This thesis deals with the estimation of unobserved variables or states from a time series of noisy observations. Approximate minimum variance filters for a class of discrete time systems with both additive and multiplicative noise, where the measurement might be delayed randomly by one or more sample times, are investigated. The delayed observations are modelled by up to N sample times by using N Bernoulli random variables with values of 0 or 1. We seek to minimize variance over a class of filters which are linear in the current measurement (although potentially nonlinear in past measurements) and present a closed-form solution. An interpretation of the multiplicative noise in both transition and measurement equations in terms of filtering under additive noise and stochastic perturbations in the parameters of the state space system is also provided. This filtering algorithm extends to the case when the system has continuous time state dynamics and discrete time state measurements. The Euler scheme is used to transform the process into a discrete time state space system in which the state dynamics have a smaller sampling time than the measurement sampling time. The number of sample times by which the observation is delayed is considered to be uncertain and a fraction of the measurement sample time. The same problem is considered for nonlinear state space models of discrete time systems, where the measurement might be delayed randomly by one sample time. The linearisation error is modelled as an additional source of noise which is multiplicative in nature. The algorithms developed are demonstrated throughout with simulated examples.

Active noise control (ANC) is a real-time process in which a system measures an external, unwanted sound source and produces a canceling waveform. The cancellation is due to destructive interference by a perfect copy of the received signal phase-shifted by 180 degrees. Existing active noise control systems process the incoming and outgoing audio on a sample-by-sample basis, requiring a high-speed digital signal processor (DSP) and analog-to-digital converters (ADCs) with strict timing requirements on the order of tens of microseconds. These timing requirements determine the maximum sample rate and bit size as well as the maximum attenuation that the system can achieve. In traditional noise cancellation systems, the general assumption is that all unwanted sound is indeterminate. However, there are many instances in which an unwanted sound source is predictable, such as in the case of a song. This thesis presents a method for active acoustic cancellation of a known audio signal using the frequency characteristics of the known audio signal compared to that of a sampled, filtered excerpt of the same known audio signal. In this procedure, we must first correctly locate the sample index for which a measured audio excerpt begins via the cross-correlation function. Next, we obtain the frequency characteristics of both the known source (WAVE file of the song) and the measured unwanted audio by taking the Fast Fourier Transform (FFT) of each signal, and calculate the effective environmental transfer function (degradation function) by taking the ratio of the two complex frequency-domain results. Finally, we attempt to recreate the environmental audio from the known data and produce an inverted, synchronized, and amplitude-matched signal to cancel the audio via destructive interference. Throughout the process, we employ many signal conditioning methods such as FIR filtering, median filtering, windowing, and deconvolution. We illustrate this frequency-domain method in Native Instruments’ LabVIEW running on the Windows operating system, and discuss its reliability, areas for improvement, and potential future applications in mobile technologies. We show that under ideal conditions (unwanted sound is a known white noise source, and microphone, loudspeaker, and environmental filter frequency responses are all perfectly flat), we can achieve a theoretical maximum attenuation of approximately 300 dB. If we replace the white noise source with an actual song and the environmental filter with a low-order linear filter, then we can achieve maximum attenuation in the range of 50-70 dB. However, in a real-world environment, with additional noise and imperfect microphones, speakers, synchronization, and amplitude-matching, we can expect to see attenuation values in the range of 10-20 dB.