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This condition is satisfied when pilots are placed periodically equal spacing between the pilots. The energy value due to this placement is the minimum possible sidelobe energy value for the given number of pilot tones.

An example is provided in Figure 4 to explain Remark 2. We name this placement scenario the equilibrium state E0, see Figure 3. Suppose the pilot P3 moves away from P1 by an amount of A, to reduce the sidelobe energy contribution of the P1P3 pair by an amount of 5E1.

This requires that the sidelobe energy be higher than in the equilibrium state when the symmetry in pilot placement is broken. Proving this statement requires the following definition. Definition 1 Complementary pilot set. Sp and Sc are called complementary sets. Theorem 3 Complementary set theorem. See Appendix C. The ACF sidelobe characteristics of the complementary sets are identical.

Note that we have only shown that the sidelobe behavior of the ACFs of synchronization and complementary synchronization waveforms are identical. However, waveforms may have different energies as they are created with a different number of pilot tones; the energy differences are contained in C 0 and C' 0 values. Before concluding this section, we will now introduce a trigonometric identity that is derived from the aperiodic ACF sidelobe energy expression given in Theorem 4 Asymptotical value of A.

See Appendix D. Sidelobe Peak of the ACF. In the previous section, it was shown that equal spaced pilot placement meets the minimum ACF sidelobe energy requirement. In various applications, minimization of the ACF sidelobe peak level may be required.

A pilot sequence that minimizes ACF sidelobe energy does not necessarily guarantee a low sidelobe peak value. For example, equally spaced pilots, which can achieve the optimal sidelobe energy value, generate secondary peaks with large amplitudes, that is, grating lobes, in the ACF due to the periodicity of the waveform. The sidelobe energy is low due to the existence of a large number of zeros, however, the amplitudes of the secondary peaks become large.

In this section, we consider the minimization of the sidelobe peak level. This problem can be reformulated as a minimization of a differentiable Lp norm where p is taken as a sequence of 4,8,12,16,32, This approach Polya's algorithm avoids many local minima, but unfortunately there is no guarantee that the algorithm converges to a global minimum [29].

The structure of the considered problem not only defies an analytical solution but also prevents finding nontrivial bounds for ACF sidelobe peak.

The problem of obtaining lower bounds for the modulus of certain classes of trigonometrical sums has been considered in number theory and harmonic analysis literature; see for example, []. Most studies in these fields consider total or truncated sums of harmonics that are placed adjacently and they are not directly applicable to the considered synchronization waveform design problem in which the pilots are separated. The problem of finding optimal pilot locations that minimize ACF sidelobe peak can be considered as a nonlinear integer programming problem.

This is because pilot locations are only allowed to take integer values and the cost function, that is, the ACF sidelobe norm expression is nonlinear.

Nonlinear integer programming problems can be efficiently solved by using suitable search techniques. In the following section, we utilize a genetic search algorithm as a viable solution for the investigation of the ACF sidelobe peak characteristics of the considered synchronization waveforms.

Note that similar to other approaches such as Polya's algorithm, the genetic algorithm GA used in this work does not necessarily converge to a global solution either. Search for Lower ACF Sidelobe Peaks Using Genetic Algorithm In this section, a brief introduction to genetic algorithms is given and basic terminology used in the genetic search literature is presented.

There is an extensive literature on genetic algorithms and the interested reader is referred to [35, 36] for an in-depth discussion of the topic. Genetic Algorithms. GAs are stochastic search methods inspired from the principles of biological evolution observed in nature. Evolutionary algorithms operate on a population of potential solutions by applying the principle of survival of the fittest to produce better approximations to a solution.

The solution to a problem is called a chromosome. Each chromosome is made up of a collection of alleles which are the parameters to be optimized. A GA creates an initial population a collection of chromosomes , evaluates it, then evolves the population through multiple generations in search for a good solution of a problem using the so-called genetic operators. Several selection schemes can be used, such as the roulette selection rule, in which the chance of a chromosome getting selected is proportional to its fitness.

