CN101667911B - Secret communication method based on fraction order Fourier transform order-multiplexing - Google Patents
Secret communication method based on fraction order Fourier transform order-multiplexing Download PDFInfo
- Publication number
- CN101667911B CN101667911B CN2009100937244A CN200910093724A CN101667911B CN 101667911 B CN101667911 B CN 101667911B CN 2009100937244 A CN2009100937244 A CN 2009100937244A CN 200910093724 A CN200910093724 A CN 200910093724A CN 101667911 B CN101667911 B CN 101667911B
- Authority
- CN
- China
- Prior art keywords
- mrow
- msub
- centerdot
- msup
- mfrac
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired - Fee Related
Links
- 238000004891 communication Methods 0.000 title claims abstract description 39
- 238000000034 method Methods 0.000 title claims abstract description 22
- 230000005540 biological transmission Effects 0.000 claims abstract description 12
- 238000005070 sampling Methods 0.000 claims abstract description 7
- 239000013598 vector Substances 0.000 claims description 22
- 239000011159 matrix material Substances 0.000 claims description 19
- 230000015572 biosynthetic process Effects 0.000 claims description 5
- 238000003786 synthesis reaction Methods 0.000 claims description 5
- 238000001914 filtration Methods 0.000 claims description 3
- 230000008569 process Effects 0.000 claims description 3
- 238000000605 extraction Methods 0.000 claims 1
- 238000013461 design Methods 0.000 abstract description 4
- 230000008859 change Effects 0.000 abstract description 2
- 238000011084 recovery Methods 0.000 abstract 1
- 238000005516 engineering process Methods 0.000 description 5
- 238000012545 processing Methods 0.000 description 3
- 238000004088 simulation Methods 0.000 description 2
- 230000007547 defect Effects 0.000 description 1
- 238000011161 development Methods 0.000 description 1
- 238000010586 diagram Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 229940050561 matrix product Drugs 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 230000003287 optical effect Effects 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 230000002441 reversible effect Effects 0.000 description 1
- 208000011580 syndromic disease Diseases 0.000 description 1
- 230000009466 transformation Effects 0.000 description 1
- 238000012795 verification Methods 0.000 description 1
Images
Landscapes
- Complex Calculations (AREA)
Abstract
The invention relates to a secret communication method based on fraction order Fourier transform order-multiplexing, belonging to the information safety field. The secret communication method comprises the following steps: establishing linear equations by using the relation among filters in a multiplex transmission multiplexer accurately reestablished by fraction order Fourier domains with different orders and selecting discrete-sampling discrete fraction order Fourier transform as a basic tool, realizing the demodulation of the fraction order Fourier transform order-multiplexing; and realizing effective recovery of transmission information by using the multiplexed fraction order Fourier transform order as a secret in the information decryption. The secret communication method based on the fraction order Fourier transform order-multiplexing completely utilizes the order change characteristic of the fraction order Fourier transform, enlarges the secret space relative to the traditional secret communication method based on fraction order Fourier transform, improves the safety of the system, can realize secret communication under a multi-carrier mode, and can provide a new way for the design of a multi-user communication system.
Description
Technical Field
The invention relates to a secret communication method based on fractional order Fourier transform order multiplexing, and belongs to the technical field of information security.
Background
With the rapid development of modern data communication services, the phenomenon of illegal interception of information occurs occasionally, and therefore, the secure communication technology is developed accordingly. In a secure communication system, both parties of communication perform encryption processing before information transmission based on a key for realizing an agreement, and decrypt the information using a set key after the information is received, and the original information cannot be recovered if the decryption key is wrong. Therefore, the size of a secret communication system key space directly determines the security of the system.
The concept of fractional fourier transform was proposed as early as 1929, and was applied to the optical field in the 80 s of the 20 th century, and it became one of the research hotspots in the signal processing field from the 90 s. Fractional order fourier transform is a generalized form of fourier transform, which performs signal processing on a uniform time-frequency domain, is more flexible than conventional fourier transform, and can give all the characteristics of signal transformation from time domain to frequency domain by selecting orders from 0 to 1. Therefore, the fractional Fourier transform is widely applied to the field of information security, and a few scholars propose an image encryption technology and a secret communication technology based on the fractional Fourier transform, and use parameters of the fractional Fourier transform which are excessive relative to the Fourier transform as keys in information decryption, so that the key space is enlarged, and the security of the system is improved. For example, some researchers have designed a secure communication system using the fractional Hilbert transform based on the fractional fourier transform, which expands the key space and improves the security of the system compared with the secure communication system based on the fractional Hilbert transform.
