Chaotic Signals in Digital Communications


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Skickas inom vardagar specialorder. Chaotic Signals in Digital Communications combines fundamental background knowledge with state-of-the-art methods for using chaotic signals and systems in digital communications. The book builds a bridge between theoretical works and practical implementation to help researchers attain consistent performance in realistic environments.

It shows the possible shortcomings of the chaos-based communication systems proposed in the literature, particularly when they are subjected to non-ideal conditions. It also presents a toolbox of techniques for researchers working to actually implement such systems. A Combination of Tutorials and In-Depth, Cutting-Edge ResearchFeaturing contributions by active leading researchers, the book begins with an introduction to communication theory, dynamical systems, and chaotic communications suitable for those new to the field.

This lays a solid foundation for the more applied chapters that follow. N are mapped to a weighted number system representation by determining a series of digits in the weighted number system based on the RNS solutions No. For example, a digit can be a particular bit of a binary sequence. According to another aspect of the invention, the RNS solutions No. N are mapped to a weighted number system representation by identifying a number in the weighted number system that is defined by the RNS solutions No.

According to yet another aspect of the invention, the RNS solutions No. N are mapped to a weighted number system representation by identifying a truncated portion of a number in the weighted number system that is defined by the RNS solutions No. The truncated portion can include any serially arranged set of digits of the number in the weighted number system. The truncated portion can also be exclusive of a most significant digit of the number in the weighted number system. The truncated portion can also be a segment including a defined number of digits extracted from a chaotic sequence.

The truncated portion can further be a result of a partial mapping of the RNS solutions No. N to a weighted number system representation. According to an embodiment of the invention, a mixed-radix conversion method is used for mapping RNS solutions No. To be consistent with said reference, the following discussion of mixed radix conversion utilizes one 1 based variable indexing instead of zero 0 based indexing used elsewhere herein. A set of moduli are also chosen so that a mixed-radix system and a RNS are said to be associated. The mixed-radix conversion process described here may then be used to convert from the [RNS] to the mixed-radix system.

The a i are determined sequentially in the following manner, starting with a 1. Hence, a 1 is just the first residue digit. Furthermore, m 1 is relatively prime to all other moduli, by definition. Hence, the division remainder zero procedure [Division where the dividend is known to be an integer multiple of the divisor and the divisor is known to be relatively prime to M] can be used to find the residue digits of order 2 through N of.

Inspection of. In this way, by successive subtracting and dividing in residue notation, all of the mixed-radix digits may be obtained. From the preceding description it is seen that the mixed-radix conversion process is iterative. The conversion can be modified to yield a truncated result. The CRT arithmetic operation can be defined by a mathematical equation 6 [returning to zero 0 based indexing]. The b j 's enable an isomorphic mapping between an RNS N-tuple value representing a weighted number and the weighted number. However without loss of chaotic properties, the mapping need only be unique and isomorphic.

As such, a weighted number x can map into a tuple y. The tuple y can map into a weighted number z. Thus for certain embodiments of the present invention, all b j 's can be set equal to one or more non-zero values without loss of the chaotic properties. As such, the chaotic sequence output can be represented as a binary sequence. Each bit of the binary sequence has a zero 0 value or a one 1 value. The chaotic sequence output can have a maximum bit length MBL defined by a mathematical equation 7.

In this regard, it should be appreciated that M represents a dynamic range of a CRT arithmetic operation. It should also be appreciated that the CRT arithmetic operation generates a chaotic numerical sequence with a periodicity equal to the inverse of the dynamic range M. The dynamic range requires a Ceiling[Log 2 M ] bit precision.


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According to an embodiment of the invention, M equals three quadrillion five hundred sixty-three trillion seven hundred sixty-two billion one hundred ninety-one million fifty-nine thousand five hundred twenty-three 3,,,,, As such, the chaotic sequence output is a fifty-two 52 bit binary sequence having an integer value between zero 0 and three quadrillion five hundred sixty-three trillion seven hundred sixty-two billion one hundred ninety-one million fifty-nine thousand five hundred twenty-two 3,,,,, , inclusive. In such a scenario, the chaotic sequence output can have a bit length less than Ceiling[Log 2 M ].

It should be noted that while truncation affects the dynamic range of the system it has no effect on the periodicity of a generated sequence. As should be appreciated, the above-described chaotic sequence generation can be iteratively performed. In such a scenario, a feedback mechanism e. In a first iteration, n equals one 1 and x is selected as two 2 which is allowable in a residue ring.

