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Systems and Vectors I
Poles and Zeros
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algebraic amplitude analytic approximation bandpass function bandwidth Butterworth function Chebyshev Chebyshev function coefficients complex plane complex-conjugate components constant contour integration convergent curve decibels defined density function derivative determined differential equation discontinuities discussed electric network evaluation example excitation exponential F(jw factor finite poles finite zeros Fourier series Fourier transform function F(p Gaussian hence Hurwitz polynomial infinite infinity interpretation inversion jw axis Laplace transform left half-plane limit linear locus low-pass function magnitude and phase mathematical mean-square value method multiplier negative nepers number of finite obtained odd function operational optimum origin partial-fraction expansion phase angle phasor Plancherel's theorem plot pole-zero poles and zeros positive Prob ratio rational fraction reactance real and imaginary real axis residue result right half-plane semicircle shown in Fig signal simple pole sine wave solution specific spectrum square-magnitude Taylor series theorem they'to axis transfer function U(jw vector waveform yields