However, the existing secure communication system based on fractional fourier transform is only a simple extension of the traditional secure communication system based on fourier transform, that is, fractional fourier transform or fractional fourier domain is adopted to replace fourier transform or fourier domain in the traditional system, and the order of fractional fourier transform is not fully utilized.
Disclosure of Invention
The invention aims to provide a secret communication method based on fractional Fourier transform order multiplexing in order to improve the security of a system and fully utilize the order of fractional Fourier transform. The secret communication method selects discrete sampling type discrete fractional Fourier transform and fractional circumferential convolution theorem as basic tools, utilizes a fractional Fourier domain to accurately reconstruct a multiplexer to realize fractional Fourier transform order multiplexing, utilizes the multiplexed fractional Fourier transform order as a secret key, overcomes the defect that the existing secret communication system based on the fractional Fourier transform is only based on simple expansion of a Fourier transform system and cannot fully utilize the fractional Fourier transform order, expands the space of the secret key, and improves the safety of the system.
The invention relates to a secret communication method based on fractional Fourier transform order multiplexing, which comprises two parts of information encryption and information decryption;
the steps for realizing information encryption are as follows:
step one, constructing a fractional Fourier domain accurate reconstruction N-channel transmission multiplexer shown in figure 1. According to the length L of the information to be transmitted, an N-channel multiplexer with the Fourier domain length of NxL and the time domain sampling interval of delta t is selected, and a comprehensive filter bank of the N-channel multiplexer is composed of { G }l(k) 0, 1, N-1, and an analysis filter bank is denoted by { H }l(k)}l＝0，1，...，N-1And (4) showing. P is designed by, for example, formula (1) and formula (2) according to the number M of users in the systemiOrder (i ═ 1, 2, …, M2) Fractional order Fourier domain multiplexer
Wherein,
is piA first-path synthesis filter in the fractional-order Fourier domain N-channel multiplexer,
is piThe first path analysis filter in the fractional order fourier domain N-channel multiplexer,
is piFractional Fourier domain sampling interval of order, and fractional Fourier domain order piSatisfies the relationship shown in the formula (3)
Step two, modulating the information to be transmitted by adopting the fractional order Fourier domain multiplexer comprehensive filter group obtained in the information encryption step one to obtain M fractional order Fourier transform order multiplexing signals
M is 1, 2,.. times.m, and its mathematical expression is shown in formula (4)
Wherein,the ith channel transmission information representing the ith user, and a matrix thetaint∈□NL×LRepresents an interpolation operation, which is defined as shown in equation (5)
Matrix array
Represents piAn order fractional Fourier domain synthesis filtering operation defined as shown in equation (6)
Matrix array
Represents piDiscrete fractional order Fourier transform of order defined as shown in equation (7)
Step three, M fractional Fourier transform order multiplexing signals obtained in the information encryption step two
And sending out in sequence.
The steps for realizing information decryption are as follows:
step one, received M fractional Fourier transform order multiplexing signalsSequentially by a selected p by an information encryption step(m-1)M+1M is 1, 2,.. M, an analysis filter bank of an order fractional order fourier domain multiplexer to obtain M output signal vectors;
step two, extracting k (k is 0, 1, and L-1) elements of M output signal vectors obtained in the step one of information decryption to form a vector
By matrix multiplication
Demodulating the vector formed by the k-th element in all the transmitted signal vectors
The m-th element in (b) is the m-th userThe kth element of the l channel input signal vector;
step three, using all input signal vectors obtained in the information decryption step two
Resuming sending information
The decryption process is completed.
Advantageous effects
Compared with the traditional secret communication technology based on fractional order Fourier transform, the secret communication method provided by the invention fully utilizes the order change characteristic of the fractional order Fourier transform, enlarges the key space and improves the safety of the system;
the secret communication method is realized by utilizing a fractional Fourier domain multiplexer, and compared with the traditional secret communication technology based on fractional Fourier transform, the secret communication under a multi-carrier transmission mode can be realized;
the secret communication method provided by the invention utilizes the multiplexed fractional Fourier transform order to distinguish different users, and provides a new mode for the design of a future multi-user communication system.