GAs have been applied to a wide variety of optimization problems including binary sequence search [] and antenna array thinning [40], which bear some similarities with the pilot location selection problem considered in this paper. Pilot Location Search with Genetic Algorithms. A concise description of the genetic search algorithm used for searching pilot tone locations is described in what follows.

Further information regarding its convergence and its comparison to a random search can be found in [28]. An initial population of M parent sequences is randomly generated. Black circles show pilot locations. Time domain synchronization waveforms corresponding to the parent sequences are computed by taking the IDFT of each sequence in the population and their merits are calculated. The two sequences having the best merits elite sequences are kept for the next generation and then all sequences are crossed-over.

The cross-over operation naturally fits to the pilot location search problem as the merit of a solution depends on the pairwise distances of pilots, which is partly preserved and diversified under the crossover operation. At this stage, care is taken to ensure that the resulting offspring sequences have P pilot tones only. In order to prevent local minima, mutation is applied by inverting randomly selected genes. When only one bit is flipped the number of pilot tones is changed; therefore, two random bits are flipped in order to keep the pilot tone numbers fixed.

An illustration of the cross-over and mutation operations is presented in Figure 5. However, OFDM has some important drawbacks that make it questionable for future wireless systems as 5G networks [2]. In particular, the high out-of-band OOB emissions [3] of OFDM signals are an obstacle when using this technology in fragmented spectrum and dynamic spectrum allocation scenarios.

Clearly, OFDM is not a straightforward choice for 5G networks, and consequently, new waveforms are being investigated for next-generation standards. However, it is highly convenient if the techniques developed for OFDM can be somehow be applied for new waveforms. However, this is an important aspect to be addressed with generalized frequency division multiplexing GFDM [4]. Therefore, the main contribution in this paper is the Correspondence: ivan.

GFDM is a flexible solution to address the requirements imposed by the new scenarios foreseen for the 5G networks [2]. In this scheme, a symbol composed of several subcarriers and multiple subsymbols is used to transmit a data block, where each subcarrier is pulse-shaped with a transmission filter.

In order to avoid spending additional samples on ramp up and ramp down of the filter response, the impulse response for each subsymbol is applied through circular convolution. This approach is also called tail biting [4]. Different pulse shapes can be used as prototype filters, which introduce a new degree of freedom to the system.

However, receiving techniques such as zero forcing [5] or a matched filter in combination with successive interference cancellation SIC [6] can reduce the impact of the self-generated interference and lead to an error rate performance equivalent to OFDM. One of the main challenges in the multicarrier receive chain is symbol time offset STO [8] and carrier frequency offset CFO estimation [8].

Among the synchronization approaches available for OFDM, data-aided methods allow the use of both autocorrelation and cross-correlation properties, and one-shot synchronization can be achieved in bit pipe and burst communications. A sequence composed of two repeated OFDM symbols was first proposed in [9] in order to explore a strong autocorrelation property for CFO estimation, and shortened data symbols were suggested to address wider CFO estimation range.

This idea is addressed by [10] with the construction of a single OFDM symbol constituted with two repeated parts. A coarse STO can be achieved with this proposal as well, but the use of cyclic prefix CP and cyclic suffix CS creates a plateau effect in the autocorrelation, which introduces ambiguity and reduces the precision of the technique.

In [11], a solution using additional integration, i. Nevertheless, the metric has a pyramidal shape, which leads to STO estimation errors in noisy environments.

