Symbol Frequency Research Articles

Implementation of preamble based GFDM prototype for robust 5G systems

AbstractGeneralized frequency division multiplexing (GFDM) is a flexible block‐structured multi‐carrier scheme recently proposed for next‐generation wireless communication systems. There are various approaches suggested for its analysis and implementation via simulations but testing in real‐time environments is not heavily investigated. This paper carries out the real‐time implementation of the GFDM system utilizing software‐defined radio (SDR) by emphasizing mainly channel estimation and synchronization. Symbol timing, frequency offset, and channel estimate algorithms are applied using a windowed preamble with two identical halves to satisfy low egress noise requirements. Time and frequency estimation is evaluated in terms of residual offsets along with symbol error rate over frequency selective channels. This algorithm is extended to a preamble composed of multiple identical parts. This facilitates a large frequency estimation range at the cost of complexity. For practical validation of the above concepts, the National Instruments (NI) universal software radio peripheral (USRP) 2953R is employed as hardware and it is interfaced with LabVIEW.

Sparcl: A language for partially invertible computation

Abstract Invertibility is a fundamental concept in computer science, with various manifestations in software development (serializer/deserializer, parser/printer, redo/undo, compressor/decompressor, and so on). Full invertibility necessarily requires bijectivity, but the direct approach of composing bijective functions to develop invertible programs is too restrictive to be useful. In this paper, we take a different approach by focusing on partially invertible functions—functions that become invertible if some of their arguments are fixed. The simplest example of such is addition, which becomes invertible when fixing one of the operands. More involved examples include entropy-based compression methods (e.g., Huffman coding), which carry the occurrence frequency of input symbols (in certain formats such as Huffman tree), and fixing this frequency information makes the compression methods invertible. We develop a language Sparcl for programming such functions in a natural way, where partial invertibility is the norm and bijectivity is a special case, hence gaining significant expressiveness without compromising correctness. The challenge in designing such a language is to allow ordinary programming (the “partially” part) to interact with the invertible part freely, and yet guarantee invertibility by construction. The language Sparcl is linear-typed and has a type constructor to distinguish data that are subject to invertible computation and those that are not. We present the syntax, type system, and semantics of the language and prove that Sparcl correctly guarantees invertibility for its programs. We demonstrate the expressiveness of Sparcl with examples including tree rebuilding from preorder and inorder traversals, Huffman coding, arithmetic coding, and LZ77 compression.

Combined intra symbol frequency domain averaging channel estimation for polarization division multiplexed coherent optical OFDM/OQAM systems

Intrinsic imaginary interference (IMI) induced by chromatic dispersion (CD) and polarization mode dispersion (PMD) seriously degrades the performance of optical orthogonal frequency division multiplexing offset-quadrature amplitude modulation (O-OFDM/OQAM) systems. Therefore, effective channel estimations are required for such systems to maintain good transmission performance. In the previous publications, both the full loaded (FL) and the half loaded (HL) frequency-domain channel estimation methods for polarization-division-multiplexed (PDM) coherent optical OFDM/OQAM (CO-OFDM/OQAM) systems have been proposed. However, for these frequency-domain methods, the channel frequency-domain responses caused by the correlation of the amplified spontaneous emission (ASE) noise are not considered. In this paper, we systematically discuss the frequency-domain channel transmission model for PDM CO-OFDM/OQAM systems. With the analysis of the correlation of the ASE noise, we propose a combined intra symbol frequency-domain averaging (CISFDA) channel estimation method for PDM OFDM/OQAM systems. Compared with the full-loaded frequency-domain channel estimation (FL-FD) method, the CISFDA method promotes the system robustness against the ASE noise and fiber nonlinear effect significantly. The proposed method is validated by numerical Monte Carlo simulations of the PDM CO-OFDM/OQAM system. Our new algorithm outperforms the FL-FD method in the 36 Gb/s PDM OFDM CO-OFDM/OQAM system, and the transmission distance is improved by 500 km with the bit-error-rate (BER) target at 3.8×10-3.

