By O. Kulhánek
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Additional resources for Introduction to Digital Filtering in Geophysics
11] which is the complex response function of the filter. By moving the complex variable ζ along the unit circle in the z-plane, or in other words by varying the angular frequency ω in the interval of interest, the amplitude and phase response of the filter are determined. To obtain the system function in a polynomial form and the unit-impulse response, we perform the long division a/(l + bz- 1) or determine the output sequence corresponding to the input impulse χ = (1, 0, 0 , . . ) . The response to the input sequence χ is the impulse response, and its ζ transform is H(z).
The real- and imaginary-part separa tions shown in the example above are usually rather laborious even for simple systems. Further and more importantly, the frequency selectivity of the response function is not immediately visible from either of these two definitions. This of course complicates the design. In the following section we shall discuss the socalled pole-zero technique which in general is an effective method with a direct physical interpretation. 2 we have considered system functions which were either polynomials or rational fractions of polynomials in z~ l.
Plane, σ = 0 and the absolute value of ζ is unity. This implies, as we have already discussed, that the section of the imaginary axis of the />-plane lying between ω = 0 and ω = 2π/Δί = ω8 is mapped into the unit circle of the z-plane. Other parts of the imaginary axis just overlap on the unit circle. 37] that any line, σ = constant, in the p-plane is mapped into the z-plane as a circle with its centre at ζ = 0 and radius toAt. ρ(σ < 0)'e oM For any point in the left halfplane of < 1 and | ζ J < 1.
Introduction to Digital Filtering in Geophysics by O. Kulhánek