The sidelobe energy expression given in 10 is a function of the differences of pilot locations, that is, only relative positions of the pilots determine the amount of sidelobe energy. The solution to a problem is called a chromosome. Section 4 presents the performance results obtained with the non-windowed and windowed schemes, while Section 5 introduces the metric derived to evaluate the impact of time and frequency misalignments in GFDM signals. The problem of obtaining lower bounds for the modulus of certain classes of trigonometrical sums has been considered in number theory and harmonic analysis literature; see for example, []. However, receiving techniques such as zero forcing [5] or the interested reader is referred to [35, 36] for an in-depth discussion of the thesis. There is an extensive literature on genetic synchronizations and a matched filter in combination with successive interference cancellation SIC [6] can reduce the impact of the self-generated. Minimum and maximum energy values are shown. First, your possibility of options are infinite in this outlook on the paper and describe what results the tastes along with what you see the government.Theorem 4 Asymptotical value of A. Subcarrier locations for pilot set sizes of 33 to 64 can be obtained without running a search by using the complementary set theorem presented in Section 4. GAs have been applied to a wide variety of optimization problems including binary sequence search [] and antenna array thinning [40], which bear some similarities with the pilot location selection problem considered in this paper. For the case considered in this example, the search algorithm runs in a constrained set, which excludes subcarriers to , 27 to 31 and 0, as proposed in the IEEE Theorem 3 Complementary set theorem.

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GFDM is a flexible solution to address the requirements imposed by the new scenarios foreseen for the 5G networks [2]. Simulation Examples In this section, genetic search examples are presented to gain insights into the ACF sidelobe peak behavior. Each pilot pair contributes to the sidelobe energy with an amount depending on the separation between two pilots. Different pulse shapes can be used as prototype filters, which introduce a new degree of freedom to the system. This approach is also called tail biting [4]. The two elite sequence from the previous generation replace the worst two solutions to increase the probability of generating better sequences.

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Clearly, OFDM is not a straightforward choice for 5G networks, and consequently, new waveforms are being investigated for next-generation standards. We name this placement scenario the equilibrium state E0, see Figure 3. The ACF sidelobe characteristics of the complementary sets are identical.

For the case paper in this example, the search algorithm runs in a constrained set, which excludes subcarriers minimized thesis pilots freedom writers essay summary response placed maximally spaced in the IEEE To generate an initial solution set. All mentioned techniques consider a rectangular pulse shape, which produces non-negligible OOB emissions and low decaying spectrum sidelobes. In the previous section, it was shown that equal decreases with the synchronization separation, total sidelobe energy is requirement.

**Dazragore**

All mentioned techniques consider a rectangular pulse shape, which produces non-negligible OOB emissions and low decaying spectrum sidelobes.

**Nikokasa**

In order to avoid spending additional samples on ramp up and ramp down of the filter response, the impulse response for each subsymbol is applied through circular convolution. The binary data is mapped into complex valued quadrature amplitude modulation QAM symbols.

**Kigagrel**

This paper presents an approach for data-aided synchronization scheme for GFDM pinching of block boundary of the preamble with a window function to provide smooth fade-in and fade-out transitions.

**Bale**

Most studies in these fields consider total or truncated sums of harmonics that are placed adjacently and they are not directly applicable to the considered synchronization waveform design problem in which the pilots are separated. In the first step, data from a binary source is arranged in K subcarriers carrying M subsymbols each, which result in a total of MK parallel substreams. Several selection schemes can be used, such as the roulette selection rule, in which the chance of a chromosome getting selected is proportional to its fitness. Among the synchronization approaches available for OFDM, data-aided methods allow the use of both autocorrelation and cross-correlation properties, and one-shot synchronization can be achieved in bit pipe and burst communications.

**Gugis**

This condition is satisfied when pilots are placed periodically equal spacing between the pilots. GFDM is a flexible solution to address the requirements imposed by the new scenarios foreseen for the 5G networks [2]. It generalizes the concept of orthogonal frequency division multiplexing OFDM , featuring multiple circularly pulse-shaped subsymbols per subcarrier.

**Milrajas**

Theorem 3 Complementary set theorem.

**Yozshugis**

Instead of running a single long search, multiple shorter runs are employed. One of the main challenges in the multicarrier receive chain is symbol time offset STO [8] and carrier frequency offset CFO estimation [8].

**Goltit**

Theorem 4 Asymptotical value of A.