Precision Mining of Gene-Disease Associations via Frequent Itemset Analysis and Bioinformatics Integration

Biomedical text mining involves the extraction of relevant information from biomedical datasets. It plays a crucial role in genetic research, especially in the development of new drugs where understanding the relationships between genes and diseases is vital. This study introduces a method for generating sets of candidate genes associated with diseases, employing frequent itemset mining for analysis. Genes are ranked based on parameters such as maximum frequent itemset size and gene symbol frequency. This approach aims for precision and efficiency compared to traditional laboratory-based methods, providing highly accurate associations and uncovering novel relationships. Unlike time-consuming laboratory methods, our proposed approach leverages data from the NCBI (National Centre for Biotechnology Information database )via Entrez and utilizes bioinformatics tools like blast for indirect gene associations. Genes exhibiting single nucleotide polymorphisms are identified as indirect genes. The outcomes of this research are anticipated to contribute significantly to biomedical research by offering precise and valuable associations, thereby advancing our understanding of gene-disease relationships..

Blind estimation of modulation parameters for PCMA signals using frame cyclic features

Blind receiver technologies for paired carrier multiple access (PCMA) signals have always been a challenging task with many technical difficulties, among which the estimation of modulation parameters is a fundamental but important element. Despite some achievements in previous studies, more systematic and sophisticated estimation methods have not been adequately investigated. In this paper, schemes for the blind estimation of the symbol timing phase, amplitude attenuation, frequency offset, and carrier phase for PCMA signals in satellite communications are proposed. The data flow transmitted in satellite communication often has a certain frame structure, the most important of which is the synchronization data, namely the so-called cycle features. The proposed schemes assume that the modulated signals have fixed frame length and frame sync code and that the symbol rate has been estimated when the signals are encoded asynchronously. Distinct from the previous methods, our schemes exploit the sync waveform and the overlapping waveform, which are estimated via singular value decomposition (SVD) (using the frame cyclic features) and interference cancelation, together with their demodulation results as aid data, for the estimation of the desired parameters. The simulation results demonstrate that the schemes are effective in the parameters estimation of PCMA signals and outperform the comparison algorithms.

Adaptive DLMS equalizer based on modern FPGA

A simple and effective way to implement adaptive equalization schemes in digital communication systems is to implement them on general-purpose microcontrollers. However, increasing the order of the equalizer, which leads to increase the time to obtain the output sample, at the same time leads to decreasing maximum allowable symbol rate. It makes the implementation based on microcontrollers impractical for use in modern high-speed communication protocols. A more suitable computing platform for implementing LMS algorithms is FPGA, where there is no dependence of output sample obtaining time on equalizer order. In order to derive the error signal at the equalizer output from the last (actual) output sample, the estimation of the error signal in the FPGA is required to occur during the same clock period as the calculation of the output signal sample. Trying to fit all the necessary operations into one clock period leads to decrease maximum allowable clock frequency and inefficient using of computing resources within the FPGA. The solving of this problem is to use the Delayed LMS (DLMS) algorithm, invented in the late 80s. The essence of DLMS algorithm is to pipeline both the blocks for calculating the equalizer coefficients and the output samples. The meaning of pipelining is to install synchronous delay elements (D-flip-flops) between the elements of combinatorial logic, which makes it possible to increase the clock and symbol frequency of the equalized signal. The disadvantage of this method is the presence of the synchronous delay of several clock cycles and, as consequence, using of irrelevant error signal for updating coefficients. Retiming procedure is used to optimally distribute synchronous delays to maximize system clock frequency. This article describes the DLMS equalizer and shows the effectiveness of DLMS algorithms implementation on FPGA. The effectiveness of algorithms, which have been used for the retiming procedure is shown. Diagrams of BER(SNR) and BER (Doppler frequency shift) of LMS and DLMS equalization circuits are presented. Obtained results facilitate transfer of demodulator circuit of existing modem from the Zynq microcontroller part built into the Xilinx XC7Z020 FPGA to its programmable logic part in order to increase the performance of the modem receiver to solve current and future challenges.

