Phase and Amplitude Distortion Methods for Digital Synthesis of Classic Analogue Waveforms

An essential component of digital emulations of subtractive synthesizer systems are the algorithms used to generate the classic oscillator waveforms of sawtooth, square and triangle waves. Not only should these be perceived to be authentic sonically, but they should also exhibit minimal aliasing distortions and be computationally efficient to implement. This paper examines a set of novel techniques for the production of the classic oscillator waveforms of Analogue subtractive synthesis that are derived from using amplitude or phase distortion of a mono-component input waveform. Expressions for the outputs of these distortion methods are given that allow parameter control to ensure proper bandlimited behavior. Additionally, their implementation is demonstrably efficient. Lastly, the results presented illustrate their equivalence to their original Analogue counterparts

Antiderivative antialiasing for memoryless nonlinearities

Aliasing is a commonly encountered problem in audio signal processing, particularly when memoryless nonlinearities are simulated in discrete time. A conventional remedy is to operate at an oversampled rate. A new aliasing reduction method is proposed here for discrete-time memoryless nonlinearities, which is suitable for operation at reduced oversampling rates. The method employs higher order antiderivatives of the nonlinear function used. The first-order form of the new method is equivalent to a technique proposed recently by Parker et al. Higher order extensions offer considerable improvement over the first antiderivative method, in terms of the signal-to-noise ratio. The proposed methods can be implemented with fewer operations than oversampling and are applicable to discrete-time modeling of a wide range of nonlinear analog systems.Peer reviewe

A Low Noise Sub-Sampling PLL in Which Divider Noise Is Eliminated and PD-CP Noise Is not multiplied by N^2

This paper presents a 2.2-GHz low jitter sub-sampling based PLL. It uses a phase-detector/charge-pump (PD/CP)that sub-samples the VCO output with the reference clock. In contrast to what happens in a classical PLL, the PD/CP noise is not multiplied by N2 in this sub-sampling PLL, resulting in a low noise contribution from the PD/CP. Moreover, no frequency divider is needed in the locked state and hence divider noise and power can be eliminated. An added frequency locked loop guarantees correct frequency locking without degenerating jitter performance when in lock. The PLL is implemented in a standard 0.18- m CMOS process. It consumes 4.2 mA from a 1.8 V supply and occupies an active area of 0.4 X 0.45 m

Hardward and algorithm architectures for real-time additive synthesis

Additive synthesis is a fundamental computer music synthesis paradigm tracing its origins to the work of Fourier and Helmholtz. Rudimentary implementation linearly combines harmonic sinusoids (or partials) to generate tones whose perceived timbral characteristics are a strong function of the partial amplitude spectrum. Having evolved over time, additive synthesis describes a collection of algorithms each characterised by the time-varying linear combination of basis components to generate temporal evolution of timbre. Basis components include exactly harmonic partials, inharmonic partials with time-varying frequency or non-sinusoidal waveforms each with distinct spectral characteristics. Additive synthesis of polyphonic musical instrument tones requires a large number of independently controlled partials incurring a large computational overhead whose investigation and reduction is a key motivator for this work. The thesis begins with a review of prevalent synthesis techniques setting additive synthesis in context and introducing the spectrum modelling paradigm which provides baseline spectral data to the additive synthesis process obtained from the analysis of natural sounds. We proceed to investigate recursive and phase accumulating digital sinusoidal oscillator algorithms, defining specific metrics to quantify relative performance. The concepts of phase accumulation, table lookup phase-amplitude mapping and interpolated fractional addressing are introduced and developed and shown to underpin an additive synthesis subclass - wavetable lookup synthesis (WLS). WLS performance is simulated against specific metrics and parameter conditions peculiar to computer music requirements. We conclude by presenting processing architectures which accelerate computational throughput of specific WLS operations and the sinusoidal additive synthesis model. In particular, we introduce and investigate the concept of phase domain processing and present several “pipeline friendly” arithmetic architectures using this technique which implement the additive synthesis of sinusoidal partials

A Perceptual Comparison of “Black Box” Modeling Algorithms for Nonlinear Audio Systems

