In this study, we investigate the usage of generative adversarial networks for modelling a collection of sounds. The proposed method incites an interpretation of musical sound synthesis based on audio collections rather than synthesizer component controls. This promises the generation of arbitrarily complex sounds without the restrictions of traditional synthesizer components. Furthermore,
the method promises to introduce non-linear interpolations within abritrarily varied collections of sounds. These two elements motivate a new approach in creating musical instruments. Here, we introduce a proof of principle method with qualifications and quantifactions of the results. First, we cover the imagelike audio signal representation and neural network architectures that compose a trainable system capable of producing audio signals. Despite some artifacts, the trained system is able to produce structural similarities in the spectral information compared to the training data set. Furthermore, we introduce a metric to quantitatively compare signal characteristics between two sets of signals. The difference between characteritics appears to decline throughout the training of the system.

In this thesis we have looked into the complexity of neural networks. Especially convolutional neural networks (CNNs), which are useful for image recognition, are looked into. In order to better understand the process in the neural networks, in the first half of this report a mathematical foundation for neural networks and CNNs is constructed. After this, during different experiments on a simple CNN, a better understanding of the variables that can be chosen for the network is gained. In the first experiment the network was trained on two translated images of a cat in order to test the common believe that CNNs are translation invariant. The result of this experiment refuted this believe and led to the understanding of translation equivariance, with which translation invariance often is confused. After this, the network was trained to learn the difference between a cat and a dog. From this, it appeared that the network can become translation invariant if the right training data is chosen. Also, a look was taken at the kernels of the convolution layer and the feature maps they produce. Finally, there was looked into an other transformation of the images, namely rotation. This led to the idea that the resolution of the images has influence on classifying the rotated cats and dogs.

Vector spherical harmonics are a set of basis functions for vector fields derived from the spherical harmonic functions. They are commonly used in spectral methods in certain areas of applied mathematics. In most of the existing literature they are defined in a way that is heavily dependent on the used coördinate system. In this thesis we define vector spherical harmonics in a coördinate free manner. We first do this on a sphere, using two tensor operators: the exterior derivative and the Hodge-star. We then extend these functions to vector functions in 3-dimensional Euclidian space and define a last type of vector function to form a complete basis for vector fields. We do this in such a manner that the vector spherical harmonics inherit some of the nice properties of the spherical harmonic functions with respect to rotations and rotational symmetric operators.

The transcription of voice using neural networks is a technique that deserves attention, as speech assistants are becoming increasingly popular. Neural networks have often difficulty with determining the differences between a talking person and noise. Humans have a much better understanding of this and could possibly apply their knowledge of the structure of the signals to improve the understanding of the neural network. A problem that is extremely difficult for a neural network is understanding and transcribing the lyrics of a song.

This thesis analyzes signal-processing techniques that can be applied to a song to improve the understanding of a speech-recognition algorithm. It is mainly focused on filtering the fore- ground lyrics from the accompaniment. Some basic filtering methods are described including a low-amplitude filter and a band-pass filter. But also two more complicated filters which make use of the periodicity of the background music will be treated.
The first filter is a method of voice separation using the two-dimensional Fourier transform. This method, proposed by Prem Seetharaman, Fatemeh Pishdadian, Bryan Pardo in 2017 [15], combines techniques of signal-processing and image-processing by finding periodic repetitions in a signal by identifying peaks in the two-dimensional Fourier transform of the spectrogram of the signal.
The second filter is a newly proposed method that can be used for the separation of foreground from background music. The algorithm compares columns in the spectrogram and classifies columns as overlapping if there are multiple occurrences of columns similar to the selected col- umn (repetitions). The frequency components, the different frequencies obtained from a discrete short-time Fourier transform, of overlapping columns are afterward compared with components of the same frequency in other columns. Under certain circumstances, overlapping frequency components are subtracted from components in other columns of the spectrogram. This removes repetitions of that frequency throughout the song. The components of the spectrogram that re- main after several iterations of this method are most likely to correspond to the least repetitive parts of the song.

The decisions that are made while constructing the method of comparing spectrogram columns are discussed and are compared with steps performed in the method that uses the two-dimensional Fourier transform. An implementation and demonstration are also attached. From the research it is expected that the two-dimensional Fourier transform perform better on strict periodic accompaniment, while the method that compares spectrogram columns is more likely to perform better on songs with a less tight rhythm.

