of '1', a normal distribution noise generator generates a mean of- μ and a variance of σ2Is normally distributed and randomly ordered as { xk，k＝1，...，n}～N(-μ，σ2)；
1-2: a pseudo-random code generator for generating an orthogonal or quasi-orthogonal pseudo-random code sequence with length n in one bit period, a single-polarity and double-polarity conversion for converting unipolar '0' and '1' into bipolar '+ 1' and '1', respectively, and a pseudo-random sequence after the single-polarity and double-polarity conversion is marked as { mk，k＝1，...，n}；
1-3: multiplying bipolar pseudo-random code sequence by normal distribution sequence modulated by hidden bit to obtain hidden signal sequence { sk＝mkxkK is 1,.. multidot.n }, the concealment signal exhibits the characteristic of zero-mean normal distribution, i.e. the mean value of each bit concealment signal sequence is 0;
the information recovery process of the covert communication system receiver comprises the following steps:
2-1: in case of synchronization, the receiver generates a pseudo-random sequence that is identical to the transmitter; the pseudo-random sequence is converted into unipolar '0' and unipolar '1' through single-polarity and bipolar '1' and bipolar pseudo-random code { mk，k＝1，...，n}；
2-2: n random sequences r received by the receiver in one bit periodk＝sk+ wk, k 1., n, where wk, k 1., n is an additive white gaussian noise sequence with a mean of 0 and a variance of 0
Figure FDA0002398293090000025
Then rkK 1.. multidata, n is synchronized with the pseudo-random sequence of the synchronous bipolarity { m }kMultiplying k by 1.. multidot.n to obtain { y }k＝rkmkThe sequence k 1.. multidot.n is fed to the mean estimator, and then the length n is used as the length n of the sequence ykObtaining the estimated value of the parameter of mean value in random sequence
Figure FDA0002398293090000021
The estimator is as follows:
Figure FDA0002398293090000022
2-3: obtaining estimates of concealed bits using hard decisions
Figure FDA0002398293090000023
The hard decision rule is:
Figure FDA0002398293090000024
in covert communication systems, a pair of synchronized pseudo random code generators are required, where the pseudo random codes function: firstly, the transmitted concealed signal presents the characteristic of zero mean value; secondly, the method is used for distinguishing different users, and the receiver can recover the non-0-mean normal random sequence modulated by the concealed bit from the received signal only if the receiver is consistent with the pseudo-random code of the transmitter; thirdly, the eavesdropper must master the structure of the covert communication system and the synchronous pseudo-random code to break the covert communication.
2. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 1, characterized in that: the pseudo-random codes pick approximately orthogonal Gold codes.
3. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 2, characterized in that: the pseudo-random code sequence, if used to distinguish users, is the same at each concealed bit.
4. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: non-0-mean normal distribution as a carrierWhen the ratio of the mean value mu to the standard deviation sigma is less than 0.4, namely mu is less than or equal to K sigma (K is more than 0 and less than or equal to 0.4), the hidden signal obtained after multiplication with the bipolar pseudo-random code sequence is approximate to the normal distribution random noise with zero mean value, and the variance is
Figure FDA0002398293090000031
Specifically, the random variables X to N (μ, σ)2) The distribution column P (M ═ 1) ═ 1/2, M and X are independent from each other, and when S is MX, the probability density function of S is
Figure FDA0002398293090000032
When mu is not more than k sigma (k is more than 0 and not more than 0.4), fSG for(s)S(s) approximation
Figure FDA0002398293090000033
In the formula
Figure FDA0002398293090000034
5. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: mean estimator
Figure FDA0002398293090000035
Is composed of
Figure FDA0002398293090000036
Accordingly, when the pseudorandom code sequence of the receiver is synchronized with the pseudorandom code sequence of the transmitter, the estimator has an average value of
Figure FDA0002398293090000037
The variance of the estimator is
Figure FDA0002398293090000038
In the formula
Figure FDA0002398293090000041
Is the variance of additive white gaussian noise.
6. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: when the hidden bit '0' and '1' are synchronous and the pseudo-random code sequences of the transmitting and receiving parties are synchronous, the bit error rate rho under the additive white Gaussian channel is
Figure FDA0002398293090000042
The complementary error function in the formula is defined as
Figure FDA0002398293090000043
The formula indicates the relationship between the bit error rate and the number of samples n, the ratio r of the mean value mu and the standard deviation of the concealed signal and the signal-to-noise ratio r, when the pseudo-random code sequences of the two parties of the transmitter and the receiver are not synchronous, the bit error rate is different
Figure FDA0002398293090000044
This means that an eavesdropper cannot acquire any valid information if it cannot synchronize with the pseudorandom code of the transmitter.
7. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 6, wherein: with the increase of the signal-to-noise ratio, the bit error rate is reduced, but there is a bit error rate platform error floor, when the signal-to-noise ratio tends to be infinite, the bit error rate platform value is the bit error rate when the ideal channel is equal to the bit error rate when the signal-to-noise ratio is equal to the ideal channel
Figure FDA0002398293090000045
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