barcode lib ssrs Impulse Invariance in Software

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Impulse Invariance
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FILTER DESIGN
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With the inzpulse invariance method, a digital filter is designed by sampling the impulse response of an analog filter: h ( n )= h,,(nT,) From the sampling theorem, it follows that the frequency response of the digital filter, H(eJW), related to the is frequency response H,(jR) of the analog filter as follows:
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More generally, this may be extended into the complex plane as follows:
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The mapping between the s-plane and the z-plane is illustrated in Fig. 9-1 I . Note that although the jR-axis maps onto the unit circle, the mapping is not one to one. In particular, each interval of length 2 n / T , along the jR-axis is mapped onto the unit circle (i.e., the frequency response is aliased). In addition, each point in the left-half s-plane is mapped to a point inside the unit circle. Specifically, strips of width 2 n / T , map onto the z-plane. If the frequency response of the analog filter, H,,(jR). is sufficiently bandlimited, then
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Although the impulse invariance may produce a reasonable design in some cases, this technique is essentially limited to bandlimited analog filters.
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To see how poles and zeros of an analog filter are mapped using the impulse invariance method, consider an analog filter that has a system function
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The impulse response, hu(t), is
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FILTER DESIGN
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 s-plane to the z-plane 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 left-half s-plane inside the unit circle and maps the jC2-axis in a one-to-one manner onto the unit circle. However, the relationship between the jR-axis and the unit circle is highly nonlinear and is given by thefr-equency 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 jR-axis 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 low-pass 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 low-pass filter. With T, = 2, the prewarping function is
Design an analog low-pass 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 first-order digital low-pass filter with a 3-dB 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
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Because the 3-dB 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 low-pass filters from analog low-pass filters. There are two approaches that may be used to design other types of frequency selective filters, such as high-pass, bandpass, or bandstop filters. The tirst is to design an analog low-pass 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 s-plane to z-plane mapping. Table 9-5 provides a list of some analog-to-analog transformations.
Table 9-5 The 'kansformation of an Analog Low-pass Filter with a 3-dR Cutoff Frequency L2, to Other F'req"ency selective Filters
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