Probabilistic System Assignment

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Probability, random variables and random processes have applications in a wide range of fields. One application of probability theory is in the stock market where investors take advantage of the random fluctuations to maximize their benefits. These random fluctuations can be described using probabilistic theory [1]. Another discipline that makes use of probability is the insurance business. Car insurance premiums are calculated based on the probability and inherent risks of drivers of a certain class that are most likely to cause accidents.
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Research is being conducted mainly in the field of computer recognition of speech. The aim is to recognize commands spoken to a computer by making use of a procedure known as template matching. A discrete set of words are defined for the computer vocabulary. The template of the word is then created as the word is spoken. The template may be stored as a time waveform, spectrum of the word or a vector of features of the word [3].
Speech recognition by computers is a challenging task due to the factors that need to be considered. These include interference from the background, speaker’s pronunciation and pitch, the duration of the spoken word and variability of amplitude, just to mention a few. The application of stochastic processes and probability theory then become essential in developing models for speech production and recognition [3]. These models may be used to design systems for speech recognition. From speech recognition, a computer can be used to predict the speech of a user. This is done by predicting the future values of a waveform using current and previous samples. This is used mainly in speech coders where the signal-to-qauntization noise associated with the quantizer can be increased by quantizing the prediction error. The figure below illustrates the block diagram of a speech coder using differential pulse code
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This variable represents the samples of each speech waveform. These inputs go through the predictor which is tasked to best estimate Xn given that the past samples are known. This process is achieved using linear prediction, causing the predictor to output linear combinations of the samples. Assuming that the predictor uses the last m samples to determine its estimates, it would have the output as shown in equation 1.
Y_n=∑_(i=1)^m▒〖a_i X_(n-i) 〗 (1)

The constants ai are selected to optimize the performance of the predictor. The difference between Xn and Yn result in the quantity Zn which is known as the predictor error and should be kept as low as possible. Zn is passed through the level 2b quantizer where each level is encoded with a b bit code word [3]. The original samples can be recovered from the binary representation from the quantizer inputs

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