CH19 - CHAPTER Recursive Filters 19 Recursive filters are...

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319 CHAPTER 19 Recursive Filters Recursive filters are an efficient way of achieving a long impulse response, without having to perform a long convolution. They execute very rapidly, but have less performance and flexibility than other digital filters. Recursive filters are also called Infinite Impulse Response (IIR) filters, since their impulse responses are composed of decaying exponentials. This distinguishes them from digital filters carried out by convolution, called Finite Impulse Response (FIR) filters. This chapter is an introduction to how recursive filters operate, and how simple members of the family can be designed. Chapters 20, 26 and 33 present more sophisticated design methods. The Recursive Method To start the discussion of recursive filters, imagine that you need to extract information from some signal, . Your need is so great that you hire an old x [ ] mathematics professor to process the data for you. The professor's task is to filter to produce , which hopefully contains the information you are x [ ] y [ ] interested in. The professor begins his work of calculating each point in y [ ] according to some algorithm that is locked tightly in his over-developed brain. Part way through the task, a most unfortunate event occurs. The professor begins to babble about analytic singularities and fractional transforms, and other demons from a mathematician's nightmare. It is clear that the professor has lost his mind. You watch with anxiety as the professor, and your algorithm, are taken away by several men in white coats. You frantically review the professor's notes to find the algorithm he was using. You find that he had completed the calculation of points through y [0] , and was about to start on point . As shown in Fig. 19-1, we will y [27] y [28] let the variable, n , represent the point that is currently being calculated. This means that is sample 28 in the output signal, is sample 27, y [ n ] y [ n & 1] is sample 26, etc. Likewise, is point 28 in the input signal, y [ n & 2] x [ n ]
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The Scientist and Engineer's Guide to Digital Signal Processing 320 y [ n ] a 0 x [ n ] % a 1 x [ n & 1] % a 2 x [ n & 2] % a 3 x [ n & 3] % & y [ n ] a 0 x [ n ] % a 1 x [ n & 1] % a 2 x [ n & 2] % a 3 x [ n & 3] % & % b 1 y [ n & 1] % b 2 y [ n & 2] % b 3 y [ n & 3] % & EQUATION 19-1 The recursion equation. In this equation, is x [ ] the input signal, is the output signal, and the y [ ] a 's and b 's are coefficients. is point 27, etc. To understand the algorithm being used, we ask x [ n & 1] ourselves: "What information was available to the professor to calculate , y [ n ] the sample currently being worked on?" The most obvious source of information is the input signal , that is, the values: . The professor could have been multiplying each point x [ n ], x [ n & 1], x [ n & 2], & in the input signal by a coefficient, and adding the products together: You should recognize that this is nothing more than simple convolution, with the coefficients: , forming the convolution kernel. If this was all the a 0 , a 1 , a 2 , & professor was doing, there wouldn't be much need for this story, or this chapter.
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CH19 - CHAPTER Recursive Filters 19 Recursive filters are...

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