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Frequency Response and Filter Design By Poles and Zeros Positioning Dr. Mohamed Bingabr University of Central Oklahoma Slides For Lathi’s Textbook Provided by Dr. P. Cheung

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Frequency Response of a LTI System u We have seen that LTI system response to x(t)=e st is H(s)e st. We represent such input-output pair as: u Instead of using a complex frequency, let us set s = j , this yields: u It is often better to express H(j ) in polar form: u Therefore Frequency Response Amplitude Response Phase Response

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h(t) = 3e -5t |H(s)|

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Frequency Response Example (1) u Find the frequency response of a system with transfer function: u Then find the system response y(t) for input x(t)=cos2t and x(t)=cos(10t-50 ) u Substitute s=j

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Frequency Response Example (2)

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Frequency Response Example (3) u For input x(t)=cos2t, we have: u u Therefore

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Frequency Response Example (4) u For input x(t)=cos(10t-50 ), we will use the amplitude and phase response curves directly: u u Therefore

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Frequency Response of delay of T sec u H(s) of an ideal T sec delay is: u Therefore u That is, delaying a signal by T has no effect on its amplitude. u It results in a linear phase shift (with frequency), and a gradient of –T. u The quantity: u is known as Group Delay.

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Frequency Response of an ideal differentiator u H(s) of an ideal differentiator is: u Therefore u This agrees with: u That’s why differentiator is not a nice component to work with – it amplifies high frequency component (i.e. noise!).

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Frequency Response of an ideal integrator u H(s) of an ideal integrator is: u Therefore u This agrees with: u That’s why integrator is a nice component to work with – it suppresses high frequency component (i.e. noise!).

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Filter Design By Poles and Zeros Positioning

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Effects of Poles & Zeros on Frequency Response (1) u Consider a general system transfer function: The value of the transfer function at some complex frequency s = p is:

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Effects of Poles & Zeros on Frequency Response (2) Therefore the magnitude and phase at s = p are given by:

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Effects of Poles & Zeros on Frequency Response (3) u Frequency Response of a system is obtained by evaluating H(s) along the y-axis (i.e. taking all value of s=j ). u Consider the effect of two complex system poles on the frequency response. Near to a pole ENHANCES amplitude

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Effects of Poles & Zeros on Frequency Response (4) u Consider the effect of two complex system zeros on the frequency response. Near to a zero SUPPRESSES amplitude

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Poles & Low-pass Filters u Use the enhancement and suppression properties of poles & zeros to design filters. u Low-pass filter (LPF) has maximum gain at =0, and the gain decreases with . u Simplest LPF has a single pole on real axis, say at (s=- c ). Then u To have a “brickwall” type of LPF (i.e. very sharp cut-off), we need a WALL OF POLE as shown, the more poles we get, the sharper the cut-off. Chebyshev Filter x x x x x x x jcjc -j c Re 0 x x x x x x x Butterworth Filter N = 2

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Poles & Band-pass Filter u Band-pass filter has gain enhanced over the entire passband, but suppressed elsewhere. u For a passband centred around 0, we need lots of poles opposite the imaginary axis in front of the passband centre at 0.

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Notch Filter u Notch filter could in theory be realised with two zeros placed at ±j 0. However, such a filter would not have unity gain at zero frequency, and the notch will not be sharp. u To obtain a good notch filter, put two poles close to the two zeros on the semicircle as shown. Since both pole/zero pair are equal-distance to the origin, the gain at zero frequency is exactly one. Same for =± .

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Notch Filter Example u Design a second-order notch filter to suppress 60 Hz hum in a radio receiver. u Make 0 =120 . Place zeros are at s = ±j 0, and poles at - 0 cos ± j 0 sin . u We get: HW7_Ch4: 4.8-1(a,b), 4.8-2 (a,b,d), 4.10-1, 4.10-3, 4.10-5, 4.10-8, 4.10-9

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