In the thesis we discuss frequency hopping multiple access (FHMA) systems and construction of optimal frequency hopping sequence and applications. For these systems, it is desirable to use frequency-hopping sequences (FHSs) with low Hamming correlation to reduce the multiple-access interference. In general, optimal FHSs with respect to the Lempel–Greenberger bound do not always exist for all lengths and frequency set sizes.
Therefore, it is an important problem to verify whether an optimal FHS with respect to the Lempel-Greenberger bound exists or not for a given length and a given frequency set size. I constructed FHS that optimally satisfies with respect to the Lempel-Greenberger bound and Peng-Fan bound for efficiency of available frequency. In the construction and theorem, I used these mathematical backgrounds in preliminary (i.e. finite field, primitive element, primitive polynomial, frequency hopping sequence, multiple frequency shift keying, DS/CDMA) to prove mathematically.
In addition, a new method is presented to increase the data rate or the number of users. Using this method, sequences are split into two times of length and satisfy Lempel-Greenberger bound and Peng-Fan bound.
Introduction
For these properties we use a finite field or Galois field that contains a finite number of elements. There exists a finite field of order q if and only if the order q is a prime number (where p is a prime number and k is a positive integer). In a field of order, adding p copies of an element always results in zero; that is, the attribute of the field is p.
In a finite field of order q, the polynomial − has all q elements in the finite field as roots. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. The coefficient of the highest degree term in the polynomial must be 1, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.
Moreover, we choose a primitive polynomial is the minimal polynomial of a primitive element of the field of finite extension. In the case of FHS, we use an FHS for the frequency diversity and low correlation of the set of sequences.
Preliminaries
- Integral domain and Euclidean Domains
- Properties of Finite Fields
- Primitive Polynomial in Finite Field
- Frequency Hopping Systems
- Multiple frequency-shift keying
- Properties of Frequency Hopping
- DS/CDMA
Lagrange's theorem shows that the number of elements in a subgroup is always a divisor of the number of elements in. An element of multiplicative order − 1, i.e. a generator of the cyclic group F∗= F − {0}, is called a primitive root of the field. The minimal polynomial of a value α is the polynomial of lowest degree with coefficients of a specified type such that it is a root of the polynomial.
It is used as a multiple access method in the code division multiple access (CDMA) scheme, frequency hopping code division multiple access (FH-CDMA). Adaptive Frequency-hopping spread spectrum (AFH) (as used in Bluetooth) improves resistance to radio frequency interference by avoiding crowded frequencies in the hopping range. The position of the M-ary signal set is mapped pseudo-randomly over a hopping bandwidth by the LFSR. A typical FH/MFSK system block diagram is shown in Figure 2.5.
In the case of the FH/MFSK system, the data symbol modulates the carrier frequency, which is pseudo-randomly determined by the LFSR. We indicate the bandwidth of the frequency jump and the smallest frequency gap between each jump position. From Figure 2.5, we know that the receiver is the reverse side of the transmitter's signal processing.
In Figure 2.7 we extend the example from Figure 2.6 with the additional property of a chip repetition factor of N = 4. At the top of the figure we see the same data series, with = 150 bps, as in the earlier example; and we see the same 3-bit partition to form the 8-ary symbols. As in the previous example, Figure 2.7 illustrates that the center of the data band is shifted with each new chip time.
The position of the solid line is in the same relationship to the dashed line for each of the tokens associated with the given symbol. In the case of direct sequence and spread spectrum systems, the term "chip" refers to the PN (pseudorandom number) code symbol (shortest duration symbol in the DS system). For SFH, the shortest continuous waveform in the system is a data symbol; however, for FFH, the shortest continuous waveform is a spike.
As in Figure 2.7, the dashed line in each column corresponds to the center of the data band, and the solid line to the symbol frequency. The transmission signal occupies a bandwidth equal to the spreading factor multiplied by the bandwidth of the user data.
Frequency Hopping Sequence
- Sequence with minimal polynomial over finite field
- A fast algorithm for complexity of binary sequence with period 2
- Correlation and Bound
- Optimal Frequency-Hopping Sequences
- BlueBee (BLE and Zigbee)
- BlueBee design
- GFSK and OQPSK
In case it is a power of 2, we can know the complexity of LFSR with a number. In the case of a power of 2, the complexity can be determined in steps using the much simpler algorithm presented in this correspondence. For a given sequence of arbitrary period N, Massey's algorithm [43] sequentially takes the sequence and at each stage computes the connecting polynomial for the shortest LFSR that generates the found part of the sequence.
The Massey algorithm may need to run through more than one length period of the sequence before it stabilizes on the correct connection polynomial. In practice, additional iterations are required to ensure that the algorithm has actually stabilized. The algorithm given in this correspondence works only for a sequence of length period = 2 and computes the complexity in = steps.
The storage requirements of the Massey algorithm depend directly on the ultimate complexity of the sequence, whereas the current algorithm must always store a single period of the sequence, making the algorithm inappropriate for very long periods. To minimize such interference, it is necessary to design an FHS set with layers ( ) and . We call , an FHS set, optimal if ( ) reaches one of the bounds in Lemma 3.5, that is, is optimal with respect to the Peng-Fan bound.
The above nested construction is based on the information that every integer x in ℤ can be uniquely represented as ( ) and ( ) when and are relatively prime. On the other hand, the following nested construction can be applied even when and are not relatively first. In the following propositions and conclusions, we calculate the Hamming correlation between two FHSs of FHSs located in construction A1 or A2.
Cross-Technology Communication is a promising solution recently proposed to the coexistence problem of heterogeneous wireless technologies in the ISM bands. BLE Transmitter: BLE uses Gaussian Frequency Shift Keying (GFSK) modulation, which is normally achieved by phase shifting over time. ZigBee Receiver: BlueBee enables BLE to send emulated ZigBee packets that can be demodulated by any standard ZigBee device via the standard Offset Quadrature Phase Shift Keying (OQPSK) demodulation procedure.
BlueBee enables BLE to transmit emulated ZigBee packets, which can be demodulated by any commodity ZigBee device through the standard phase shift quadrature keying (OQPSK) demodulation procedure. Due to the difference of these sampling frequencies, the signal '1' is translated to '11' in ZigBee.
One-Coincidence Frequency Hopping Sequence
- Definition of OC-FHS
- New Construction of OC-FHS Sets
- Calculation of Hamming Correlation Values
- Separating of Sequence
If we have a good SNR, we can reduce the period and increase the number of sequences. Sarwate, "Reed-Solomon codes and the design of sequences for spread spectrum multiple-access communication," Reed-Solomon codes and their applications, S. Siddiqi, "Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings," IEEE Trans.
Titlebaum, “Time-frequency hopping signals Part I: Coding based on the theory of linear congruences,” IEEE Trans. Fukshansky, "A new family of one-coincidence sets of sequences with distributed elements for frequency hopping cdma systems", Jan Polydoros, "Coherent detection of frequency-hopping quadrature modulations in the presence of jamming-Part I: QPSK and QASK; Part II : QPR class I modulation,” IEEE Trans.Commun., vol.
Key, “An analysis of the structure and complexity of nonlinear binary sequence generators,” IEEE Trans. Gong, “New designs for signal sets with low cross-correlation, balance properties, and large linear span: ( ) case,” IEEE Trans. Gong, “A new class of sequences with zero or low correlation zone based on interleaving technique,” IEEE Trans.