Nonlinear systems identification is a widespread topic of interest, particularly within the audio industry, as these techniques are employed to synthesize black box models of nonlinear audio effects. Given the myriad approaches to black box modeling, questions arise as to whether an “optimal” approach exists, or one that achieves valid subjective results as a model with minimal computational expense. This thesis uses ABX listening tests to compare black box models of three hardware audio effects using two popular nonlinear implementations, along with two proposed modified implementations. Models were constructed in the Hammerstein form using sine sweeps and a novel measurement technique for the filters and nonlinearities, respectively. Testing revolved around null hypotheses assuming no change in model identification regardless of the device modeled, implementation used, or program material of the model stimulus. Results provide clear evidence of an effect on all of these accounts, and support a full rejection of the null hypotheses. Outcomes demonstrate a preferable implementation out of the algorithms tested, and suggest the removal of certain implementations as valid approaches altogether

Computationally efficient music synthesis : methods and sound design

Tässä diplomityössä esitetään musiikkisyntetisaattorin suunnittelua systeemille, jonka laskentateho ja muistikapasiteetti ovat rajoitettuja. Ensiksi kerrataan mahdollisia synteesitekniikoita sekä arvioidaan niiden käyttökelpoisuutta laskennallisesti tehokkaassa musiikkisynteesissä. Käytännössä käyttökelpoiset tekniikat ovat lisäävä ja lähde-suodinsynteesit, ja erikoistapauksissa taajuusmodulaatio-, aaltotaulukko- ja samplaussynteesit. Tämän jälkeen käyttökelpoisten tekniikoiden rakenteiden suunnittelua esitetään tarkemmin, sekä esitetään näiden rakenteiden ominaisuuksia ja suunnitteluongelmia. Suurin ongelma kohdataan digitaalisessa lähde-suodinsynteesissä, jossa klassisten aaltomuotojen, kuten saha-aallon käyttö lähdesignaalina on ongelmallista laskostumisen takia, joka johtuu aaltomuodossa olevista epäjatkuvuuksista. Olemassa olevia kaistarajoitettuja aaltomuotosynteesimenetelmiä kerrataan, ja polynomimuotoiseen kaistarajoitetuun askelfunktioon perustuvaa menetelmää esitellään tarkemmin antamalla suunnittelusääntöjä käyttökelpoisille polynomeille. Menetelmää testataan lisäksi kahdella kolmannen asteen polynomilla. Nämä polynomit vähentävät laskostumista korkeilla taajuuksilla enemmän verrattuna ensimmäisen asteen polynomiin, mutta pienillä taajuksilla ensimmäisen asteen polynomi tuottaa parempia tuloksia. Lisäksi kerrataan muita mahdollisia ääniefektialgoritmeja ja arvioidaan niiden käyttökelpoisuutta laskennallisesti tehokkaassa musiikkisynteesissä. Useasti äänisynteesisysteemin täytyy pystyä generoimaan musiikkia, jossa käytetään monia erilaisia ääniä, jotka ulottuvat oikeista akustisista soittimista elektronisiin soittimiin ja luonnon ääniin. Siksi tällainen systeemi tarvitsee huolellista äänten suunnittelua. Tässä diplomityössä esitetään suunnittelusääntöjä erilaisten äänien imitoimiseksi. Lisäksi esitellään synteesimenetelmien parametrien vaikutus äänivarianttien suunnitteluun.In this thesis, the design of a music synthesizer for systems suffering from limitations in computing power and memory capacity is presented. First, different possible synthesis techniques are reviewed and their applicability in computationally efficient music synthesis is discussed. In practice, the applicable techniques are limited to additive and source-filter synthesis, and, in special cases, to frequency modulation, wavetable and sampling synthesis. Next, the design of the structures of the applicable techniques are presented in detail, and properties and design issues of these structures are discussed. A major implementation problem is raised in digital source-filter synthesis, where the use of classic waveforms, such as sawtooth wave, as the source signal is challenging due to aliasing caused by waveform discontinuities. Methods for existing bandlimited waveform synthesis are reviewed, and a new approach using polynomial bandlimited step function is presented in detail with design rules for the applicable polynomials. The approach is also tested with two different third-order polynomials. They reduce aliasing more at high frequencies, but at low frequencies their performance is worse than with the first-order polynomial. In addition, some commonly used sound effect algorithms are reviewed with respect to their applicability in computationally efficient music synthesis. In many cases the sound synthesis system must be capable of producing music consisting of various different sounds ranging from real acoustic instruments to electronic instruments and sounds from nature. Therefore, the music synthesis system requires careful sound design. In this thesis, sound design rules for imitation of various sounds using the computationally efficient synthesis techniques are presented. In addition, the effects of the parameter variation for the design of sound variants are presented

