## Paarmann L.D.

## Chapter 4

## CHEBYSHEV TYPE I FILTERS - all with Video Answers

## Educators

Chapter Questions

Given the defining equations for a Chebyshev Type I response, (4.1) and (4.2), and given that $\omega_c=1000 \mathrm{rad} / \mathrm{s}, A_p=1.8 \mathrm{~dB}$, and $N=3$ :

(a) Determine the value of $\varepsilon$.

(b) Determine the value of $\omega_p$.

(c) Determine the value of $\left|{ }_p H_{(j)}\right|_{\text {min }}^2$.

(d) Determine the frequencies of the peaks in the passband.

(e) Determine the frequencies of the valleys in the passband.

(f) Accurately sketch the magnitude frequency response. Use only a calculator for the necessary calculation. Use a vertical scale in $d B(0$ to $-50 \mathrm{~dB}$ ). and a linear radian frequency scale from 0 to $5000 \mathrm{rad} / \mathrm{s}$.

(g) Accurately sketch the magnitude frequency response. Use only a calculator for the necessary calculations. Use a linear vertical scale from 0 to 1 , and a linear radian frequency scale from 0 to $2000 \mathrm{rad} / \mathrm{s}$.

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Given the defining equations for a Chebyshev Type I response, (4.1) and (4.2), and given that $\omega_c=1000 \mathrm{rad} / \mathrm{s}, A_p=1.5 \mathrm{~dB}$, and $N=6$ :

(a) Determine the value of $\varepsilon$

(b) Determine the value of $\omega_p$.

(c) Determine the value of $|H(j \omega)|_{\min }^2$.

(d) Determine the frequencies of the peaks in the passband.

(e) Determine the frequencies of the valleys in the passband.

(f) Accurately sketch the magnitude frequency response. Use only a calculator for the necessary calculations. Use a vertical scale in $d B(0$ to $-100 \mathrm{~d} \mathrm{~B}$ ), and a linear radian frequency scale from 0 to $5000 \mathrm{rad} / \mathrm{s}$.

(g) Accurately sketch the magnitude frequency response. Use only a calculator for the necessary calculations. Use a linear vertical scale from 0 to 1 , and a linear radian frequency scale from 0 to $1200 \mathrm{rad} / \mathrm{s}$.

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In Figure 4.1 it may appear that the response fall-off is greater than $N \times 20 \mathrm{~dB} /$ decade. However, this is not the case for very high frequencies. Demonstrate that for very high frequencies the fall-off is indeed $-N \times 20 \mathrm{~dB} /$ decade.

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Determine the value of Filter Selectively for the Chebyshev Type I filter specified in Problem 4.2. Compare this value with the Filter Selectivity for a Butterworh filter with similar specifications: $\quad \omega_c=1000 \mathrm{rad} / \mathrm{s}$, and $N=6$

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Determine the value of the Shaping Factor for the Chebyshev Type I filter specified in Problem 4.2, for $a=3 d B$ and $b=80 d B$. Compare this value with the Shaping Factor for a Butterworth filter with similar specifications: $\omega_c=1000 \mathrm{rad} / \mathrm{s}$, and $N=6$.

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Estimate the Shaping Factor for a Chebyshev Type I filter with $N=5$, $\omega_c=1$, and $A_p=1 d B$, from Figure 4.1. Compare your results with that obtained from (4.12).

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Suppose filter specifications are stated as follows: $f_c=3500 \mathrm{~Hz}$, $f_s=7000 \mathrm{~Hz}$, and $A_s=60 \mathrm{~dB}$. Note that (4.14) requires knowledge of $\omega_p$ to determine the required order, and when it is $\omega_c$ that is specified the order is required to determine $\omega_p$ from $\omega_c$ via (4.7). However, it is very practical to use (4.14), using $\omega_c$ in place of $\omega_p$. This is justified since, (1) in practical problems there is usually not much difference between $\omega_c$ and $\omega_p$, especially for higher order filters, (2) since the orders are restricted to integers it seldom makes any difference, and (3) since $\omega_c>\omega_p$, any error will result in exceeding the specifications rather than failing to meet the specifications. Making use of (4.14), determine the required filter order to meet the given specifications:

(a) For a Chebyshev Type I filter with $0.1 d B$ of ripple.

(b) For a Chebyshev Type I filter with $0.5 d B$ of ripple.

(c) For a Chebyshev Type I filter with $1.0 \mathrm{~dB}$ of ripple.

(d) For a Chebyshev Type I filter with $1.5 \mathrm{~dB}$ of ripple.

(e) For a Chebyshev Type I filter with $2.0 \mathrm{~dB}$ of ripple.

(f) For a Chebyshev Type I filter with $2.5 d B$ of ripple.

(g) For comparison purposes, for a Butterworth filter.

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Given that $N=4, \omega_p=10$, and $\varepsilon=0.4$, express $|H(j \omega)|^2$ in polynomial form similar to Example 4.4, and demonstrate that it satisfies the Analog Filter Design Theorem.

