For functions that vary with time, let st be a continuous function or signal to be sampled, and let sampling be performed by measuring the value of the continuous function every t seconds, which is called the sampling interval or the sampling period. It is a little bit trickier than the first one, though, so here is a hint. In this lecture, we look at sampling in the frequency domain, to explain why we must sample a signal at a frequency greater than the nyquist frequency. Sampling theorem article about sampling theorem by the free. The essential content of the theorem is that you cant make money in expectation by. An example of a doobs process is an edge exposure martingale, which helps to. Then the sampled function is given by the sequence. The theorem that a signal that varies continuously with time is completely determined by its values at an infinite sequence of equally spaced times if the frequency of these sampling times is greater than twice the highest frequency component of the signal. A continuous time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to the twice. Doobs optional stopping theorem the doobs optional. This result follows easily because, as we have seen mtn.
The chapter finishes with the investigation of random stopping times with an infinite time horizon. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem. It was the french version of this book that francis pitard digested and shortened to produce his volume pierre gys sampling theory and sampling practice, heterogeneity, sampling correctness and statistical process control. Fundamental inequalities, convergence and the optional. In probability theory, the optional stopping theorem or doob s optional sampling theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Optional sampling let x be a submartingale with respect to fn.
Lecture 18 the sampling theorem university of waterloo. In probability theory, the optional stopping theorem or doobs optional sampling theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. Sampling is the process of selecting a subset of observations from an entire population of interest so that characteristics from the subset sample can be used to draw conclusion or making. Sampling theory representing continuous signals with discrete numbers roger b. The second edition of this book has become a world famous publication used by many practitioners and is taught in. The sampling theorem a1 123 experiment taking samples in the first part of the experiment you will set up the arrangement illustrated in figure 1.
The shannon sampling theorem and its implications gilad lerman notes for math 5467 1 formulation and first proof the sampling theorem of bandlimited functions, which is often named after shannon, actually predates shannon 2. Sampling theory in signal and image processing c 2005 sampling publishing vol. An overview of pierre gys contribution to sampling ausimm. Since martingales can be used to model the wealth of a gambler participating in a fair game.
Theorem doobs stopping theorem let be a filtration defined on a probability space and let be a stochastic process that is adapted to the filtration, whose paths are right continuous and locally bounded. The sampling theorem is of vital importance when processing information as it means that we can take a series of samples of a continuously varying signal and use those values to represent the entire signal without any loss of the available information. For each of the following choices of fo and 0, determine x,t. The sampling theorem and the bandpass theorem by d. Lecture 18 1 doobs martingale process 2 applications of the. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. In order also to obtain these results for submartingales and supermartingales, in the first section, we start with a decomposition theorem for adapted processes. Introduction to sampling theory and data analysis these notes are meant to introduce the ocean scientist and engineer to the concepts associated with the sampling and analysis of oceanographic time series data, and the effects that the sensor, recorder, sampling plan and analysis can have on the results.
Mar 26, 2012 the following first theorem shows that martingales behave in a very nice way with respect to stopping times. Sampling theory in this appendix, sampling theory is derived as an application of the dtft and the fourier theorems developed in appendix c. If t is a stopping time for which pft mg 1 for some m doobs optional sampling theorem says that this equality still holds if the times are replaced by bounded stopping times. It had been called the shannon sampling theorem as early as 1954, but also just the sampling theorem by several other books in the early 1950s. P be a probability space, f ff nga ltration on, and x fx nga martingale with respect to f. The term nyquist sampling theorem capitalized thus appeared as early as 1959 in a book from his former employer, bell labs, and appeared again in 1963, and not capitalized in 1965. The sampling theorem is easier to show when applied to sampling rate conversion in discretetime, i. Sampling is the process of selecting a subset of observations from an entire population of interest so that characteristics from the subset sample can be.
Optional sampling theorem 5 2 walds identities often additional properties of thold typically et theorem holds. In the previous lecture we introduced the optional sampling theorem. A continuous time signal can be represented in its samples and can be recovered back when sampling frequency fs is. The martingale stopping theorem dartmouth mathematics. Optional sampling theorem 5 2 walds identities often additional properties of thold typically et sampling theorem statement. Nyquist sampling f d2, where dthe smallest object, or highest frequency, you wish to record. Doobs optional sampling theorem states that the properties of martingales. Application of optional sampling theorem in a proof of the burkholderdavisgundy inequalitiy. First, we must derive a formula for aliasing due to uniformly sampling a continuoustime signal. Conditions will be such that the requirements of the sampling theorem, not yet given, are met. The doobs optional stopping time theorem is contained in many basic texts on probability and martingales.
Electronic storage and transmission of signals and images has been of obvious importance in our civilization. To overcome this, the band pass theorem states that the input signal xt can be converted into its samples and can be recovered back without distortion when sampling frequency f s theorem implies that there is a sufficiently high sampling rate at which a bandlimited signal can be recovered exactly from its samples, which is an important step in the processing of continuous time signals using the tools of discrete time signal processing. Nyquistshannon sampling theorem statement of the sampling theorem. We show the optional sampling and optional stopping theorems in the second section. Consider a bandlimited signal xt with fourier transform x slide 18 digital signal processing. Dannenberg interpolationreconstruction convolve with a sinc function in other words, form the superposition of. Other applications that follow from doobs optional sampling theorem in. The nyquist theorem states that in order to adequately reproduce a signal it should be periodically sampled at a rate that is 2x the highest frequency you wish to record. And, we demonstrated the sampling theorem visually by showing the reconstruction of a 1hz cosine wave at various sampling frequencies above and below the nyquist frequency.
The following two lemmas help clarify the definition of ft. X such that the stopped process x is ui, then 3 holds. In probability theory, the optional stopping theorem says that, under certain conditions, the expected value of a martingale at a stopping. Doobs optional stopping theorem the doobs optional stopping time theorem is contained in many basic texts on probability and martingales. If f2l 1r and f, the fourier transform of f, is supported. Your expected fortune when stopping is the same as when you started. Specifically, for having spectral content extending up to b hz, we choose in form. Implementations of shannons sampling theorem, a time. The lowpass sampling theorem states that we must sample at a rate, at least twice that of the highest frequency of interest in analog signal. Mar 03, 2014 by the optional stopping theorem we have that.
Lecture 18 the sampling theorem relevant section from boggess and narcowich. Martingales and the optional stopping theorem math. From the telephone, to radio, and then to television, engineers and scientists have. Limit theorem entitles us to the assumption that the sampling distribution is gaussianeven if the population from which the samples are drawn does not follow a gaussian distributionprovided we are dealing with a large enough sample. For a statistician, large enough generally means 30 or greater as a rough rule of thumb although. I believe we should now take the expectation of the expression i derived and set it equal to one by optional sampling theorem, but i dont know what follows. Dannenberg professor of computer science, art, and music. I would like to ask the reader to try to answer the second question. Sampling theorem sampling theorem a continuoustime signal xt with frequencies no higher than f max hz can be reconstructed exactly from its samples xn xnts, if the samples are taken at a rate fs 1ts that is greater than 2f max.
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