Design Of Polynomial-based Filters For Continuously Variable Sample Rate Conversion With Applications In Synthetic Instrumentati

In this work, the design and application of Polynomial-Based Filters (PBF) for continuously variable Sample Rate Conversion (SRC) is studied. The major contributions of this work are summarized as follows. First, an explicit formula for the Fourier Transform of both a symmetrical and nonsymmetrical PBF impulse response with variable basis function coefficients is derived. In the literature only one explicit formula is given, and that for a symmetrical even length filter with fixed basis function coefficients. The frequency domain optimization of PBFs via linear programming has been proposed in the literature, however, the algorithm was not detailed nor were explicit formulas derived. In this contribution, a minimax optimization procedure is derived for the frequency domain optimization of a PBF with time-domain constraints. Explicit formulas are given for direct input to a linear programming routine. Additionally, accompanying Matlab code implementing this optimization in terms of the derived formulas is given in the appendix. In the literature, it has been pointed out that the frequency response of the Continuous-Time (CT) filter decays as frequency goes to infinity. It has also been observed that when implemented in SRC, the CT filter is sampled resulting in CT frequency response aliasing. Thus, for example, the stopband sidelobes of the Discrete-Time (DT) implementation rise above the CT designed level. Building on these observations, it is shown how the rolloff rate of the frequency response of a PBF can be adjusted by adding continuous derivatives to the impulse response. This is of great advantage, especially when the PBF is used for decimation as the aliasing band attenuation can be made to increase with frequency. It is shown how this technique can be used to dramatically reduce the effect of alias build up in the passband. In addition, it is shown that as the number of continuous derivatives of the PBF increases the resulting DT implementation more closely matches the Continuous-Time (CT) design. When implemented for SRC, samples from a PBF impulse response are computed by evaluating the polynomials using a so-called fractional interval, µ. In the literature, the effect of quantizing µ on the frequency response of the PBF has been studied. Formulas have been derived to determine the number of bits required to keep frequency response distortion below prescribed bounds. Elsewhere, a formula has been given to compute the number of bits required to represent µ to obtain a given SRC accuracy for rational factor SRC. In this contribution, it is shown how these two apparently competing requirements are quite independent. In fact, it is shown that the wordlength required for SRC accuracy need only be kept in the µ generator which is a single accumulator. The output of the µ generator may then be truncated prior to polynomial evaluation. This results in significant computational savings, as polynomial evaluation can require several multiplications and additions. Under the heading of applications, a new Wideband Digital Downconverter (WDDC) for Synthetic Instruments (SI) is introduced. DDCs first tune to a signal\u27s center frequency using a numerically controlled oscillator and mixer, and then zoom-in to the bandwidth of interest using SRC. The SRC is required to produce continuously variable output sample rates from a fixed input sample rate over a large range. Current implementations accomplish this using a pre-filter, an arbitrary factor resampler, and integer decimation filters. In this contribution, the SRC of the WDDC is simplified reducing the computational requirements to a factor of three or more. In addition to this, it is shown how this system can be used to develop a novel computationally efficient FFT-based spectrum analyzer with continuously variable frequency spans. Finally, after giving the theoretical foundation, a real Field Programmable Gate Array (FPGA) implementation of a novel Arbitrary Waveform Generator (AWG) is presented. The new approach uses a fixed Digital-to-Analog Converter (DAC) sample clock in combination with an arbitrary factor interpolator. Waveforms created at any sample rate are interpolated to the fixed DAC sample rate in real-time. As a result, the additional lower performance analog hardware required in current approaches, namely, multiple reconstruction filters and/or additional sample clocks, is avoided. Measured results are given confirming the performance of the system predicted by the theoretical design and simulation