Taemong Technology : Symbolic Frequencies in Contemporary Korean Birth Dreams

In his 1989 study Oriental Birth Dreams, Fred Jeremy Seligson presented statistical figures on the frequencies of primary dream symbols in conception dreams (taemong) indicating the genders, personality characteristics and future careers of the children they represent. One of the most rapidly shifting and developing socio-economic landscapes in the world, South Korea has gone from one of its poorest countries in 1953 (post-Korean War) to the tenth largest economy in the world today. While the operative dream symbols in the birth dreams collected by Seligson thirty-three years ago reflect a society still highly attuned to the natural world, the author asks: what might occur if the symbolic frequencies of the dream images shift in accordance with Korea’s socio-cultural, technological, and environmental changes? As South Koreans continue their exodus from the countryside to the urban center of the Seoul metro area the IT capital of the world and now the site of residence of approximately half the country’s population (26 million people) the nature-based conception images (such as animals, fruits and flowers) of their ancestors may be increasingly supplanted by the unnatural objects of their immediate surroundings (such as smart phones, computers and characters from video games). With attention to the spiritual, psychological and biological aspects of birth dreams, their medical applications, epistolary, and poetic forms, this paper examines their nature as personal mythologies. It then concludes with a close reading of a recently discovered digital birth dream in order to understand how it might affect the oral tradition of taemong, the lives of those who dream them and those who are dreamt.

Stepped-Carrier OFDM With a Nonlinear Hopping Pattern for Joint Radar and Communications

In this article, a stepped-carrier orthogonal frequency-division multiplexing (OFDM) system with a nonlinear hopping pattern is developed for joint radar and communication (RadCom). In the proposed OFDM-based RadCom system, the carrier frequencies of OFDM symbols for multiple RadCom nodes are orthogonally switched to avoid interference while also maximizing the signal-to-radar interference ratio (SRIR). Differently from conventional linear stepped-carrier OFDM radar that does not consider the communication performance, we first develop a subband allocation method that considers both the radar and the communication performance for multiple RadCom nodes. Channel state information (CSI) of multiple nodes is shared with a centralized server, which then computes the subband allocation strategy that gives the maximum SRIR. Furthermore, a distributed subband allocation strategy based on stateless Q-learning is also proposed, which is suitable when the RadCom nodes do not share their CSI. We also develop the radar signal processing algorithm for range and velocity estimation when the stepped-carrier OFDM waveforms are transmitted through multiple subbands with a nonlinear hopping pattern. Through computer simulations, we confirm that the proposed stepped-carrier OFDM with the nonlinear hopping pattern exhibits a higher achievable sum rate than conventional linear stepped OFDM while not sacrificing the radar performance of target parameter estimation.

Robust timing and frequency joint synchronization method for QNSC-enabled security optical communications system based on CO-OFDM

The security of optical fiber communication network has been paid more and more attention, and Quantum Noise Stream Cipher (QNSC) can guarantee its physical layer security. However, the QNSC system realized with CO-OFDM technology is very susceptible to timing synchronization error and carrier frequency offset. Existing symbol timing and frequency synchronization algorithms based on preambles have problems of timing metric platform, interference of side lobes, and low timing and frequency synchronization accuracy especially in optical signal transmission scenario of high-order modulation and low SNR. Furthermore, the preamble is not encrypted, which can be easily identified by illegal receiver. In order to solve above problems, a robust timing and frequency joint synchronization algorithm is proposed. The algorithm is composed of the preamble interleaved by conjugate, negative and image sequences of original element sequence. The preamble conforms to the preamble structure of Schmidl, Minn and CNI by adding negative and conjugate operator at the specific interval of the preamble. The correspond timing metric method, i.e. combination of Schmidl, Minn and CNI timing metric strategies, can fully extract characteristics of the preamble, and then timing and frequency joint synchronization is accurately obtained. For timing synchronization, experiments in 10Gbit/s CO-OFDM based QNSC system have illustrated that proposed algorithm has better timing synchronization performance than baseline algorithms. The detection probability is augmented and timing offset mean is decreased than that of baseline algorithms especially at low SNR. The simulation illustrates that proposed algorithm has more accurate frequency offset (FO) estimation performance compared with the baseline algorithms even at low SNR. Furthermore, the simulation illustrates the security performance of CO-OFDM based QNSC system is improved by encrypting the preamble with secret key.

Secure OFDM transmission scheme based on chaotic encryption and noise-masking key distribution.

In this Letter, we propose a secure orthogonal frequency division multiplexing (OFDM) transmission scheme based on chaotic encryption and noise-masking key distribution. With the implementation of a three-dimensional digital chaotic system, the security performance is effectively enhanced by scrambling the phase, symbol, and subcarrier frequency of the OFDM signal. The proposed noise-masking key distribution can mask the key information of the chaotic system into noise and transmit it with the chaotic encrypted signal simultaneously. By this mechanism, the legal receiver can realize uninterrupted authentication and decryption even if the key is constantly updated. Transmission of a 62.2-Gb/s quadrature phase shift keying (QPSK) and 124.4-Gb/s 16 quadrature amplitude modulation (16QAM) OFDM signal over a 2-km 7-core fiber using the proposed scheme is experimentally demonstrated. The results show that the proposed scheme can realize security enhancement and cost-effective key distribution without significant bit error ratio (BER) performance degradation.

