This letter presents a novel approach for \mbox{efficiently} computing time-index powered weighted sums of the form $\sum_{n=0}^{N-1} n^{K} v[n]$ using cascaded accumulators. Traditional direct computation requires $K{\times}N$ general multiplications, which become prohibitive for large $N$, while alternative strategies based on lookup tables or signal reversal require storing entire data blocks. By exploiting accumulator properties, the proposed method eliminates the need for such storage and reduces the multiplicative cost to only $K{+}1$ constant multiplications, enabling efficient real-time implementation. The approach is particularly useful when such sums need to be efficiently computed in sample-by-sample processing systems.

Complexity reduction is one of the main issues of digital signal processing (DSP) algorithms, especially in communication systems where each new generation brings new requirements towards increasing data rates and improved accuracy positioning, leading to the growth of power consumption and chip area. To meet these requirements and at the same time find a trade-off between high performance and low implementation cost, more sophisticated DSP algorithms need to be developed. Recent communication standards require flexible, adaptive systems capable of real-time frequency-domain tuning. Variable digital filters (VDFs) address these needs by enabling "on-the-fly" frequency response adjustments without the need for online filter design. The key feature of VDFs is that they require only an adjustment of one or a few parameters to change their characteristics, without the need for extensive additional computations. Most VDF coefficients remain fixed after the initial design, allowing for efficient hardware implementation. This makes VDFs essential for modern adaptive communication technologies.

This thesis primarily focuses on the design and low-complexity implementation techniques of VDFs and presents three main contributions. Firstly, it proposes three VDF realizations for simultaneous lowpass filtering and equalization using polynomial channel models, with systematic design procedures based on minimax optimization for all the proposed structures. In addition, a fast design method for the VDFs with several variable parameters, which can substantially decrease the design time, is presented. Secondly, it introduces frequency-domain implementations of VDFs using the overlap-save technique. Based on the assumption that these filters have been designed using a common design approach based on optimizing the impulse response coefficients, the filter DFT coefficients are proposed to be implemented as fixed, hybrid, or variable weights. Lastly, the thesis presents an efficient design approach for a variable-bandwidth digital filter implemented in the frequency domain using the overlap-save method. The proposed approach is based on a hybrid of frequency sampling and optimization, allowing for direct optimization of the DFT coefficients considering the filter frequency-domain implementation and thereby noticeably reducing the cost of implementation and an online update of the DFT filter coefficients when the bandwidth is varied.

Reduktion av komplexitet är en av huvudfrågorna för digital signalbehandling (DSP) algoritmer, särskilt i kommunikationssystem där varje ny generation ställer nya krav på att öka datahastigheter och förbättrad noggrannhet positionering, vilket leder till en ökning av strömförbrukningen och kretsytan. För att möta dessa krav och samtidigt hitta en avvägning mellan hög prestanda och låg implementeringskostnad behöver mer sofistikerade DSP-algoritmer utvecklas. Senaste kommunikationsstandarder kräver flexibla, adaptiva system som kan frekvensdomäninställning i realtid. Variabla digitala filter (VDF) tillgodoser dessa behov genom att möjliggöra "on-the-fly" frekvenssvarsjusteringar utan behov av onlinefilterdesign. Nyckelegenskapen hos VDF:er är att de bara kräver en justering av en eller ett fåtal parametrar för att ändra deras egenskaper, utan behov av omfattande ytterligare beräkningar. De flesta VDF-koefficienter förblir fixerade efter den ursprungliga designen, vilket möjliggör effektiv hårdvaruimplementering. Detta gör VDF:er väsentliga för modern adaptiv kommunikationsteknik.

Den här avhandlingen fokuserar främst på design och implementeringstekniker med låg komplexitet för VDF:er och presenterar tre huvudsakliga bidrag. För det första föreslår den tre VDF-realiseringar för samtidig lågpassfiltrering och utjämning med användning av polynomkanalmodeller, med systematiska designprocedurer baserade på minimax optimering för alla föreslagna strukturer. Dessutom presenteras en snabb designmetod för VDF:erna med flera variabla parametrar, som avsevärt kan minska designtiden. För det andra introducerar den frekvensdomänimplementationer av VDF:er med överlappningssparateknik. Baserat på antagandet att dessa filter har utformats med användning av en gemensam designmetod baserad på optimering av impulssvarskoefficienterna, föreslås filtrets DFT-koefficienter implementeras som fasta, hybrida eller variabla vikter. Slutligen presenterar avhandlingen en effektiv designansats för ett digitalt filter med variabel bandbredd implementerat i frekvensdomänen med användning av överlappningssparametoden. Det föreslagna tillvägagångssättet är baserat på en hybrid av frekvenssampling och optimering, vilket möjliggör direkt optimering av DFT-koefficienterna med tanke på implementeringen av filterfrekvensdomänen och därigenom märkbart minska kostnaden för implementering och en onlineuppdatering av DFT-filterkoefficienterna när bandbredden är varierande.

This paper introduces an efficient design approach for a fast-convolution-based variable-bandwidth (VBW) filter. The proposed approach is based on a hybrid of frequency sampling and optimization (HFSO), that offers significant computational complexity reduction compared to existing solutions for a given performance. The paper provides a design procedure based on minimax optimization to obtain the minimum complexity of the overall filter. A design example includes a comparison of the proposed design-based VBW filter and time-domain designed VBW filters implemented in the time domain and in the frequency domain. It is shown that not only the implementation complexity can be reduced but also the design complexity by excluding any computations when the bandwidth of the filter is adjusted. Moreover, memory requirements are also decreased compared to the existing frequency-domain implementations. Index Terms—Variable bandwidth filter, fast convolution, frequency-domain design, time-varying systems, overlap-save, multirate filter banks.

The paper introduces frequency-domain implementations of variable digital filters using the overlap-save method. Expressions for implementation and design complexities are derived for real-valued impulse responses. Design examples include implementations of a variable bandwidth (VBW) filter alone as well as a cascade of a VBW filter and a variable fractional delay(VFD) filter. Compared to a time-domain implementation and a filter bank approach, the proposed structures can reduce the implementation complexity significantly and achieve savings up to 95% in the multiplication rate and up to 89% in the addition rate.

This paper introduces realizations of a reconfigurable finite-impulse-response (FIR) filter for simultaneous equalization and lowpass filtering. The proposed structures employ properties of both a variable bandwidth (VBW) filter and a variable equalizer (VE) with an adjustable coefficient using the Farrow structure, therefore they consist of weighted combinations of fixed subfilters. Design procedures using minimax optimization technique are provided. The paper includes a design example and complexity comparisons between the proposed structures of the reconfigurable lowpass equalizer (RLPE) and a common approach of using a regular FIR equalization filter requiring online redesign.