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Unformatted text preview: 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. 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 )) dω 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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This note was uploaded on 02/27/2011 for the course 332 222 taught by Professor Staff during the Spring '08 term at Rutgers.
- Spring '08