Drawings
FIG. 1-a fractional Fourier domain multiplexer;
FIG. 2-two-user secure communication system based on fractional Fourier transform order multiplexing;
FIG. 3-two-user secure communication system encrypted signal constellations based on fractional Fourier transform order multiplexing;
FIG. 4-decrypted signal constellation under wrong key;
figure 5(a) -decrypted signal mean square error with decryption key p'1And p'2Variogram map, (b) -decrypted signal mean square error with decryption key p'3And p'4A variable surface map;
FIG. 6(a) -decrypted Signal error Rate with decryption Key p'1And p'2Graph of varying surface, (b) -decrypted signal bit error rate as a function of decryption key p'3And p'4A variable surface map;
Detailed Description
The secret communication method based on the fractional Fourier transform comprises the following steps
Design of fractional Fourier domain multiplexer
According to the number M of users in the designed system, M satisfying the reversible condition of the matrix C in the formula (3) is selected2Fractional order Fourier transform order, multiplexing order p discussed belowiThe determinant value of the matrix C in the formula (3) can be expressed as shown in the formula (8)
Wherein,
as can be seen from the basic rules of mathematical operations,
not zero, then to make | C | not zero, then p is choseniUsing determinant | C1The simplest method is to construct the non-zero | into a vandermonde determinant, i.e. to select p Mi+11, and
wherein
Are different from each other;
selecting a corresponding Fourier domain accurate reconstruction filter bank according to the size of the data volume transmitted by each user and the number of channels required by information transmission of each user;
thirdly, accurately reconstructing a filter bank according to the fractional Fourier domain order and the sampling interval selected in the first step and the Fourier domain selected in the second step, and obtaining an integrated filter bank and an analysis filter bank of each order fractional Fourier domain accurately-reconstructed multiplexer through the formulas (1) and (2);
secondly, an encryption step
Modulating a signal to be transmitted by utilizing the comprehensive filter group of the different-order fractional Fourier domain multiplexer obtained in the step (III) of designing the fractional Fourier domain multiplexer, and then obtaining M fractional Fourier transform order multiplexing signals through a formula (4)
m＝1，2，...，M；
(II) the fractional Fourier domain order multiplexing signal obtained in the step (I)
And sending out in sequence.
Step three, decryption step
Firstly, the received M fractional Fourier transform order multiplexing signals
Respectively through p corresponding to the encryption terminal(m-1)M+1An order, M ═ 1, 2., M, fractional order fourier domain multiplexer analysis filter bank, to obtain M groups of demodulated signals;
(II) taking out the vector formed by the kth element in all the signal vectors obtained in the step (I)
By passing
Obtaining a signal vector consisting of kth elements in all the signal vectors;
and (III) reconstructing all the signal vectors obtained in the step (II) to recover the transmitted information, and finishing the decryption process.
The following is a detailed theoretical verification of the embodiments in connection with the basic principles of the fractional fourier domain multiplexer.
In the fractional fourier domain multiplexer system shown in fig. 1, the output signal at the end of the system synthesis filter bank can be expressed in matrix form as shown in formula (11)
Accordingly, the output signal at the system analysis filter bank end can be expressed in the form as shown in equation (12)
Wherein Hl，p∈□NL×NLRepresenting a fractional Fourier domain analysis filtering operation, i.e.
Hi，p(k) Is hi(n) discrete fractional Fourier transform of order p, matrix thetadec∈□L×NLRepresents the decimation operation of the signal, which is defined as
To satisfy the exact reconstruction conditions of a fractional Fourier domain multiplexer, i.e.
Then
θdec·F-p·Hl，p·Gl，p·Fp·θint＝IN (16)
When l ≠ m,
θdec·F-p·Hl，p·Gm，p·Fp·θint＝0N (17)
wherein, INIs an N × N identity matrix, 0NIs an N multiplied by N zero matrix.
