## On the Frequency-domain Responses of Causal Digital Filters |

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### Contents

FREQUENCYDOMAIN ANALYSIS | 13 |

ON THE EQUIVALENCE OF | 58 |

The FrequencyTransfer Function of a | 129 |

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### Common terms and phrases

all-pass analogue approximation problem analogue equalizer analogue filters analogue systems angular frequency application attenuation band-pass filters band-stop filter bilinear transformation bounded input causal digital filters Chebychev complex numbers convolution method corresponding analogue approximation denoted difference equation digital approximation problem digital system elliptic filters expansion with real finite finite-energy digital filter Fourier coefficients Fourier series frequency transform frequency-domain responses frequency-transfer function group delay distortion group delay equalization Hardy space holomorphic imaginary implies input sequence Laplace transform linear linear-phase filtering low-pass minimum differences minimum-phase digital filter minimum-phase function nonrecursive filters normalized group delay open unit disc outer function output sequence phase distortion phase equalization physical implementation poles power-series expansion problem for digital problems for analogue Proof properties radial limit rational function real coefficients recursive digital filters recursive filters result solution stable digital filter techniques Theorem thesis transfer function transfer function H(z unit circle weighting sequence zeros