Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

General Characteristics of Filters 1-15With v ¼ s=j in Equation 1.18, the functionL s 2 ¼D(s)D( s)N(s)N( s)can be formed. This is called the loss function of the filter and, as is evident, its zeros are the poles of H(s)and their negatives, whereas its poles are the zeros of H(s) and their negatives.1.5 Ideal and Practical FiltersAn ideal low-pass filter is one that will pass only low-frequency components. Its loss characteristic isgiven byA(v) ¼ 0 for 0 v < v c1 for v c < v < 1The frequency ranges 0 to v c and v c to 1 are the passband and stopband, respectively. The boundarybetween the passband and stopband, namely, v c , is the cutoff frequency. An ideal high-pass filter will passall components with frequencies above the cutoff frequency and reject all components with frequenciesbelow the cutoff frequency, i.e.,A(v) ¼ 1 for 0 v < v c0 for v c < v < 1Idealized loss characteristics can similarly be identified for bandpass and bandstop filters as8< 1 for 0 v < v c1A(v) ¼ 0 for v c1 < v < v:c21 for v c2 v < 1and8< 0 for 0 v < v c1A(v) ¼ 1 for v c1 < v < v:c20 for v c2 v < 1respectively.Practical filters differ from ideal ones in that the passband loss is not zero, the stopband loss is notinfinite, and the transition between passband and stopband is gradual. Practical loss characteristics forlow-pass, high-pass, bandpass, and bandstop filters assume the forms( A pA LP (v)for 0 v < v p A a for v a v 1( A a for 0 v v aA HP (v) A p for v p v < 18 A a>: A a for v a2 v 1