By substituting the value of n and x into the stated polynomial equation f x nT , a first solution having a value forty-six 46 is obtained. In a second iteration, n is incremented by one and x equals the value of the first solution, i. In a third iteration, n is again incremented by one and x equals the value of the second solution. In step , a plurality of polynomial equations f 0 x nT ,. After step , step is performed where a determination for each polynomial equation f 0 x nT ,.

In step , a modulus is selected for each polynomial equation f 0 x nT ,. The modulus is selected from the moduli identified in step It should also be appreciated that a different modulus must be selected for each polynomial equation f 0 x nT ,. In step , a constant C m is selected for each polynomial equation f 0 x nT ,. Each constant C m corresponds to the modulus selected for the respective polynomial equation f 0 x nT ,. Each constant Cm is selected from among the possible constant values identified in step for generating an irreducible form of the respective polynomial equation f 0 x nT ,.

After step , method continues with step In step , a value for time increment T is selected. Thereafter, an initial value for the variable x of the polynomial equations is selected. The initial value for the variable x can be any value allowable in a residue ring. Notably, the initial value of the variable x defines a sequence starting location. As such, the initial value of the variable x can define a static offset of a chaotic sequence.

In step , RNS arithmetic operations are used to iteratively determine RNS solutions for each of the stated polynomial equations f 0 x nT ,. In step , a series of digits in a weighted number system are determined based in the RNS solutions. Step can involve performing a mixed radix arithmetic operation or a CRT arithmetic operation using the RNS solutions to obtain a chaotic sequence output. After completing step , method continues with a decision step Subsequently, method returns to step If the chaos generator is terminated :YES , then step is performed where method ends.

Chaos generators 1 ,. As such, the following discussion of chaos generator is sufficient for understanding chaos generators 1 ,. Accordingly, chaos generator is comprised of computing processors 0 ,. Each computing processor 0 ,. As such, each computing processor 0 ,. Mapping processor can be coupled to an external device not shown via a data bus The external device not shown includes, but is not limited to, a communications device configured to combine or modify a signal in accordance with a chaotic sequence output.

The constant value can be selected so that a polynomial equation f 0 x nT ,. Each of the solutions can be expressed as a unique residue number system RNS N-tuple representation. In this regard, it should be appreciated that the computing processors 0 ,. Each of the computing processors 0 ,. The computing processors 0 ,. In this regard, it should be appreciated that the feedback mechanisms 0 ,.

Accordingly, the feedback mechanisms 0 ,. In this regard, the computing processors 0 ,. Such RNS-to-binary conversion methods are generally known to persons having ordinary skill in the art, and therefore will not be described herein. However, it should be appreciated that any such RNS-to-binary conversion method can be used without limitation. It should also be appreciated that the residue values expressed in binary number system representations are hereinafter referred to as moduli solutions No. N comprising the elements of an RNS N-tuple. According to an embodiment of the invention, computing processors 0 ,.

The table address is used to initiate the chaotic sequence at the start of an iteration.

The result is a series of digits in the weighted number system based on the moduli solutions No. P is a fewest number of bits required to achieve a binary representation of the weighted numbers. In this regard, it should be appreciated that mapping processor can employ a weighted-to-binary conversion method.

Chaotic Signals in Digital Communications - CRC Press Book

Weighted-to-binary conversion methods are generally known to persons having ordinary skill in the art, and therefore will not be described herein. However, it should be appreciated that any such weighted-to-binary conversion method can be used without limitation. RNSs are well known to those having ordinary skill in the art, and therefore will not be described herein. A different value for at least one of the listed parameters can be changed during each of two or more timeslots of a TDM frame. The different value causes causing a cyclic shift in a spreading sequence or a change from a first spreading code to a second spreading code.

All of the apparatus, methods, and algorithms disclosed and claimed herein can be made and executed without undue experimentation in light of the present disclosure. While the invention has been described in terms of preferred embodiments, it will be apparent to those having ordinary skill in the art that variations may be applied to the apparatus, methods and sequence of steps of the method without departing from the concept, spirit and scope of the invention.

More specifically, it will be apparent that certain components may be added to, combined with, or substituted for the components described herein while the same or similar results would be achieved. All such similar substitutes and modifications apparent to those having ordinary skill in the art are deemed to be within the spirit, scope and concept of the invention as defined. Effective date : Year of fee payment : 4. Systems and methods for selectively controlling access to data streams communicated from a first communication device FCD using a timeslotted shared frequency spectrum and shared spreading codes.

