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barcode lib ssrs so the minimum order is N = 8. Finally, for the elliptic filter, we first evaluate in Software
so the minimum order is N = 8. Finally, for the elliptic filter, we first evaluate Code128 Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generate USS Code 128 In None Using Barcode creator for Software Control to generate, create Code 128A image in Software applications. where
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Code128 Creator In Visual Studio .NET Using Barcode printer for ASP.NET Control to generate, create Code128 image in ASP.NET applications. Code 128B Printer In .NET Using Barcode generator for .NET Control to generate, create Code 128 Code Set B image in .NET framework applications. With k = 0.6682, we have
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Making GTIN  128 In ObjectiveC Using Barcode creation for iPhone Control to generate, create EAN / UCC  13 image in iPhone applications. Painting GTIN  13 In None Using Barcode printer for Font Control to generate, create GTIN  13 image in Font applications. the impulse response is ha(t) = tesk'u(t) where u(t) is the unit step function. Sampling h,(t) with a sampling period T,, we have Using the ztransform property nx(n) and the ztransform pair cunu(n) c4t it follows that the ztransform of h(n) is dX(z) dz I
 az' Therefore, for a secondorder pole, we have the mapping
Suppose that we would like to design and implement a lowpass filter with
( a ) What order FIR equiripple filter is required to satisfy these specifications (b) Repeat part ( a ) for an elliptic filter. (c) Compare the complexity of the implementations for the equiripple and elliptic filters in terms of the number of coefficients that must be stored, the number of delays that are required, and the number of multiplications necessary to compute each output sample y ( n ) . (a) With a transition width of Aw = 0.027r. an estimate of the required filter order for an FIR equiripple filter is [CHAP. 9
(b) For an elliptic filter, we have
With
then o r N = 10.
(c) For an FIR filter of order N = 254, the output y(n) is
Therefore, implementing this filter requires N = 254 delays. Since this filter has linear phase, exploiting the symmetry of the unit sample response, h(n) = h(254  n) it follows that we must only provide storage for 128 filter coefficients, h(O), h(l), . . . , h(127). In addition, we may simplify the evaluation of y(n) as follows, y(n) = ) h(k)x(n  k) = ) h(k)[x(n  k) + x(n  254 + k)] + h(127)x(n  127) A LO
Thus, 128 multiplications are required to compute each value of y(n). For a 10thorder elliptic filter, Therefore, it follows that 21 memory locations are required to store the coefficients a(k) and b(k), and 10 delays are required for a canonic implementation. In addition, we see that 21 multiplications are necessary to evaluate each value of y(n). However, since the zeros of H ( z ) lie on the unit circle, the coefficients b(k) are symmetric, By exploiting this symmetry, we may eliminate five multiplications per output point and five memory locations. The input x a ( t ) and output ya(t) of a continuoustime filter with a rational system function are related by a linear constant coefficient differential equation of the form Suppose that we sample x,(t) and ya(t), and approximate a first derivative with the first backward difference, CHAP. 91
FILTER DESIGN
Approximations to higherorder derivatives are then defined as follows, Applying these approximations to the differential equation gives the following approximation to the differential equation: The first backward difference def nes a mapping from the splane to the zplane that is given by
,y= 1  z' Determine the characteristics of this mapping, and compare it to the bilinear transformation. Is this a good mapping to use Explain why or why not. As with the bilinear transformation, the first backward difference will map a rational function of s into a rational function of z. To see how points in the splane map to points in the zplane, let us write the mapping as follows, Note that with s = u
+jR, .and it follows that points in the lefthalf splane (a c 0) are mapped to points inside the unit circle, lzl < I . Thus, stable analog filters are mapped to stable digital filters. Now, let us look at how the jRaxis is mapped to the zplane. With s = j R , we see that which is an equation for a circle of radius r = f centered at z =
i. To see this, note that
Thus. The properties of the mapping are illustrated in the following figure.
Since the jRaxis does not map onto the unit circle, the frequency response of the digital filter produced with this mapping will not, in general. be an accurate representation of the frequency response of the analog filter except when w is close to zero. In other words, the frequency response of a continuoustime filter will be well preserved only for low frequencies.

