It also has the best stop band attenuation The filter designed using firpm has

# It also has the best stop band attenuation the filter

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It also has the best stop band attenuation.The filter designed usingfirpmhas the lowest order and it meetsthe specifications.We had to adjust the initial design to match the specifications withthe lowest order for all of the examples.C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201993 / 248
Simulating the DifferentiatorThere are several different approaches to the design of IIR digitalfilters [1].We will discuss some of these in this section.A very common practice is to design discrete–time systems usingcontinuous–time prototypes by modeling the differentiator.The differentiator can be represented in the Laplace Transformdomain by the Laplace variables.C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201994 / 248
Simulating the DifferentiatorFor example, the differentiator may be modeled using thedifference operatordy(t)dtt=nT=y(nT)-y(nT-T)T(41)or equivalentlys=1-z-1T(42)C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201995 / 248
Simulating the DifferentiatorWe can continue in this manner to determine the secondderivative asd2y(t)dtt=nT=ddtdy(t)dtt=nT=y2(nT)-2y(nT-T) +y(nT-2T)T2(43)or equivalentlys2=(1-z-1)2T2(44)C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201996 / 248
Simulating the DifferentiatorThis approach works reasonably well when the sampling intervalis relatively small.Figure 19 shows the plot of the magnitude response for the idealdifferentiatorsjΩwith the difference operator forT=1 wheres1-e-jω.C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201997 / 248
Simulating the Differentiator00.511.522.533.500.511.522.533.5Normalized frequency (radians)MagnitudeIdealDiff. OperatorFigure:Magnitude response of the ideal differentiator compared to themagnitude response of the difference operator.C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201998 / 248
Simulating the DifferentiatorLet us further consider the consequences of mapping from thes–plane to the Z–plane using [1]s1-z-1T(45)or equivalentlyz11-sT(46)If we replacesbyjΩ, we obtainz=11-jΩT=11+ Ω2T2+jΩT1+ Ω2T2(47)C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 201999 / 248
Simulating the DifferentiatorAsΩgoes from-∞to, we see that the corresponding locus ofpoints in the Z–plane is a circle of radius12with center atz= (12,0).This is shown if Figure 20 where the imaginary axis in the S–planeis mapped inside the unit circle in the Z–Plane to the dotted circlewith centerz= (12,0)and radius12.C. Williams & W. Alexander (NCSU)DESIGN OF DIGITAL FILTERSECE 513, Fall 2019100 / 248
Simulating the Differentiator-1-0.8-0.6-0.4-0.200.20.40.60.81-1-0.8-0.6-0.4-0.200.20.40.60.81Real PartImaginary PartIdealDiff. OperatorFigure:Magnitude response of the ideal differentiator compared to themagnitude response of the difference operator.