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

A model is designed for solving gravitational profiles of self-gravitating and rotating planets via the use of Poisson's equation for total gravity, i.e., the sum of the gravitational and rotational potential. Poisson's equation is a partial differential equation that is solved with the usage of Green's functions. This is a major advantage because Earth's surface is an equipotential, which leads to a Dirichlett boundary condition. We apply this methodology to a spherical Earth, where the Green's function is closed. It is demonstrated that, when this Green’s functions based methodology is applied to a homogeneous density, a linear spherical density and the PREM density profiles (where all density profiles are spherically symmetric), then discontinuities occur in the accelerationof gravitation across the surface of the Earth, in the range of 0.01 %- 0.13 %. Such an effect is a symptom of incompatibility with the laws of physics for geostaticalequilibrium, and it is certainly significant as compared to the numerical accuracy of the solution. Moreover, the discrepancy is dependent on geographical latitude, as was to be expected, because it somehow reflects the centrifugal effect. The conclusion is made that this method is indeed capable of detecting, and even quantifying, the incompatibility of spherically symmetric mass distributions with the fundamental laws of physics for self-gravitating and rotating planets. This opens a way -and this is one of the main results- to computing corrections to spherically symmetric mass distributions, such as the PREM model. Based on this method, a further way forward to this end is suggested.

In this thesis we classify samples of music according to the genre that the music belongs to using neural networks. We divide this task into four parts. In the first part, we prepare the audio files to be used as input to a neural network. Specifically, we examine ways to create spectrograms. We then optimise the spectrograms by reducing and normalising them. The second part consists of theoretical information regarding neural networks. We initially look at the perceptron, the building blocks of any neural network, and then extend this notion to various networks, such as the multilayer perceptron, the recurrent network and the convolutional network. In the third part, we apply the theoretical knowledge that we gained in part two and implement a standard neural network, a recurrent neural network, a convolutional neural network, and a combination of
recurrent and convolutional neural networks. We examine various network structures and we evaluate them based on what the networks can learn, how fast they can learn it and how accurate its classifications are. Simultaneously, we focus on creating an efficient network, using the fewest amounts of computational resources possible. We train each of the networks with data that we created in part one, and compare the performance of the networks with each other. In terms of accuracy and loss measures, we find that the best performing network is the combination of the recurrent and convolutional neural network. This network is able to determine which of six considered genres a 3 second sound sample belongs to with an accuracy of 90%. However, in terms of computational resources required to train the models, the convolutional neural network with many kernels during training converges using least computational cost. We then experiment with the amount of kernels in the convolutional layers, and find that a layer with many kernels learns faster, but does not necessarily yield better results. This is because networks with fewer kernels eventually learn the same kernels that are significant. Finally, we consider the impact of varying sound sample lengths on the performance of the networks. For a 1 second sound sample, we see that the recurrent network outperforms the other networks in terms of accuracy of the predictions. However, the larger we make the sound samples between 1 and 3 seconds, the better each network performs. Part four consists of an explanation of each component in the system, and presents a complete system built with Google’s TensorFlow [15], in which all the components work together to create an end-to-end classification system.

In this thesis we examine various aspects of the output signal produced by the electrical pickup of an electric guitar. We do this by formulating a model of this output signal based on the one dimensional damped wave equation for the motion of one guitar string and (initially) a simple linear model for the output signal based on this motion. This model is compared to measurements of the output signal of a real guitar and their corresponding spectra. From this comparison it is concluded that the damping term in the damped wave equation is sufficient to roughly describe the decrease in amplitude over time of the envelope in the output signal of an electrical guitar.
The measured guitar signal also displays beating which has been considered to be caused by the interference of parallel and perpendicular modes on the guitar string with slightly differing frequencies. It is shown that the order of magnitude of this difference in frequencies between parallel and perpendicular modes is neither caused by different effective values of the damping coefficient nor by the influence of co-vibrating string boundaries. The last chapter gives a very simple method to introduce beating into the predicted signal. This is done by describing the motion of the string as occurring in a rotating plane in which the radial direction is given by the above mentioned model.