Protected data signals 1 ,. The first modulated signals are combined with first chaotic spreading codes to form digital chaotic signals. The digital chaotic signals are additively combined to form a protected data communication signal PDCS. Statement of the Technical Field The invention concerns communication systems. Description of the Related Art Multiple access communication systems permit multiple users to re-use a portion of a shared transmission spectrum for simultaneous communications.

Acquisition Mode: In acquisition mode, the re-sampling filters , 1 ,. Steady State Demodulation Mode: In steady state demodulation mode, the correlator tracks the correlation between the received modulated signal and the locally generated chaotic sequences close to the nominal correlation peak to generate magnitude and phase information as a function of time.

CHAOS-BASED SECURE COMMUNICATION SYSTEM

We claim: 1. A method for selectively controlling access to multiple data streams which are communicated from a first communication device using a timeslotted shared frequency spectrum and shared spreading codes, comprising the steps of: performing discrete-time modulation processes using at least two protected data signals including protected data to form at least two first modulated signals;. The method according to claim 1 , further comprising selecting each of said first chaotic spreading codes to be a chaotic spreading sequence generated using a plurality of polynomial equations and modulo operations.

The method according to claim 1 , wherein each of the discrete-time modulation processes is selected from the group comprising an M-ary phase shift keying modulation process, a quadrature amplitude modulation process and an amplitude shift keying modulation process. The method according to claim 3 , wherein the second modulated signal is formed using an amplitude-and-time-discrete modulation process. The method according to claim 1 , further comprising the steps of: modulating a global data signal to form a second modulated signal; and. The method according to claim 1 , wherein the output communication signal is transmitted from the first communication device to a second communication device having at least one key to recover all of the protected data and the global data transmitted during two or more timeslots of said TDM frame.

The method according to claim 1 , wherein the output communication signal is transmitted from the first communication device to a second communication device having at least one key to recover the global data and a portion of the protected data transmitted during two or more timeslots of said TDM frame. The method according to claim 1 , wherein the output communication signal is transmitted from the first communication device to a second communication device having at least one key to recover only the global data transmitted during two or more timeslots of said TDM frame. The method according to claim 1 , wherein at least a portion of the composite protected data communication signal is transmitted in a first timeslot of said TDM frame and at least a portion of the global data communication signal is transmitted in a second timeslot different from the first timeslot of the TDM frame.

The method according to claim 1 , wherein at least a portion of the composite protected data communication signal and at least a portion of the global data communication signal are transmitted in the same timeslot of said TDM frame. A communication system configured for selectively controlling access to multiple data streams which are communicated using a timeslotted shared frequency spectrum and shared spreading codes, comprising: a first modulator configured to perform discrete-time modulation processes using at least two protected data signals including protected data to form at least two first modulated signals;.

The communication system according to claim 13 , further comprising at least one generator configured to generate each of said first chaotic spreading codes using a plurality of polynomial equations and modulo operations. The communication system according to claim 13 , further comprising: a second modulator configured to modulate a global data signal to form a second modulated signal; and.

The communication system according to claim 15 , wherein the second modulated signal is formed using an amplitude-and-time-discrete modulation process. The communication system according to claim 13 , wherein the second communication device has at least one key to recover all of the protected data and the global data transmitted during two or more timeslots of said TDM frame.

The communication system according to claim 13 , wherein the second communication device has at least one key to recover the global data and a portion of the protected data transmitted during two or more timeslots of said TDM frame. The communication system according to claim 13 , wherein the second communication device having at least one key to recover only the global data transmitted during two or more timeslots of said TDM frame. The communication system according to claim 13 , wherein at least a portion of the composite protected data communication signal is transmitted in a first timeslot of said TDM frame and at least a portion of the global data communication signal is transmitted in a second timeslot different from the first timeslot of the TDM frame.

The communication system according to claim 13 , wherein at least a portion of the composite protected data communication signal and at least a portion of the global data communication signal are transmitted in the same timeslot of said TDM frame. USB2 en. System and method for detecting and processing received signal with pulse sequence. Method and apparatus for communicating data in a digital chaos communication system. USB1 en. Method and apparatus for communicating data in a digital chaos cooperative network.