Algorithms and architectures for the multirate additive synthesis of musical tones

In classical Additive Synthesis (AS), the output signal is the sum of a large number of independently controllable sinusoidal partials. The advantages of AS for music synthesis are well known as is the high computational cost. This thesis is concerned with the computational optimisation of AS by multirate DSP techniques. In note-based music synthesis, the expected bounds of the frequency trajectory of each partial in a finite lifecycle tone determine critical time-invariant partial-specific sample rates which are lower than the conventional rate (in excess of 40kHz) resulting in computational savings. Scheduling and interpolation (to suppress quantisation noise) for many sample rates is required, leading to the concept of Multirate Additive Synthesis (MAS) where these overheads are minimised by synthesis filterbanks which quantise the set of available sample rates. Alternative AS optimisations are also appraised. It is shown that a hierarchical interpretation of the QMF filterbank preserves AS generality and permits efficient context-specific adaptation of computation to required note dynamics. Practical QMF implementation and the modifications necessary for MAS are discussed. QMF transition widths can be logically excluded from the MAS paradigm, at a cost. Therefore a novel filterbank is evaluated where transition widths are physically excluded. Benchmarking of a hypothetical orchestral synthesis application provides a tentative quantitative analysis of the performance improvement of MAS over AS. The mapping of MAS into VLSI is opened by a review of sine computation techniques. Then the functional specification and high-level design of a conceptual MAS Coprocessor (MASC) is developed which functions with high autonomy in a loosely-coupled master- slave configuration with a Host CPU which executes filterbanks in software. Standard hardware optimisation techniques are used, such as pipelining, based upon the principle of an application-specific memory hierarchy which maximises MASC throughput

Aliasing reduction in clipped signals

Most real-world audio devices, particularly those of interest in musical applications, fall under the category of nonlinear systems. Examples of these devices include overdrive and distortion circuits used by guitar and bass players, dynamic range processors, and vintage synthesizer circuits. Nonlinear algorithms are known to expand the bandwidth of the input signal by introducing harmonic and intermodulation distortion. Naive digital emulations of these systems are susceptible to aliasing due to the inherent frequency constraints of discrete systems.  This thesis focuses on new digital signal processing techniques designed to reduce the level of aliasing introduced by memoryless nonlinearities. The underlying motivation of this work is to incorporate these tools within the framework of virtual analog (VA) modeling, an area of study that concentrates on the emulation of analog audio devices in the digital domain. In VA modeling, aliasing reduction has been studied extensive for the case of synthesis of classical oscillator waveforms like those used in subtractive synthesis. However, in audio effects processing oversampling has traditionally been the only available tool to ameliorate this problem. The first part of this work proposes the use of bandlimited correction functions previously used in waveform synthesis, to reduce the aliasing caused by special nonlinearities that introduce discontinuities in the derivatives of a signal. This family of novel methods includes the use of the bandlimited ramp function (BLAMP), its efficient polynomial approximations, and its integrated form. A new VA model of a highly nonlinear wavefolder circuit, which incorporates one of these techniques, is proposed.  The second family of techniques elaborated in this thesis is that of the antiderivative method. This innovative approach to aliasing reduction is based on the discrete differentiation of integrated nonlinearities and can be applied to arbitrary explicit memoryless nonlinearities regardless of their form. The use of the antiderivative forms in VA modeling is proposed by introducing two novel transistor/diode-based wavefolder models, and two static diode clipper models that incorporate these techniques.  Results obtained show the proposed algorithms effectively reduce the level of aliasing in nonlinear processing and can help reduce, and in some cases even eliminate, the oversampling requirements of the system. The proposed algorithms are suitable for real-time software implementations of VA instruments and effects processors