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Understanding digital signal processing - Chapter 8

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Tài liệu tham khảo giáo trình tìm hiểu xử lý tín hiệu số bằng tiếng anh ( Understanding digital signal processing ) Chương 8. tín hiệu trung bình | CHAPTER EIGHT- Signal Averaging How do we determine the typical amount a valid estimate or the true value of some measured parameter In the physical world it s not so easy to do because unwanted random disturbances contaminate our measurements. These disturbances are due to both the nature of the variable being measured and the fallibility of our measuring devices. Each time we try to accurately measure some physical quantity we ll get a slightly different value. Those unwanted fluctuations in a measured value are called noise and digital signal processing practitioners have learned to minimize noise through the process of averaging. In the literature we can see not only how averaging is used to improve measurement accuracy but that averaging also shows up in signal detection algorithms as well as in low-pass filter schemes. This chapter introduces the mathematics of averaging and describes how and when this important process is used. Accordingly as we proceed to quantify the benefits of averaging we re compelled to make use of the statistical measures known as the mean variance and standard deviation. In digital signal processing averaging often takes the form of summing a series of time-domain signal samples and then dividing that sum by the number of individual samples. Mathematically the average of N samples of sequence x n denoted xave is expressed as N 2 V rM x l x 2 x 3 . x N xave z M 1 8-1 What we call the average statisticians call the mean. In studying averaging a key definition that we must keep in mind is the variance of the sequence Ơ2 defined as 319 320 Signal Averaging ữ2 ỵw-xm 2 8-2 _ x l - xave 2 x 2 - xave 2 x 3 - xave 2 . x N - xave 2 --------------------------------. 8-2 A.S explained in Appendix D the Ơ2 variance in Eqs. 8-2 and 8-2 gives us a well-defined quantitative measure of how much the values in a sequence fluctuate about the sequence s average. That s because the x l - xave value in the bracket for example is the difference between the