Suppose that the transmission information of the system utilizes p as shown in formulas (1) and (2), respectively1、p2、...、pMThe fractional order Fourier domain multiplexer implements the transmission, and pi≠pjI ≠ j, then the output signal at the transmitting end of the system can be written in the form shown in equation (18)
Then, at the receiving end of the system, if the received signal passes through pjThe demodulation end of the order fractional Fourier domain multiplexer, the output signal of the receiving endThe number may be written in the form shown in equation (19)
Theorem 1: when l ≠ m, <math>
<mrow>
<msub>
<mi>θ</mi>
<mi>dec</mi>
</msub>
<mo>·</mo>
<msub>
<mi>F</mi>
<mrow>
<mo>-</mo>
<msub>
<mi>p</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>·</mo>
<msub>
<mi>H</mi>
<mrow>
<mi>l</mi>
<mo>,</mo>
<msub>
<mi>p</mi>
<mi>j</mi>
</msub>
</mrow>
</msub>
<mo>·</mo>
<msub>
<mi>G</mi>
<msub>
<mrow>
<mi>m</mi>
<mo>,</mo>
<mi>p</mi>
</mrow>
<mi>i</mi>
</msub>
</msub>
<mo>·</mo>
<msub>
<mi>F</mi>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</msub>
<mo>·</mo>
<msub>
<mi>θ</mi>
<mi>int</mi>
</msub>
<mo>=</mo>
<msub>
<mn>0</mn>
<mi>N</mi>
</msub>
<mo>.</mo>
</mrow>
</math>
and (3) proving that: suppose an order p discrete fractional order Fourier transform matrix FpCan be written as shown in formula (20)
Fp＝Λp，uWΛp，t (20)
Wherein, Λp，u，W，Λp，t∈□NL×NLAnd satisfy the formulas (21), (22) and (23)
Correspondingly, -p discrete fractional Fourier transform matrix F-pCan be written as shown in formula (24)
F-p＝Λ-p，tWHΛ-p，u (24)
Therefore, desire to prove <math>
<mrow>
<msub>
<mi>θ</mi>
<mi>dec</mi>
</msub>
<mo>·</mo>
<msub>
<mi>F</mi>
<mrow>
<mo>-</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</mrow>
</msub>
<mo>·</mo>
<msub>
<mi>H</mi>
<mrow>
<mi>l</mi>
<mo>,</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</mrow>
</msub>
<mo>·</mo>
<msub>
<mi>G</mi>
<msub>
<mrow>
<mi>m</mi>
<mo>,</mo>
<mi>p</mi>
</mrow>
<mi>j</mi>
</msub>
</msub>
<mo>·</mo>
<msub>
<mi>F</mi>
<msub>
<mi>p</mi>
<mi>j</mi>
</msub>
</msub>
<mo>·</mo>
<msub>
<mi>θ</mi>
<mi>int</mi>
</msub>
<mo>=</mo>
<msub>
<mn>0</mn>
<mi>N</mi>
</msub>
</mrow>
</math> If it is true, only the relation shown in the formula (25) is verified
According to formula (1), formula (2), formula (6) and formula (13), i.e. the formula
Wherein, diagonal matrix Hl，Gm∈□NL×NLAnd is and
[Gm]k，k＝Gm(k) (27)
[Hl]k，k＝Hl(k) (28)
due to matrix product
And
can also be written in the form shown in formula (29) and formula (30)
Wherein, the diagonal matrix
And satisfies the formula (31)
Accurate reconstruction of the properties of the multiplexer from the fourier domain yields the relationship shown in equation (32)
Therefore, theorem 1 is used for the syndrome.
According to theorem 1, formula (19) can be further written as shown in formula (33)
According to pjThe linear equation set formed by the M vector linear equations shown in the formula (33) can be obtained through different values of the M vector linear equations.
Theorem 2: linear equation of equations
And
the correlation is linear.
And (3) proving that: according to the exact reconstruction characteristics of the Fourier domain multiplexer, the matrix product, and the equations (1), (2), (6), (13), (20), (24)
Can be written as shown in formula (36)
Multiplication of equation (34) by both ends by equation (13), equation (24) and equation (36)
At the left end of equal sign has
At the right end of equal sign have
From the formulas (37) and (38), it can be found that the formulas (34) and (35) are linearly related, and theorem 2 proves.
From theorem 2, it can be found that fractional order Fourier transform order multiplexing cannot be realized by one-time transmission, and in order to solve the problem of linear correlation between linear equations shown in theorem 2, original information is transmitted for M-1 times, so that a linear equation set shown in formula (39) can be obtained according to theorem 1
Wherein
Then, the total vector can be obtained by the equation (36) and the linear equation set (39)
The k-th element in the equation (41)
Wherein,C∈□M×Mand satisfy the formulas (42), (43) and (44)
Then, if the system can realize demodulation, it needs to reasonably select the fractional Fourier transform multiplexing order piSo that the determinant | C | is not zero. When | C | is not zero, the solution of equation (41) is as shown in equation (45)
The invention is described in detail below with reference to the following figures and examples:
in the simulation experiment, two user fractional Fourier domain order multiplexing secure communication systems composed of fractional Fourier domain two-channel multiplexer shown in FIG. 2 are adopted, and the corresponding fractional Fourier transform multiplexing order is p1＝0.12、p2＝0.62、p30.18 and p40.68. The first path of signal of the user 1 is a QPSK modulation signal, the second path of signal is an 8PSK modulation signal, the first path of signal of the user 2 is an 8QAM signal, and the second path of signal is a 16QAM signal. The constellation diagram of the two output signals output by the encryption end of the system is shown in fig. 3.
