Dorf, R.C., Wan, Z. Transfer Functions of Filters The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

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1 Dorf, R.C., Wan, Z. Transfer Functions of Filters The Electrical Engineering Handbook Ed. Richard C. Dorf oca Raton: CRC Press LLC,

2 Transfer Functions of Filters Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis. Introduction. Ideal Filters.3 The Ideal Linear-Phase Lo-Pass Filter.4 Ideal Linear-Phase andpass Filters.5 Causal Filters.6 utterorth Filters.7 Chebyshev Filters. Introduction Filters are idely used to pass signals at selected frequencies and reject signals at other frequencies. An electrical filter is a circuit that is designed to introduce gain or loss over a prescribed range of frequencies. In this section, e ill describe ideal filters and then a selected set of practical filters.. Ideal Filters An ideal filter is a system that completely rejects sinusoidal inputs of the form x(t) A cos t, < t <, for in certain frequency ranges and does not attenuate sinusoidal inputs hose frequencies are outside these ranges. There are four basic types of ideal filters: lo-pass, high-pass, bandpass, and bandstop. The magnitude functions of these four types of filters are displayed in Fig... Mathematical expressions for these magnitude functions are as follos: Ideal lo-pass: H ( ), îï, - > (.) Ideal high-pass: H ( ), îï, - < < ³ (.) Ideal bandpass: H ( ), îï, all other (.3) Ideal bandstop: H ( ), îï, all other (.4) by CRC Press LLC

3 H H (a) (b) H H (c) (d) FIGURE. Magnitude functions of ideal filters:(a) lo-pass; (b) high-pass; (c) bandpass; (d) bandstop. The stopband of an ideal filter is defined to be the set of all frequencies for hich the filter completely stops the sinusoidal input x(t) A cos t, < t <. The passband of the filter is the set of all frequencies for hich the input x(t) is passed ithout attenuation. More complicated examples of ideal filters can be constructed by cascading ideal lo-pass, highpass, bandpass, and bandstop filters. For instance, by cascading bandstop filters ith different values of and, e can construct an ideal comb filter, hose magnitude function is illustrated in Fig... FIGURE. H Magnitude function of an ideal comb filter..3 The Ideal Linear-Phase Lo-Pass Filter Consider the ideal lo-pass filter ith the frequency function H( ) ì - j t ïe d, - îï, < -, > (.5) here t d is a positive real number. Equation (.5) is the polar-form representation of H(). From Eq. (.5) e have and H ( ), - îï, < -, > td, / H ( ) - - îï, < -, > by CRC Press LLC

4 H() t d t d Slope t d FIGURE.3 Phase function of ideal lo-pass filter defined by Eq. (.5). H() t d t d Slope t d FIGURE.4 Phase function of ideal linear-phase bandpass filter. The phase function /H() of the filter is plotted in Fig..3. Note that over the frequency range to, the phase function of the system is linear ith slope equal to t d. The impulse response of the lo-pass filter defined by Eq. (.5) can be computed by taking the inverse Fourier transform of the frequency function H(). The impulse response of the ideal lopass filter is ht () Sat [ ( - t d )], - < t < p (.6) here Sa(x) (sin x)/x. The impulse response h(t) of the ideal lo-pass filter is not zero for t <. Thus, the filter has a response before the impulse at t and is said to be noncausal. As a result, it is not possible to build an ideal lo-pass filter..4 Ideal Linear-Phase andpass Filters One can extend the analysis to ideal linear-phase bandpass filters. The frequency function of an ideal linearphase bandpass filter is given by H( ) ì - j t ïe d, îï, all other here t d,, and are positive real numbers. The magnitude function is plotted in Fig..(c) and the phase function is plotted in Fig..4. The passband of the filter is from to. The filter ill pass the signal ithin the band ith no distortion, although there ill be a time delay of t d seconds. by CRC Press LLC

5 .77 - p p (a) (b) (c) (d) FIGURE.5 Causal filter magnitude functions: (a) lo-pass; (b) high-pass; (c) bandpass; (d) bandstop..5 Causal Filters As observed in the preceding section, ideal filters cannot be utilized in real-time filtering applications, since they are noncausal. In such applications, one must use causal filters, hich are necessarily nonideal; that is, the transition from the passband to the stopband (and vice versa) is gradual. In particular, the magnitude functions of causal versions of lo-pass, high-pass, bandpass, and bandstop filters have gradual transitions from the passband to the stopband. Examples of magnitude functions for the basic filter types are shon in Fig..5. For a causal filter ith frequency function H(), the passband is defined as the set of all frequencies for hich H( ) ³ H( p) 77. H( p) (.7) here p is the value of for hich H() is maximum. Note that Eq. (.7) is equivalent to the condition that H() d is less than 3 d don from the peak value H( p ) d. For lo-pass or bandpass filters, the idth of the passband is called the 3-d bandidth. A stopband in a causal filter is a set of frequencies for hich H() d is don some desired amount (e.g., 4 or 5 d) from the peak value H( p ) d. The range of frequencies beteen a passband and a stopband is called a transition region. In causal filter design, a key objective is to have the transition regions be suitably small in extent..6 utterorth Filters The transfer function of the to-pole utterorth filter is Hs () s n + ns + n Factoring the denominator of H(s), e see that the poles are located at s - n ± j n by CRC Press LLC

