IIR Filters FIR Filters Phase difficult to control,no particular linear phase always possible (grp delay） techniques available Stability can be unstable, always stable, can have limit cycles no limit cycles Order less more History derived from analog filters no analog history Others polyphase implementation possible can always be made causal The only way to achieve integer or fractional constant delays is by using FIR filters. Limit cycles are instability of a particular type due to quantization,which is severely non-linear. The number of arithmetic operations needed per unit time is directly related to filter order.#MAD stands for number of multiplications and additions.When using this as a criterion for comparing different filters,one should pay attention to whether the #MAD is measured per input sample,per output sample,or per unit time(clock cycle).It is possible that an IIR of lower order actually requires more #MAD than an FIR of higher order,because FIR filters may be implemented using polyphase structures. Different conventions exist for specifying magnitude responses for IIR and FIR filters. In particular,IIR filter specs are normalized to [1-A1,1](dB)in the passband,and [-oo,A2](dB)in the stopband;FIR filter specs are normalized to between 1+61 within the passband,and t62 in the stopband,where 01,62 are given as decimals. IIR Filter Design Historically,digital IIR filters have been derived from their analog counterparts.There are several common types of analog filters:Butterworth which have maximally flat passbands in filters of the same order,Chebyshev type I which are equiripple in the passband,Chebyshev type II which are equiripple in the stopband,and Elliptic filters which are equiripple in both the passband and the stopband.The digital version of these can be obtained from analog designs through the bilinear transformation,discussed in detail in OSB Sections 7.1.2 and 7.1.3. In short,the bilinear transformation is an algebraic mapping from the continuous frequency variable s to the discrete frequency variable z such that the imaginary axis in the s-plane corresponds to one revolution of the unit circle in the z-plane: 、1-z-1 .2w2 →j0=j斤tan2 tanw/2 8一1+27 n2’e tanwe/2 r in the digital frequency domain corresponds to infinity in the analog frequency domain.Note that the bilinear transformation is really only appropriate in mapping filters which approximate piecewise constant filters. 2IIR Filters FIR Filters Phase difficult to control, no particular linear phase always possible (grp delay) techniques available Stability can be unstable, always stable, can have limit cycles no limit cycles Order less more History derived from analog filters no analog history Others polyphase implementation possible can always be made causal • The only way to achieve integer or fractional constant delays is by using FIR filters. • Limit cycles are instability of a particular type due to quantization, which is severely non-linear. • The number of arithmetic operations needed per unit time is directly related to filter order. #MAD stands for number of multiplications and additions. When using this as a criterion for comparing different filters, one should pay attention to whether the #MAD is measured per input sample, per output sample, or per unit time (clock cycle). It is possible that an IIR of lower order actually requires more #MAD than an FIR of higher order, because FIR filters may be implemented using polyphase structures. • Different conventions exist for specifying magnitude responses for IIR and FIR filters. In particular, IIR filter specs are normalized to [1 − Δ1, 1](dB) in the passband, and [−∞, Δ2](dB) in the stopband; FIR filter specs are normalized to between 1 ± δ1 within the passband, and ±δ2 in the stopband, where δ1, δ2 are given as decimals. IIR Filter Design Historically, digital IIR filters have been derived from their analog counterparts. There are several common types of analog filters: Butterworth which have maximally flat passbands in filters of the same order, Chebyshev type I which are equiripple in the passband, Chebyshev type II which are equiripple in the stopband, and Elliptic filters which are equiripple in both the passband and the stopband. The digital version of these can be obtained from analog designs through the bilinear transformation, discussed in detail in OSB Sections 7.1.2 and 7.1.3. In short, the bilinear transformation is an algebraic mapping from the continuous frequency variable s to the discrete frequency variable z such that the imaginary axis in the s-plane corresponds to one revolution of the unit circle in the z-plane: 1 − z−1 2 ω Ω tan ω/2 s → 1 + z−1 ⇒ jΩ = j tan 2 , T Ωc → tan ωc/2 π in the digital frequency domain corresponds to infinity in the analog frequency domain. Note that the bilinear transformation is really only appropriate in mapping filters which approximate piecewise constant filters. 2