 By Dietrich Schlichthärle

The moment, considerably enlarged variation of the textbook supplies a finished perception into the features and the layout of electronic filters. It in brief introduces the speculation of continuous-time structures and the layout equipment for analog filters. Discrete-time platforms, the elemental constructions of electronic filters, sampling theorem, and the layout of IIR filters are generally mentioned. the writer devotes very important components to the layout of non-recursive filters and the results of finite sign in size. the reason of strategies like oversampling and noise shaping concludes the book.

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Extra resources for Digital Filters: Basics and Design

Example text

1) for p = jω as the starting point. 1) can never exceed unity. The characteristic function K(p) determines the kind of approximation to the ideal filter. For a lowpass filter, K(jω) has the property of assuming low values between ω = 0 and a certain cutoff frequency ω c, but increasing rapidly above this frequency. K(p) may be a polynomial as in the case of Butterworth, Chebyshev and Bessel filters. For inverse Chebyshev and Cauer filters, K(p) is a rational fractional function. In general, we can express the characteristic function as K ( p) = v( p ) .

In particular, we have K(0) = π . 34) As k approaches 1, K(k) can be approximated  by  4 K(k ) ≈ ln   2  1− k   = ln  4  . 0 k Fig. 2-12 Complete elliptic integral of the first kind Along the dotted paths in Fig. 2-11, the Jacobian elliptic function sn(u,k) assumes real values. On a horizontal line through the zeros where u = u1+j2nK ' (n integer), sn(u,k) oscillates between −1 and +1 with the period 4K. For small values of k, sn(u1+j2nK ') can, in fact, be approximated with good accuracy by a sine wave:  π u1   sn(u1 + j 2n K' , k ) ≈ sin   2 K(k )  n integer .

The first possible approach is to replace p with jω to obtain the frequency response H(jω) of the system. From the mathematics of complex numbers we know that the squared magnitude can be obtained as the product of a given complex number and its complex-conjugate. Thus we can write 2 H (jω ) = H (jω ) H ∗ (j ω ) . Since H(jω ) is a rational function of jω with real coefficients, it follows that H ∗ (j ω ) = H (− j ω ) . The squared magnitude can therefore be expressed as 2 H (jω ) = H (jω ) H (− j ω ) . 