Recognition of Visual Arabic Scripting News Ticker From Broadcast Stream

News ticker recognition is a vital area of research due to its applications such as information analysis, opinion mining and language translation for media regulatory authorities. Without automated systems, manual anatomizing is difficult. In this paper, we focus on the automatic Arabic and Urdu news ticker recognition system. It mainly consists of ticker segmentation and text recognition to generate textual data for various online services. Our work investigates character-wise explicit segmentation and syntactical models with Kufi and Nastaleeq fonts. Various network models anticipate learning of deep representations by homogenizing the classes regardless of inter-symbol correlations and linguistic taxonomy. The proposed learning model incorporates fairness by maximizing the balance among sensitive features of characters in a unified manner. Furthermore, we demonstrate the efficiency of the proposed model by carrying out experiments using customized news tickers datasets with accurate character-level and component-level labeling. Moreover, our method is evaluated on a challenging Urdu Printed Text Images (UPTI) dataset that only provides ligature based annotations. The proposed method attains 98.36%, outperforms the current state of the art method. Ablation investigations show that our technique enhances the performance of character classes with low symbol frequencies.

Combined Intra Symbol Frequency Domain Averaging Channel Estimation for Polarization Division Multiplexed Coherent Optical Ofdm/Oqam Systems

Combined Intra Symbol Frequency Domain Averaging Channel Estimation for Polarization Division Multiplexed Coherent Optical Ofdm/Oqam Systems

Design and validation of a chart-based measure of the limits of spatial contrast sensitivity.

Current chart-based tests of spatial contrast sensitivity (SCS) with fixed or narrow frequency ranges (≤18cycles/°) cannot characterise the limits of spatial contrast vision. Here we present the design and validation of a chart-based measure of the spatial contrast envelope. Following the principles of the standard visual acuity (Bailey-Lovie) and contrast sensitivity (Pelli-Robson) charts, a combined spatial-contrast and visual acuity chart was designed using a language-independent triangular symbol for a four-alternative forced-choice procedure plus chart rotation. Symbol frequencies ranged between 0.38 and 60cycles/° spaced along 10radial axes (0.55%-100% contrast). The chart was validated with reference to the Bailey-Lovie and Pelli-Robson charts; its reliability and sensitivity to changes in illumination, simulated cataract and blur was evaluated in healthy adults. The photopic SCS function could be measured in 5.5±0.5min; thresholding around the spatial contrast resolution limit reduced completion times to ~2min. There was good agreement with high-contrast visual acuity (difference=0.08±0.02logMAR) and contrast-sensitivity at 1.5 cycles/° (0.13±0.06logCS). Test-retest reliability was excellent at all spatial frequencies (ICC=0.99). Mesopic illumination or simulated cataract caused a generalised SCS loss; myopic blur reduced high-frequency sensitivity. Spatial contrast sensitivity was independent of radial axis orientation (cardinal or oblique). The chart provides a time-efficient, reliable and inexpensive measure of SCS with applications in research and clinic for detecting subtle deficits in early stages of ocular and neurological conditions that often manifest at higher frequencies. It is sensitive to vision changes occurring in dim lighting and with simulated cataract and blur. The chart is available open-access for self-printing; contrast variation in print can be controlled through user calibration and/or establishing normative SCS functions using the theoretical values.

LPI‐based cooperative tactic for satellite communication uplink signal accompanied by directed and power limited jamming

To protect the satellite communication uplink signal (SCUS) from being intercepted by an eavesdropper, this study proposes several tactics to ensure SCUS being in the low probability of intercept (LPI) state. First, based on the satellite link equation, an optimisation model for LPI SCUS is established with coding and synchronisation of symbol and carrier frequency. However, when the satellite is on the synchronisation orbit, the LPI model of SCUS is still very likely to be intercepted by an eavesdropper that might be close to the SCUS. Aiming at this problem, the second tactic is to mask LPI SCUS with directed and power limited jamming. The purpose of using directed and power limited jamming is to reduce direction-finding accuracy of the eavesdropper, and to achieve the optimal LPI performance with the minimum radiation energy. The high-order cumulate algorithm is used to predict the signal-to-noise ratio of mixed signal in the second tactic. The simulation results show that the model of LPI SCUS can effectively reduce the detection probability compared with the traditional satellite uplink signals, and the second tactic can effectively reduce the direction-finding accuracy of the SCUS by the eavesdropper.