At the system decryption end, if the fractional Fourier transform order adopted by decryption is not equal to the fractional Fourier transform order multiplexed by the encryption end, an error occurs in decryption. FIG. 4 gives p'1＝0.1、p’2＝0.6、p’30.2 and p'40.7 is the constellation of the decrypted signal obtained by the decryption key. FIGS. 5(a) and (b) show respectively the mean square error of the decrypted signal as a function of p'1、p’2And p'3、p’4Graphs of the curves, FIGS. 6(a) and (b) show the decrypted signal bit error rate as a function of p'1P '2 and p'3、p’4A varying surface plot.
From the simulation results, it can be found that when the key for signal decryption and the encrypted key have large errors, the information decryption has large errors, and therefore, the secret communication method provided by the invention has high security. In addition, the secret communication method provided by the invention can realize secret communication in a multi-carrier transmission mode and also provides a new way for the design of a multi-user communication system.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (1)
1. A secure communication method based on fractional Fourier transform order multiplexing is characterized in that: the method is divided into information encryption and decryption,
the steps for realizing information encryption are as follows:
step one, constructing a fractional Fourier domain to accurately reconstruct an N-channel multiplexer;
according to the length L of the information to be transmitted, an N-channel multiplexer with the Fourier domain length of NxL and the time domain sampling interval of delta t is selected, and a comprehensive filter bank of the N-channel multiplexer is composed of { G }l(k)}l＝0，1，...，N-1Denoted by, analysis filterbank is represented by { H }l(k)}l＝0，1，...，N-1Represents; p is designed by the following equations (1) and (2) according to the number M of users in the systemiOrder (i ═ 1, 2, …, M2) Fractional order Fourier domain multiplexer
Wherein,
is piA first-path synthesis filter in the fractional-order Fourier domain N-channel multiplexer,
is piThe first path analysis filter in the fractional order fourier domain N-channel multiplexer,
is a fractional Fourier domain sampling interval, and fractional Fourier domain order piSatisfies the relationship shown in the formula (3)
Step two, modulating the information to be transmitted by adopting the fractional order Fourier domain multiplexer comprehensive filter group obtained in the information encryption step one to obtain M fractional order Fourier transform order multiplexing signals
M is 1, 2,.. times.m, and its mathematical expression is shown in formula (4)
Wherein,
the ith channel transmission information representing the ith user, and a matrix thetaint∈□NL×LRepresents an interpolation operation, which is defined as shown in equation (5)
Matrix array
Represents piAn order fractional Fourier domain synthesis filtering operation defined as shown in equation (6)
Matrix arrayRepresents piDiscrete fractional order Fourier transform of order defined as shown in equation (7)
Step three, M fractional Fourier transform order multiplexing signals obtained in the information encryption step two
Sequentially sending out;
the steps for realizing information decryption are as follows:
step one, received M fractional Fourier transform order multiplexing signals
Sequentially by a selected p by an information encryption step(m-1)M+1M is 1, 2,.. M, an analysis filter bank of an order fractional order fourier domain multiplexer to obtain M output signal vectors;
step two,The k (k is 0, 1,.., L-1) th element of the M output signal vectors obtained in the first extraction information decryption step forms a vector
By matrix multiplication
Demodulating the vector formed by the k-th element in all the transmitted signal vectors
The mth element in the user is the kth element of the input signal vector of the mth channel of the mth user;
step three, using all input signal vectors obtained in the information decryption step two
Resuming sending informationThe decryption process is completed.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN2009100937244A CN101667911B (en) | 2009-09-25 | 2009-09-25 | Secret communication method based on fraction order Fourier transform order-multiplexing |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN2009100937244A CN101667911B (en) | 2009-09-25 | 2009-09-25 | Secret communication method based on fraction order Fourier transform order-multiplexing |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN101667911A CN101667911A (en) | 2010-03-10 |
| CN101667911B true CN101667911B (en) | 2012-05-23 |
Family
ID=41804366
