bilinear - ECE 421 - Sum 2010 Notes Set 5: Filter Design by...

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Unformatted text preview: ECE 421 - Sum 2010 Notes Set 5: Filter Design by Mappings 1 INTRODUCTION In a previous Notes Set we derived digital filters using a time-domain diffferential equa- tion which described an analog filter. In this Notes Set we instead use the analog S-domain transfer function as the basis for designing the digital filter. In this way we can use an exist- ing H s which has been found to work well in analog systems, and derive a corresponding digital filter which should work well in digital systems. The type of method we will use is sometimes called a Mapping (or Transformation) from the S-plane to the Z-plane. There are several of these mappings which are useful, but we will limit our current work to the methods called the Backward Integration Mapping and the Bilinear Mapping. In addition to providing useful digital filters, these mappings can often give us an idea of the performance of the resulting digital filters. ECE 421 - Sum 2010 Notes Set 5: Filter Design by Mappings 2 ANALOG INTEGRATOR To derive the desired mappings, we will use the application of an integrator to determine the appropriate S-to- Z formula. The analog integrator has causal input x t and output y t given by y t = Z = t = x d 1 Suppose t = t 1 in (1). Then (1) becomes y t 1 = Z = t 1 = x d 2 The physical interpretation of (2) is y t 1 is the Area under the x t curve from t = 0 to t = t 1 . The Laplace transform of equation (1) is given by Y s = 1 s X s 3 A transfer function H s is defined by the following ratio: H s = Y s X s 4 Therefore, solving (3) for the ratio H s = Y s = X s gives H s = 1 s 5 Equation (5) gives the transfer function for the analog integrator. We will next use Z- transforms to derive the transfer fuction H z of a digital integrator. We will then compare H s given in (5) with the H z for the digital integrator to determine the appropriate S-to- Z mapping. ECE 421 - Sum 2010 Notes Set 5: Filter Design by Mappings 3 BACKWARD INTEGRATION Now consider the case of computing the area under a sequence of digital samples us- ing the approximation technique called Backward Integration. Let x [ m be a set of digital samples, separated by T s seconds, and let y [ m be a computation of the area under the se- quence. For example, suppose x [ ; x [ 1 ; x [ 2 , and x [ 3 have been acquired as shown below: t . x [1] x [2] x [3] . s T 2 . s T 3 . s T x [0] Now we examine a digital integrator which uses a rectangular area approximation to com- pute the area under the waveform as a function of digital time.. The first rectangle using x [ 1 is shown below: T s [1] x [1] y . Area of Rectangle: Define y [ 1 as the area of the rectangle shown in the above figure. The computation of this area is seen to be y [ 1 = T s x [ 1 7 Now lets add a second rectangle, as shown on the following page. ECE 421 - Sum 2010 Notes Set 5: Filter Design by Mappings 4 BACKWARD INTEGRATION (cont.) Now add the area of the second rectangle created using...
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This note was uploaded on 09/07/2010 for the course ECE 421 taught by Professor Hallen during the Summer '08 term at N.C. State.

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bilinear - ECE 421 - Sum 2010 Notes Set 5: Filter Design by...

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