Broad Coverage Precoder Design for Synchronization in Satellite Massive MIMO Systems

In this paper, we investigate the massive multi-input multi-output (MIMO) transmission for the satellite communication systems equipped with a uniform rectangular array (URA) and aim to design the precoder with broad coverage radiation power pattern to improve the performance of time and frequency synchronizations. The modified Cramér-Rao vector bounds (MCRVB) are chosen as the benchmark of symbol timing offset and frequency offset estimation problem, which can be viewed as the time and frequency synchronization performance metric for the broad coverage precoder design. By considering the minimax MCRVB criterion and the per-antenna equal power constraint, the precoder design approach is formulated as the non-convex constrained minimax problem over the discrete radiation power pattern. This optimization problem is solved by the smoothing techniques combined with the manifold optimization method. To reduce the calculation complexity, the nonmonotone conjugate gradient method is applied. Simulation results show that the average and the minimum received powers in the coverage area of the proposed scheme are higher than those of the scheme based on fairness criterion. Compared with the existing omnidirectional and broad coverage schemes, the proposed scheme has the best synchronization performance among all the contrast solutions.

Predicting adverse cardiovascular outcomes in post-coronary artery bypass grafting patients using novel ECG frequency analysis of the QRS complex.

BackgroundA novel metric called Layered Symbolic Decomposition frequency (LSDf) has been shown to be an independent predictor of ventricular arrhythmia and mortality in patients receiving implantable cardioverter‐defibrillator (ICD) devices. This novel index studies the fragmentation of the QRS complex. However, its generalizability to predict cardiovascular events for other cardiac procedures is unknown. Herein, we investigated the applicability of LSDf as a predictive measure for major adverse cardiovascular events (MACE) in patients receiving coronary artery bypass grafting (CABG).Methods and ResultsOne hundred ninety‐five patients had high‐resolution ECG recorded prior to CABG surgery in 2012/2013 and were followed for a mean duration of 7.32 ± 0.32 years for postoperative cardiovascular outcomes. These outcomes were described as a modified composite of MACE defined as hospitalization for heart failure, ventricular tachycardia, ventricular fibrillation, and cardiovascular death including stroke and cardiac arrest. One hundred seventy‐two patients were included for analysis and 18 patients experienced a postoperative cardiovascular outcome. These patients had significantly increased age (71.3 vs. 64.6 years, p = .007), prolonged QRS duration (113.22 vs. 97.35 ms, p = .003), reduced left ventricular ejection fraction (42.7% vs. 56.5%, p < .001), and lower LSDf percent (13.5% vs. 16.9%, p = .002). Patients with an LSDf below 13.25% were 4.8 (OR 1.7–13.5, p < .001) times more likely to experience a MACE and up to 19.4 (OR 4.2–90.3, p < .001) times more likely to experience a MACE when older than 70 years and an ejection fraction below 50%.ConclusionLayered Symbolic Decomposition frequency may be an applicable metric to predict long‐term cardiovascular outcomes in patients with ischemic heart disease.

Метод формирующего оператора для моделирования циклостационарных случайных процессов

The paper proposes the shaping operator technique for modelling cyclostationary random process (CSRP) which belong to the class of second-order processes being described by the parametric pulse train model. Generating CSRP is expressed as the propagation result of the specially chosen elementary CSRP through the linear time-invariant system with a known impulse response. The paper describes the approach to forming the elementary CSRP based on the train made of Dirac delta-function uniformly following in time domain with constant step and defining the desired structural periodicity of the process being modelled. The analytical expressions for the main characteristics used for describing cyclostationary properties of the elementary process, including two-dimensional, cyclic and spectral corelation functions, were obtained. In addition to the case of the elementary CSRP with correlated weighting coefficients of delta-functions, the case of statistically independent coefficients was considered. It was shown that utilization of the spectral correlation function (SCF) for describing the cyclic property reveals the explicit analytical relation between characteristics of the elementary and modelled CSRPs provided the filter frequency response is known. The paper presents the comparative example which describes the modelling of two CSRPs: one of them was chosen as the train of rectangle pulses which is pulse amplitude modulated by stationary random time series, while the other is considered as a signal formed by the method of direct sequence spread spectrum. The chosen short-length Barker sequence as the code allowed performing visual comparison between absolute value of SCF components taken at the cyclic frequencies multiple of the chip frequency and cyclic frequencies multiple of the symbol frequency. The future development of the methods proposed in the paper opens the road to improving the performance of modern radar and telecommunication systems by means of utilizing cyclic frequencies which are non-random parameters describing the structural properties of signals under processing.