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN2009100937244A Expired - Fee Related CN101667911B (en) | 2009-09-25 | 2009-09-25 | Secret communication method based on fraction order Fourier transform order-multiplexing |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN101667911B (en) |
Families Citing this family (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103546261B (en) * | 2012-07-10 | 2016-10-05 | 中国矿业大学(北京) | Micro-seismic wireless high-speed transmission system applied to security field |
| CN106102045A (en) * | 2016-05-25 | 2016-11-09 | 赵京磊 | A kind of secret communication method using Fourier Transform of Fractional Order |
| CN113992384B (en) * | 2021-10-22 | 2023-10-20 | 延安大学 | Secret communication method based on fractional Fourier transform order multiplexing |
Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101431496A (en) * | 2008-12-08 | 2009-05-13 | 哈尔滨工业大学 | Modulation-demodulation method for fraction Fourier transform domain with unequal power spectral density |
-
2009
- 2009-09-25 CN CN2009100937244A patent/CN101667911B/en not_active Expired - Fee Related
Patent Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101431496A (en) * | 2008-12-08 | 2009-05-13 | 哈尔滨工业大学 | Modulation-demodulation method for fraction Fourier transform domain with unequal power spectral density |
Non-Patent Citations (2)
| Title |
|---|
| Soo-Chang Pei, Fellow, IEEE, and Wen-Liang Hsue.The Multiple-Parameter Discrete Fractional FourierTransform.《IEEE SIGNAL PROCESSING LETTERS》.2006,第13卷(第6期),第329至332页. * |
| 孟祥意 等.分数阶傅里叶域两通道滤波器组.《电子学报》.2008,第36卷(第5期),第919至926页. * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN101667911A (en) | 2010-03-10 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN100566231C (en) | The compensation of nonlinear distortion in the modem receiver | |
| Zepernick et al. | Pseudo random signal processing: theory and application | |
| CN101986632B (en) | Correlation delay-differential chaos shift keying-based modulation communication method | |
| US6317409B1 (en) | Residue division multiplexing system and apparatus for discrete-time signals | |
| JP5456169B2 (en) | Method for transmitting pre-equalized digital data and transmitter station implementing such a method | |
| JP5200139B2 (en) | Carrier interference processing method, carrier interference receiver system and program | |
| Perrins et al. | PAM decomposition of M-ary multi-h CPM | |
| CN104254984B (en) | Wireless communications method and corresponding emitter, receiver and communication system | |
| US20140153675A1 (en) | Estimation of a Time, Phase and Frequency Shift of an OQAM Multicarrier Signal | |
| Tao et al. | Transform order division multiplexing | |
| JP2001503591A (en) | Construction of prototype signal for multicarrier transmission | |
| Bodhe et al. | Design of Simulink Model for OFDM and Comparison of FFT-OFDM and DWT-OFDM | |
| CN101667911B (en) | Secret communication method based on fraction order Fourier transform order-multiplexing | |
| CN102857284A (en) | Data transmitting method, data receiving method, data transmitting device, data receiving device and system | |
| CN103117859B (en) | A kind of asymmetrical optical image encryption method based on elliptic curve | |
| CN202906963U (en) | A frequency deviation estimating system of a coherent demodulation frequency shift keying modulating signal | |
| Beygi et al. | Bilinear matrix factorization methods for time-varying narrowband channel estimation: Exploiting sparsity and rank | |
| CN108199988A (en) | Frequency domain GFDM low complex degree least mean-square error method of reseptances and receiver | |
| CN112583574A (en) | WFRFT transmission method and system based on two-dimensional chaotic mapping amplitude phase encryption | |
| US20040170280A1 (en) | Systems, methods and computer program products for encryption and decryption using wavelet transforms | |
| CN101739660B (en) | Method for encrypting and decrypting image based on multi-order fractional Fourier transform | |
| Saideh et al. | Efficient equalization for FBMC-OQAM under doubly selective channel estimation errors | |
| CN108574574A (en) | Physical layer encryption method based on multi-dimensional constellation rotation | |
| CN101222326B (en) | Communications security and decipherment method based on four-weighted score Fourier transform | |
| WO2004095723A2 (en) | Secure nonlinear keting modulation and demodulation |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| C06 | Publication | ||
| PB01 | Publication | ||
| C10 | Entry into substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| C14 | Grant of patent or utility model | ||
| GR01 | Patent grant | ||
| C17 | Cessation of patent right | ||
| CF01 | Termination of patent right due to non-payment of annual fee |
Granted publication date: 20120523 Termination date: 20130925 |