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barcode lib ssrs Impulse Invariance in Software
Impulse Invariance Decode Code 128 Code Set A In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Drawing Code 128 In None Using Barcode creator for Software Control to generate, create Code128 image in Software applications. FILTER DESIGN
Code 128 Code Set A Scanner In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Code 128A Drawer In Visual C#.NET Using Barcode generator for .NET framework Control to generate, create Code 128 Code Set A image in VS .NET applications. [CHAP. 9
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Therefore, the digital filter that is formed using the impulse invariance technique is
and the system function is
Thus, a pole at s = sk in the analog filter is mapped to a pole at z = tJkTs
in the digital filter.
The zeros, however, do not get mapped in any obvious way.
The Bilinear Transformation
The bilinear transformation is a mapping from the splane to the zplane defined by
Given an analog filler with a system function H,(s), the digital filter is designed as follows: H (z) = Hu
1(2 )+ 1
The bilinear transformation is a rational function that maps the lefthalf splane inside the unit circle and maps the jC2axis in a onetoone manner onto the unit circle. However, the relationship between the jRaxis and the unit circle is highly nonlinear and is given by thefrequency warpb~g,funcrion As a result of this warping, the bilinear transformation will only preserve the magnitude response of analog filters that have an ideal response that is piecewise constant. Therefore, the bilinear transformation is generally only used in the design of frequency selective filters. The parameter T, in the bilinear transformation is normally included for historical reasons. However, it does not enter into the design process, because i l only scales the jRaxis in the frequency warping function, and this scaling may be done in the specification of the analog filter. Therefore, T, may be set to any value to simplify the design procedure. The steps involved in the design of a digital lowpass filter with a passband cutoff frequency w,, stopband cutoff frequency w.,, passband ripple 6 , , and stopband ripple 8, are as follows: Prewarp the passband and stopband cutoff frequencies of the digital filter, w, and o,, , using the inverse of Eq. (9.12) to determine the passband and cutoff frequencies of' the analog lowpass filter. With T, = 2, the prewarping function is Design an analog lowpass filter with the cutoff frequencies found in step I and a passband and stopband ripple ap and a,, respectively. Apply the bilinear transformation to the filter designed in step 2. EXAMPLE 9.4.3 Let us design a firstorder digital lowpass filter with a 3dB cutoff frequency of w, = 0.2% by applying the bilinear transformation to the analog Butterworth filter I H,,(s) = 1 s j R,. FILTER DESIGN
[CHAP. 9
Because the 3dB cutoff frequency of the Butterworth filter is R,., for a cutoff frequency w, = 0.2% in the digital filter, we must have Therefore, the system function of the analog ti lter is
Applying the bilinear transformation to the analog filter gives
Note that the parameter T, does not enter into the design.
9.4.3 Frequency Transformations
The preceding section considered the design of digital lowpass filters from analog lowpass filters. There are two approaches that may be used to design other types of frequency selective filters, such as highpass, bandpass, or bandstop filters. The tirst is to design an analog lowpass filter and then apply a frequency transformation to map the analog filter into the desired frequency selective prototype. This analog prototype is then mapped to a digital filter using a suitable splane to zplane mapping. Table 95 provides a list of some analogtoanalog transformations. Table 95 The 'kansformation of an Analog Lowpass Filter with a 3dR Cutoff Frequency L2, to Other F'req"ency selective Filters

