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FIR Filters - IIF Filters

Thus implementation of 7 tap filter with input and coefficient as 8 bit width requires around 20x8=160 CLBs(~300 slices). Systolic array architecture even though improves timing by reducing critical path, utilizes same amount of hardware | Chapter 14 HR Filters The general form for an infinite impulse response HR filter s output j f e at time k is given by N M y n any k-n bmx k - m 14.1 n 1 m 0 This equation indicates that the filter s output is a linear combination of the present input the M previous inputs and the N previous outputs. The corresponding system function is given by Y obmz m H z -------------- 14.2 1 Y _1anz n where at least one of the an is nonzero and at least one of the roots of the denominator is not exactly cancelled by one of the roots of the numerator. For a stable filter all the poles of H z must lie inside the unit circle but the zeros can lie anywhere in the z plane. It is usual for M the number of zeros to be less than or equal to N the number of poles. Whenever the number of zeros exceeds the number of poles the filter can be separated into an FIR filter with M N taps in cascade with an IIR filter with N poles and N zeros. Therefore IIR design techniques are conventionally restricted to cases for which M N. Except for the special case in which all poles lie on the unit circle in the z plane it is not possible to design an IIR filter having exactly linear phase. Therefore unlike FIR design procedures that are concerned almost exclusively with the magnitude response IIR design procedures are concerned with both the magnitude response and phase response. 271 272 Chapter Fourteen 14.1 Frequency Response of HR Filters The frequency response of an HR filter can be computed from the coefficients an and bm as o Pm exp jZitmkIL H k -------------------------- 14.3 A exp 27tfc L m n 0 Q n N N n 0 m M M m A C function that uses 14.3 to compute the response for an IIR filter is provided in Listing 14.1. 14.2 IIR Realizations A direct realization of Eq. 14.1 is shown in Fig. 14.1 using the signal flow graph notation introduced in Sec. 4.4. The structure shown is known as the direct form I realization or direct form I structure for the IIR system represented by 14.1 . Examination of the .