Добавил:
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
Analysis and Application of Analog Electronic Circuits to Biomedical Instrumentation - Northrop.pdf
Скачиваний:
205
Добавлен:
10.08.2013
Размер:
4.41 Mб
Скачать

7

Analog Active Filters

7.1Introduction

Filter (noun) is an electrical engineering term having a broad definition. Filters are generally considered to operate on signals in the frequency domain; that is, they attenuate, stop, pass, or boost certain frequency regions in the input signal’s spectrum. These operations are called filtering (verb). They may be classified as low pass (LP); band pass (BP); high pass (HP); or band reject (BR or notch). Band-pass and band-reject filters can have narrowpass bands or reject bands. One descriptor of such sharply tuned filters is their Q, which can be defined as the BP filter’s center frequency divided by its half-power bandwidth; the narrower the bandwidth is, the higher the Q. (Bandwidth is generally measured as the frequency span between halfpower frequencies, where the filter’s frequency response function is down 3 dB or to 0.7071 times the peak pass gain.)

Filter transfer functions can be expressed in general as rational polynomials in the complex variable, s. The rational polynomials can be factored to find the poles and zeros of the transfer function. The filter’s poles are the roots of the denominator polynomial; its zeros are the roots of the numerator polynomial. Roots are the complex values of s that make a polynomial equal zero. If the highest power of s in the denominator polynomial is n, n poles will be in the complex s-plane. If the filter is stable (and all should be), all of the filter’s poles will lie in the left-half s-plane; however, in limiting cases, conjugate pole pairs can lie on the axis.

Likewise, if the highest power of s in the numerator polynomial is m, m zeros will be in the s-plane. The zeros can lie anywhere in the s-plane; if some lie in the right-half s-plane, the filter is called nonminimum phase. If all the zeros lie in the left-hand s-plane, the filter is called minimum phase. The poles and zeros can have real values or occur in complex-conjugate pairs of the form, s = α ± jβ, where β is nonnegative and −• < α < . Conjugate zeros on the axis are found in notch filters. A filter’s frequency response is found by letting s = in the filter’s transfer function and then finding 20 times the log of the magnitude of the frequency response function vs. f in Hertz and the angle of the frequency response function vector vs. f.

281

© 2004 by CRC Press LLC