Filters-notes - Frequency Sensitive Circuits Filters...

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Frequency Sensitive Circuits – Filters Introduction and Terminology Background needed from Laplace Transform Ideal filters Derivation of transfer functions of some filter circuits Cascading of filters Loading effects in passive filter circuits Computation of half power frequencies and bandwidth Bode plots A simple methodology of filter design Magnitude and frequency scaling Design of filters by cascading the same filter circuit over and over Butterworth LPF design; if time permits Butterworth HPF design; if time permits BPF design by utilizing LPF and HPF BRF design by utilizing LPF and HPF Narrowband BPF and BRF Some of the above material is in Chapter 9 of the text book by Ulaby and Maharbiz, but not all of it. Also, the order of the material covered in the class is different and follows the above indicated order.
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2 Introduction and Terminology A result due to Fourier and Laplace that revolutionized many scientific fields including Electrical Engineering is the concept that any signal can be thought of as being composed by a number (finite or infinite) of sinusoidal signals of different frequencies, amplitudes, and phase angles . The sinusoidal signals that compose a signal x ( t ) are called the frequency components of x ( t ). If a signal x ( t ) has a countably finite or infinite number of frequency components, then x ( t ) = N summationdisplay k =1 X k cos( ω k t + θ k ) , where N is a finite integer or infinite.On the other hand, if the signal x ( t ) has an uncountably infinite number of frequency components, then x ( t ) integraldisplay ω 2 ω = ω 1 X ( ω ) cos( ωt + θ ( ω )) where [ ω 1 , ω 2 ] is the range of frequencies contained in x ( t ). The above concept implies that a signal x ( t ) can be prescribed by two ways: 1. Prescribing x ( t ) directly in terms of time variable t is referred to as Time domain prescription. 2. Prescribing x ( t ) indirectly in terms of its frequency components is referred to as Frequency domain prescription. Time Domain Frequency Domain x ( t ) X k , ω k θ k for k = 1 , 2 , ··· , N x ( t ) X ( ω ) θ ( ω ) for ω 1 ω ω 2 . The plot of magnitude (amplitude) X k or X ( ω ) with respect to ω is called the magnitude (ampli- tude) frequency spectrum of the signal x ( t ). On the other hand, the plot of phase angle θ k or θ ( ω ) with respect to ω is called the phase angle frequency spectrum of the signal x ( t ). The frequency spectra of most of the practical signals occupy only a finite region along the ω axis. Such signals are said to be band-limited signals. For the purpose of transmission of signals, several band-limited signals can be combined to generate a composite signal, each signal having its frequency spectra in a specific region along the frequency axis.
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