Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N. Furthermore, we show that a further N multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split radix algorithm). The improved algorithms for DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.Comment: 9 page

Efficient FFT Algorithms for Mobile Devices

Increased traffic on wireless communication infrastructure has exacerbated the limited availability of radio frequency ({RF}) resources. Spectrum sharing is a possible solution to this problem that requires devices equipped with Cognitive Radio ({CR}) capabilities. A widely employed technique to enable {CR} is real-time {RF} spectrum analysis by applying the Fast Fourier Transform ({FFT}). Today’s mobile devices actually provide enough computing resources to perform not only the {FFT} but also wireless communication functions and protocols by software according to the software-defined radios paradigm. In addition to that, the pervasive availability of mobile devices make them powerful computing platform for new services. This thesis studies the feasibility of using mobile devices as a novel spectrum sensing platform with focus on {FFT}-based spectrum sensing algorithms. We benchmark several open-source {FFT} libraries on an Android smartphone. We relate the efficiency of calculating the {FFT} to both algorithmic and implementation-related aspects. The benchmark results also show the clear potential of special {FFT} algorithms that are tailored for sparse spectrum detection

Power-Efficient Hardware Architecture for Computing Split-Radix FFT on Highly Sparse Spectrum

RÉSUMÉ Le problème du transfert des signaaux du domaine temporel au domaine fréquentiel d'une manière efficace, lorsque le contenu du spectre de fréquences a une faible densité, est le sujet de cette thèse. La technique bien connue de la transformée de Fourier rapide (FFT) est l'algorithme de traitement de signal privilégié pour observer le contenu fréquentiel des signaux entrants à des émetteurs-récepteurs de télécommunication, tels que la radio cognitive, ou la radio définie par logiciel qu‟on utilise habituellement pour l‟analyse du spectre dans une bande de fréquences. Cela peut représenter un lourd fardeau de calcul sur des processeurs lorsque la FFT ordinaire est mise en oeuvre, ce qui peut impliquer une consommation d'énergie considérable. L'alimentation en énergie est une ressource limitée dans les appareils mobiles et, par conséquent, cette ressource peut être critique pour des dispositifs de télécommunications mobiles. Dans le but de développer un processeur économe en énergie pour les applications de transformation temps-fréquence, un algorithme de transformée de Fourier plus efficace, en termes du nombre de multiplications et d'additions complexes, est sélectionné. En effet, la Split-Radix Fast Fourier Transform (SRFFT) offre une performance meilleure que la FFT classique en termes de réduction du nombre de multiplications complexes nécessaires et elle peut donc conduire à une consommation d'énergie réduite. En appliquent le concept d'élagage des calculs inutiles, c'est-à-dire des multiplications complexes avec entrées ou sorties à zéro, tout au long de l'algorithme, on peut réduire la consommation d'énergie.Ainsi, une architecture matérielle énergétiquement efficace est développée pour le calcul de la SRFFT. Cette architecture est basée sur l'élagage des calculs inutiles. En fait, pour tirer parti du potentiel de la SRFFT, une nouvelle architecture d'un processeur de SRFFT configurable est d'abord conçue, puis l'architecture est développée afin d'éliminer les calculs inutiles. Cela se fait par l'utilisation appropriée d'une matrice d'élagage.----------ABSTRACT The problem of transferring a time domain signal into the frequency domain in an efficient manner, when the frequency contents are sparsely distributed, is the research topic covered in this thesis. The well-known Fast Fourier Transform (FFT) is the most common signal processing algorithm for observing the frequency contents of incoming signals in telecommunication transceivers. It is notably used in cognitive or software defined radio which usually demands for monitoring the spectrum in a wide frequency band. This may imply a heavy computation burden on processors when the ordinary FFT algorithm is implemented, and hence yield considerable power consumption. Power and energy supply is a limited resource in mobile devices and therefore, efficient execution of the Fourier transform has turned out to be critical for mobile telecommunication devices.With the purpose of developing a power-efficient processor for time-frequency transformation, the most computationally efficient Fourier transform algorithm is selected among the existing Fourier transform algorithms upon studying them in terms of required arithmetic operations, i.e. complex multiplications and additions. Indeed, the Split-Radix Fast Fourier Transform (SRFFT) offers a performance that is better than conventional FFT in terms of reduced number of complex multiplications and hence, can reduce power consumption.Appling the concept of pruning of the unnecessary computations, i.e. complex multiplications with either zero inputs or outputs, throughout the whole algorithm may reduce the power consumption even further

Fast Algorithms for the Real Discrete Fourier Transform

Fast algorithms for the computation of the real discrete Fourier transform (RDFT) are discussed. Implementations based on the RDFT are always efficient whereas the implementations based on the DFT are efficient only when signals to be processed are complex. The fast real Fourier (FRFT) algorithms discussed are the radix-2 decimation-in-time (DIT), the radix-2 decimation-in-frequency (DIF), the radix-4 DIT, the split-radix DIT, the split-radix DIF, the prime-factor, the Rader prime, and the Winograd FRFT algorithms

Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs

This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the well-known Cooley-Tukey FFT and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast algorithms, many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar

Low-Complexity Multicarrier Waveform Processing Schemes fo Future Wireless Communications

Wireless communication systems deliver enormous variety of services and applications. Nowa- days, wireless communications play a key-role in many fields, such as industry, social life, education, and home automation. The growing demand for wireless services and applications has motivated the development of the next generation cellular radio access technology called fifth-generation new radio (5G-NR). The future networks are required to magnify the delivered user data rates to gigabits per second, reduce the communication latency below 1 ms, and en- able communications for massive number of simple devices. Those main features of the future networks come with new demands for the wireless communication systems, such as enhancing the efficiency of the radio spectrum use at below 6 GHz frequency bands, while supporting various services with quite different requirements for the waveform related key parameters. The current wireless systems lack the capabilities to handle those requirements. For exam- ple, the long-term evolution (LTE) employs the cyclic-prefix orthogonal frequency-division multiplexing (CP-OFDM) waveform, which has critical drawbacks in the 5G-NR context. The basic drawback of CP-OFDM waveform is the lack of spectral localization. Therefore, spectrally enhanced variants of CP-OFDM or other multicarrier waveforms with well localized spectrum should be considered. This thesis investigates spectrally enhanced CP-OFDM (E-OFDM) schemes to suppress the out-of-band (OOB) emissions, which are normally produced by CP-OFDM. Commonly, the weighted overlap-and-add (WOLA) scheme applies smooth time-domain window on the CP- OFDM waveform, providing spectrally enhanced subcarriers and reducing the OOB emissions with very low additional computational complexity. Nevertheless, the suppression perfor- mance of WOLA-OFDM is not sufficient near the active subband. Another technique is based on filtering the CP-OFDM waveform, which is referred to as F-OFDM. F-OFDM is able to provide well-localized spectrum, however, with significant increase in the computational com- plexity in the basic scheme with time-domain filters. Also filter-bank multicarrier (FBMC) waveforms are included in this study. FBMC has been widely studied as a potential post- OFDM scheme with nearly ideal subcarrier spectrum localization. However, this scheme has quite high computational complexity while being limited to uniformly distributed sub- bands. Anyway, filter-bank based waveform processing is one of the main topics of this work. Instead of traditional polyphase network (PPN) based uniform filter banks, the focus is on fast-convolution filter banks (FC-FBs), which utilize fast Fourier transform (FFT) domain processing to realize effectively filter-banks with high flexibility in terms of subcarrier bandwidths and center frequencies. FC-FBs are applied for both FBMC and F-OFDM waveform genera- tion and processing with greatly increased flexibility and significantly reduced computational complexity. This study proposes novel structures for FC-FB processing based on decomposition of the FC-FB structure consisting of forward and inverse discrete Fourier transforms (DFT and IDFT). The decomposition of multirate FC provides means of reducing the computational complexity in some important specific scenarios. A generic FC decomposition model is proposed and analyzed. This scheme is mathematically equivalent to the corresponding direct FC imple- mentation, with exactly the same performance. The benefits of the optimized decomposition structure appear mainly in communication scenarios with relatively narrow active transmis- sion band, resulting in significantly reduced computational complexity compared to the direct FC structure. The narrowband scenarios find their places in the recent 3GPP specification of cellular low- power wide-area (LPWA) access technology called narrowband internet-of-things (NB-IoT). NB-IoT aims at introducing the IoT to LTE and GSM frequency bands in coexistence with those technologies. NB-IoT uses CP-OFDM based waveforms with parameters compatible with the LTE. However, additional means are needed also for NB-IoT transmitters to improve the spec- trum localization. For NB-IoT user devices, it is important to consider ultra-low complexity solutions, and a look-up table (LUT) based approach is proposed to implement NB-IoT uplink transmitters with filtered waveforms. This approach provides completely multiplication-free digital baseband implementations and the addition rates are similar or smaller than in the basic NB-IoT waveform generation without the needed elements for spectrum enhancement. The basic idea includes storing full or partial waveforms for all possible data symbol combinations. Then the transmitted waveform is composed through summation of needed stored partial waveforms and trivial phase rotations. The LUT based scheme is developed with different vari- ants tackling practical implementations issues of NB-IoT device transmitters, considering also the effects of nonlinear power amplifier. Moreover, a completely multiplication and addition- free LUT variant is proposed and found to be feasible for very narrowband transmission, with up to 3 subcarriers. The finite-wordlength performance of LUT variants is evaluated through simulations