6 H() One-pole filter H(s) s + To-pole filter ith z To-pole utterorth filter Passband FIGURE.6 Magnitude curves of one- and to-pole lo-pass filters. Note that the magnitude of each of the poles is equal to n. Setting s j in H(s), e have that the magnitude function of the to-pole utterorth filter is H ( ) + ( / ) n 4 (.8) From Eq. (.8) e see that the 3-d bandidth of the utterorth filter is equal to n. For the case n rad/s, the frequency response curves of the utterorth filter are plotted in Fig..6. Also displayed are the frequency response curves for the one-pole lo-pass filter ith transfer function H(s) /(s + ), and the topole lo-pass filter ith z and ith 3-d bandidth equal to rad/s. Note that the utterorth filter has the sharpest cutoff of all three filters..7 Chebyshev Filters The magnitude function of the n-pole utterorth filter has a monotone characteristic in both the passband and stopband of the filter. Here monotone means that the magnitude curve is gradually decreasing over the passband and stopband. In contrast to the utterorth filter, the magnitude function of a type Chebyshev filter has ripple in the passband and is monotone decreasing in the stopband (a type Chebyshev filter has the opposite characteristic). y alloing ripple in the passband or stopband, e are able to achieve a sharper transition beteen the passband and stopband in comparison ith the utterorth filter. The n-pole type Chebyshev filter is given by the frequency function H ( ) T n + ( / ) (.9) here T n (/ ) is the nth-order Chebyshev polynomial. Note that e is a numerical parameter related to the level of ripple in the passband. The Chebyshev polynomials can be generated from the recursion T n (x) xt n (x) T n (x) here T (x) and T (x) x. The polynomials for n, 3, 4, 5 are T (x) x(x) x T 3 (x) x(x ) x 4x 3 3x T 4 (x) x(4x 3 3x) (x ) 8x 4 8x + T 5 (x) x(8x 4 8x + ) (4x 3 3x) 6x 5 x 3 + 5x (.) by CRC Press LLC

7 H() To-pole filter Three-pole filter Passband (a) H() To-pole filter Three-pole filter (b) FIGURE.7 phase curves. Frequency curves of to- and three-pole Chebyshev filters ith c.5 rad/s: (a) magnitude curves; (b) Using Eq. (.), the to-pole type Chebyshev filter has the folloing frequency function H( ) + [( / ) - ] For the case of a 3-d ripple ( ), the transfer functions of the to-pole and three-pole type Chebyshev filters are Hs () Hs () s 5. c cs c 3 5. c 3 3 c c c s s s + 5. here c 3-d bandidth. The frequency curves for these to filters are plotted in Fig..7 for the case c.5 rad. The magnitude response functions of the three-pole utterorth filter and the three-pole type Chebyshev filter are compared in Fig..8 ith the 3-d bandidth of both filters equal to rad. Note that the transition from passband to stopband is sharper in the Chebyshev filter; hoever, the Chebyshev filter does have the 3- d ripple over the passband. by CRC Press LLC

8 H() Three-pole utterorth Three-pole Chebyshev Passband FIGURE.8 to.5 rad/s. Magnitude curves of three-pole utterorth and three-pole Chebyshev filters ith 3-d bandidth equal Defining Terms Causal filter: A filter of hich the transition from the passband to the stopband is gradual, not ideal. This filter is realizable. 3-d bandidth: For a causal lo-pass or bandpass filter ith a frequency function H(j): the frequency at hich H() d is less than 3 d don from the peak value H( p ) d. Ideal filter: An ideal filter is a system that completely rejects sinusoidal inputs of the form x(t) A cos t, < t <, for ithin a certain frequency range, and does not attenuate sinusoidal inputs hose frequencies are outside this range. There are four basic types of ideal filters:lo-pass, high-pass, bandpass, and bandstop. Passband: Range of frequencies for hich the input is passed ithout attenuation. Stopband: Range of frequencies for hich the filter completely stops the input signal. Transition region: The range of frequencies of a filter beteen a passband and a stopband. Related Topics 4. Lo-Pass Filter Functions 4.3 Lo Pass Filters. Introduction 9. Synthesis of Lo-Pass Forms References R.C. Dorf, Introduction to Electrical Circuits, 3rd ed., Ne York: Wiley, 996. E.W. Kamen, Introduction to Signals and Systems, nd ed., Ne York: Macmillan, 99. G.R. Cooper and C.D. McGillem, Modern Communications and Spread Spectrum, Ne York: McGra-Hill, 986. Further Information IEEE Transactions on Circuits and Systems, Part I: Fundamental Theory and Applications. IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing. Available from IEEE. by CRC Press LLC