Injection Coding with Spectral Nulls for Even Codeword Lengths

With a view to adapting information signals to suit the frequency characteristics of digital transmission channels, it is pertinent to design channel coding schemes that can exhibit spectral shaping capabilities after being processed with a suitable modulation technique. This involves concentrating the frequency energy of the coded and modulated information signal towards a predetermined range of the frequency spectrum, or making it have low power content at such frequencies. One modulation scheme that is suitable for achieving this is pulse amplitude modulation (PAM). We thus present a permutation coding (PC) system with injections that can exhibit spectral nulls at rational sub-multiples of the symbol frequency. Such injections are achieved by strategically removing δ columns from the source PC. We also present a mathematical expression that allows for the prediction of zero energy positions in the codebook’s spectrum. Due to the way the injections are introduced, it gives the scheme an advantage of achieving higher symbol rate, when compared with conventional PC systems. The work is however limited to cases where δ =2 and the lengths of the codewords involved are even numbers.

Injection Coding With Spectral Shaping Capability

In digital transmission systems that involve pulse amplitude modulation (PAM), channel coding is useful for spectral shaping by concentrating the frequency energy of the transmitted information towards a predetermined range of the frequency spectrum, or making it have low power content at such frequencies, so as to suit the frequency characteristics of the communication channel. We thus present a permutation coding system with injections that can exhibit spectral shaping at rational sub-multiples of the symbol frequency. We also present a mathematical expression that allows for the prediction of low energy positions in the codebook's spectrum. Upper and lower bounds for the low energy spectrum were also determined. Due to the way the injections are introduced, it gives the scheme an advantage of achieving higher symbol rate, when compared with conventional permutation coding systems.

A Simple Symmetry Present in Every Sequence Having Maximum Entropy Can Help Us to Overcome the Limit Defined by Shannon’s First Theorem

ABSTRACT: To explain the method described in this article let us start with a simple consideration. Given a base 10 sequence of 10 elements, exist 10!≈3.6 ∙10^6 sequences having maximum entropy. So, theoretically, it is possible to define a biunivocal correspondence between a base 10 sequence of 10 elements having maximum entropy and a number between 1 and 3.6 ∙10^6 . Therefore, our 10 element sequence can be compressed into no more than 7 elements in base 10. We obtain this result, by exploiting the fact that these sequences represent a small percentage of all base 10 sequences of 10 elements. Generalizing, given a sequence of length n having C symbols, the sequences having a given symbol frequency will be only a small percentage of all the possible permutations, that we know to be 𝐶^𝑛. Consequently, with this compression method, the gain obtained depends on how much it is possible to compress the sequence of values that identify the symbol frequencies. Therefore, the sequences with maximum entropy are the sequences that can be most compressed. In fact, in this case, this information can be compressed because we know that all the symbols have the same frequency, so we just need to code a single frequency. We conclude with a practical example: let us take a sequence of 100 elements in base 100 having maximum entropy. If we want to compress this sequence in base 10, according to Shannon's first theorem, we need 200 values in base 10. Given a sequence of 100 elements in base 100, there are 100!≈9.3∙10^157sequences with maximum entropy. Thus, by defining a correspondence between our sequence and a value between 1 and 9.3∙10^157, we can compress our sequence with no more than 158 values in base 10. Obtaining, in this way, a gain of 42 values in base 10. Finally, we must also send to the decoder the information regarding the frequencies of the various symbols. In this case, we have 100 symbols with a frequency of 1/100; this information can be compressed in this way 10001, using only 5 numbers in base 10. Consequently, 163 values in base 10 will be sent to the decoder. Therefore, we are able to compress our sequence into less than 200 values, exceeding the limit defined by Shannon's first theorem.