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Direct sequence spread spectrum transmission process, with generation and optimization of sequences. Apparatus for providing improved encryption protection in a communication system. Secret key cryptosystem and method utilizing factorizations of permutation groups of arbitrary order 2l. Method and apparatus for performing arithmetic operations on Galois fields and their extensions. Chaotic dynamics based apparatus and method for tracking through dropouts in symbolic dynamics digital communication signals.

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Communication by chaotic signals: the inverse system approach

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Transmitting and receiving apparatuses for reducing a peak-to-average power ratio and an adaptive peak-to-average power ratio controlling method thereof. System and method for variable rate multiple access short message communications. System for using rapid acquisition spreading codes for spread-spectrum communications.

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Performance evaluation of chaotic spreading sequences on software-defined radio

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Knuth, D. Kolumban, et al. Lai, X. The source emits M samples for each information bit in addition to the initial reference signal. Thus, the average bit energy transmitted can be found as. Information decoding is performed by correlating each incoming signal ri with its delayed version and the correlation product is averaged over M.

Signal energy estimation can be obtained only by taking the mean value of first term in 7. Ideally, this will be either zero or V ar x , all other terms are the zero mean. However, the distance between signal elements average value of the correlator for each transmitted bit is half compared to that in DCSK. Despite all that, the information can be decoded according to the following rule. For all the systems, it can be clearly observed that BER performance is decreased by increment of spreading factor M.

This is due to the nonlinear contribution of the last term in 3 , 7 , 10 , and 17 with respect to other terms which exhibit linear contribution with respect to M. This is due to two fundamental reasons: 1 number of cross terms in CDSK correlator is more than in DCSK and 2 incomplete orthogonality between intra-signal terms [ 1 , 6 , 9 ], which can affect the correlator output negatively.

The fact behind this is the reduction in average bit from 2 MVar x to 3 M 2 Var x which result in improvement by 1. However, this improvement is vanished due to signal to signal contribution. Clearly, there is an acceptable matching between theoretical expression and simulated version. However, these expressions are derived based on GA approximation method, which is suitable for the system operating in large spreading factor. To have more accurate derivation, it is preferred to implement integration method [ 15 ]. Single reference segment is used as a reference to modulate and demodulate multiple successive bits in Ref.

Average bit energy is reduced with bit error rate enhancement. However, the system is not suitable for secure communications due to easy spectrum prediction in addition to the need for multiple delay elements in both transmitter and receiver which increase the system complexity. Chaotic signals have fluctuated energy due to randomness nature of the signal. Permutation between chaotic samples is implemented to destroy the similarity between the reference signal and information signal in DCSK. Sending both reference and information bearing signal in separate time slot causes a reduction in bandwidth efficiency of differential coherent systems such as DCSK.

Hence, many systems have been designed to combine both reference signal and information bearing signal in one time slot. System is based on using Walsh code to combine reference signal and information bearing signal in single time slot rather than sending them separately. Another scheme which is based on mapping series of bits into two channels and each encoded output is consider as an initial condition value for the sequence generator pairs and their outputs are added and up converted [ 20 ]. Implementation of delay diversity scheme as a basic building block for space time block coder STBC is suggested in ref.

Here, bits stream is converted from series to parallel; an each bit in parallel channel is modulated by DCSK modulators and followed by analogue space time block coder STBC. Efficiency of multicarrier modulation has been used to send multiple bits of modulating each information bits with subcarrier using multicarrier modulation-DCSK. The system provides a considerable saving in bandwidth [ 22 ]. However, the cost which needs to pay is the complexity of having multiple carrier multipliers in the transmitter side and bank of matched filter on the receiver side.

Transmitting reference signal followed by information bearing signal is the common signal format for most of the differential coherent spread spectrum systems which can be affected by fast fading channel. A suggested scheme to send only one sample form reference signal followed directly by one sample from information bearing signal is analyzed and tested in Ref.

The system provides immunity against fading in continuous mobility environment. System block diagram is almost similar to standard DCSK except for switching timing. Major drawback of DCSK system is the addition of channel random noise in both signal segments reference and information bearing signal. Therefore, a noise reduction technique has been introduced to reduce the noise variance by sending a repeated subsegment of samples inside one bit duration rather than sending continuous stream of samples.

At the receiver, averaging operation is performed over the repeated segment before the standard correlation procedure [ 24 ]. This enhances the BER performance over other newly developed segments. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. Edited by Kais A. Al Naimee. Edited by Jan Awrejcewicz. We are IntechOpen, the world's leading publisher of Open Access books.

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