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Prove that the magnitude-squared frequency response of Problem 4.1 satisfies the Analog Filter Design Theorem.

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Prove that the magnitude-squared frequency response of Problem 4.2 satisfies the Analog Filter Design Theorem.

Determine the 9-th and 10-th order Chebyshev polynomials.

Matthew Allco*ck

Numerade Educator

Sketch the square of the 4-th order Chebyshev polynomial given in Table 4.2 for $\omega$ from 0 to $1.1 \mathrm{rad} / \mathrm{s}$. Compute the square of (4.2) over this same radian frequency range for $\omega_p=1$, and verify that it is numerically the same as the Chebyshev polynomial.

Arun Bana

Numerade Educator

Determine the poles of the filter specified in Problem 4.7 (a).

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Determine the poles of the filter specified in Problem 4.7 (f).

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Determine the poles of the filter specified in Problem 4.8.

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Determine the transfer function $H(s)$ for the Chebyshev Type I filter specified in Problem 4.1. Express the denominator of $H(s)$ in two ways: (1) As a polynomial in $s$. (2) As the product of a second-order polynomial in $s$, the roots of which being complex conjugates, and a first-order term. State the numerical values of the three poles. Sketch the six poles of $H(-s) H(s)$ on an $s$ plane, and include a sketch of the ellipse that the poles fall on. State the numerical values of the major and minor semiaxes of the ellipse.

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Determine the transfer function $H(s)$ for the Chebyshev Type I filter specified in Problem 4.2. Express the denominator of $H(s)$ in two ways: (1) As a polynomial in $s$. (2) As the product of three second-order polynomials in $s$, the roots of each second-order polynomial being complex conjugates. State the numerical values of the six poles. Sketch the twelve poles of $\mathrm{H}(-s) H(s)$ on an $s$ plane, and include a sketch of the ellipse that the poles fall on. State the numerical values of the major and minor semiaxes of the ellipse.

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Suppose an anti-aliasing filter is needed prior to an analog-to-digital converter to be used in a speech processing system. Suppose a sampling rate of 8000 samples/s is to be used, and therefore it is decided that the anti-aliasing filter should have a minimum attenuation of $60 \mathrm{~dB}$ at $4000 \mathrm{~Hz}$. Suppose a Chebyshev Type I filter is to be used.

(a) If $f_p=3000 \mathrm{~Hz}$ and $A_p=1 d B$, what is the minimum order required?

(b) If $A_p=1 d B$ and $N=10$, what is maximum value of $f_p$ ?

(c) If $f_p^p=3000 \mathrm{~Hz}$ and $N=10$, what is the minimum value of $A_p$ ?

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Under the conditions of part (c) of Problem 4.18, determine the transfer function $H(s)$, and give numerical values for all the poles.

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Sketch the step response of a 10-th order Chebyshev Type I filter with $f_c=1000 \mathrm{~Hz}$ and $A_p=1 \mathrm{~dB}$. Refer to Figure 4.9 and make use of the scaling property of Fourier transforms. What would the maximum group delay be for this filter, and at what frequency would it occur? At what time would the peak of the unit impulse response of this filter be, and what would be the value of that peak?

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Using the MATLAB functions cheblap, impulse and step:

(a) Determine the transfer function in polynomial form, and also factored to indicate the poles, of a Chebyshev Type I filter with $\omega_p=1$, $A_p=1.2 d B$, and $N=6$.

(b) Determine the impulse response and the step response for the filter of part (a).

(c) By multiplying the pole vector found in part (a) by $2 \pi 1000$ determine the transfer function of a Chebyshev Type I filter with $f_p=1000 \mathrm{~Hz}$, $A_p=1.2 d B$, and $N=6$.

(d) Determine and plot the magnitude frequency response of the filter of part (c) by using the MATLAB function freqs. Use a vertical scale in $d B$ and a linear horizontal scale from 0 to $5000 \mathrm{~Hz}$. Also determine and plot the phase response over this same frequency range. Use the MATLAB function unwrap to display the smooth phase response rather than the principle phase.

(e) By appropriately scaling the impulse response and the step response of part (b), determine and plot the impulse response and the step response of the filter of part (c). That is, the time axis for the step response needs to scaled by $1 /(2 \pi 1000)$, and the unit impulse response needs the same time-axis scaling and requires an amplitude scaling of $2 \pi 1000$.

(f) Determine and plot the phase delay of the filter of part (c). Note that this is easily obtained from the phase response of part (d).

(g) Determine and plot the group delay of the filter of part (c). Note that this also is easily obtained from the phase response of part (d): $t_{g d}(n) \cong-[\phi(n)-\phi(n-1)] / S_s$, where $\phi(n)$ is the phase in radians at step $n$, and $S_s$ is the step size in $\mathrm{rad} / \mathrm{s}$.

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