barcode lib ssrs so the minimum order is N = 8. Finally, for the elliptic filter, we first evaluate in Software

Generator USS Code 128 in Software so the minimum order is N = 8. Finally, for the elliptic filter, we first evaluate

so the minimum order is N = 8. Finally, for the elliptic filter, we first evaluate
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where
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CHAP. 91
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With k = 0.6682, we have
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FILTER DESIGN
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and Therefore. for the filter order. we find
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9 = 0.0369
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With impulse invariance, a first-order pole in H,(s) at s = sk is mapped to a pole in H ( z ) at z = e s k T s :
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Determine how a second-order pole is mapped with impulse invariance.
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If the system function of a continuous-time filter is
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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 z-transform property nx(n) and the z-transform pair cunu(n) c4-t it follows that the z-transform of h(n) is
dX(z) dz I
- az-'
Therefore, for a second-order pole, we have the mapping
Suppose that we would like to design and implement a low-pass 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 10th-order 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 continuous-time 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 higher-order 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 s-plane to the z-plane 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 s-plane map to points in the z-plane, let us write the mapping as follows,
Note that with s = u
+jR,
.and it follows that points in the left-half s-plane (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 jR-axis is mapped to the z-plane. 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 jR-axis 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 continuous-time filter will be well preserved only for low frequencies.
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