The specifications for a low-pass filter areα(0)=20 dBsΩp = 1500

The specifications for a low-pass filter are

α(0)=20 dBs

Ωp = 1500 rad/sec,           α1 = 20.5 dBs

Ωs = 3500 rad/sec,           α2 = 50 dBs

(a) Determine the minimum order of the low-pass Butterworth and Chebyshev filters, and determine which is smaller.

(b) Give the transfer function of the designed low-pass Butterworth and Chebyshev filters (make sure the dc loss is as specified).

(c) Determine the half-power frequency of the designed filters by letting α(Ωp) = αmax.

(d) Find the loss function values provided by the designed filters at Ω­p and ­Ωs. How are these values related to the αmax and αmin specifications? Explain. Which of the two filters provides more attenuation in the stop-band?

(e) If new specifications for the passband and stopband frequencies are ­Ωp = 750(rad/sec) and ­Ωs = 1750(rad/sec), are the minimum order the same as before? Explain.

Design an analog low-pass filter satisfying the following magnitude specifications:αmax

Design an analog low-pass filter satisfying the following magnitude specifications:

αmax = 0.5 dB                      αmin = 20 dB

Ωp =1000 rad/sec             Ωs =2000 rad/sec

(a) Use the Butterworth method. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify by plotting the loss function that the specifications are satisfied.

(b) Use Chebyshev method (cheby1). Plot the poles and zeros and the magnitude and phase of the designed filter. Verify by plotting the loss function that the specifications are satisfied.

(c) Use the elliptic method. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify by plotting the loss function that the specifications are satisfied.

(d) Compare the three filters and comment on their differences.

Control systems attempt to follow the reference signal at the

Control systems attempt to follow the reference signal at the input, but in many cases they cannot follow particular types of inputs. Let the system we are trying to control have a transfer function G(s), and the feedback transfer function be H(s). If X(s) is the Laplace transform of the reference input signal, Y(s) the Laplace transform of the output, and

(a) Find an expression for E(s) in terms of X(s), G(s), and H(s).

(b) Let x(t) = u(t), and the Laplace transform of the corresponding error be E1(s). Use the final value property of the Laplace trans-form to obtain the steady-state error e1ss.

(c) Let x(t) = tu(t), i.e., a ramp signal, and E2(s) be the Laplace transform of the corresponding error signal. Use the final value property of the Laplace transform to obtain the steady-state error e2ss. Is this error value larger than the one above? Which of the two inputs u(t) and r(t) is easier to follow?

(d) Use MATLAB to find the partial fraction expansions of E1(s) and E2(s) and use them to find e1(t) and e2(t). Plot them.

Consider a filter with frequency responseor a sinc function in

Consider a filter with frequency response

or a sinc function in frequency.

(a) Find the impulse response h(t) of this filter. Plot it and indicate whether this filter is a causal system or not.

(b) Suppose you wish to obtain a band-pass filter G(jΩ) from H(jΩ). If the desired center frequency of |G(jΩ)|is 5, and its desired magnitude is 1 at the center frequency, how would you process h(t) to get the desired filter? Explain your procedure.

(c) Use symbolic MATLAB to find h(t), g(t) and G(jΩ). Plot |H(jΩ)|, h(t), g(t), and |G(jΩ)|. Compute the Fourier transform using the integration function.

Let the signal x(t) = r(t + 1) − 2r(t)

Let the signal x(t) = r(t + 1) − 2r(t) + r(t − 1) and y(t) = dx(t)/dt.

(a) Plot x(t) and y(t)

(b) Find X(Ω) and carefully plot its magnitude spectrum. Is X(Ω) real? Explain.

(c) Find Y(Ω)(use properties of Fourier transform) and carefully plot its magnitude spectrum. Is Y(Ω) real? Explain.

(d) Determine from the above spectra which of these two signals is smoother. Use MATLAB integration function int to find the Fourier transforms. Plot 20 log10|Y(Ω)|and 20 log10|X(Ω)|and decide. Would you say in general that computing the derivative of a signal generates high frequencies, or possible discontinuities?

For signals with infinite support, their Fourier transforms cannot be

For signals with infinite support, their Fourier transforms cannot be derived from the Laplace transform unless they are absolutely integrable or the region of convergence of the Laplace transform contains the jΩ­-axis. Consider the signal x(t) = 2e−2∣t

(a) Plot the signal x(t) for −∞ < t < ∞.

(b) Use the evenness of the signal to find the integral

and determine whether this signal is absolutely integrable or not.

(c) Use the integral definition of the Fourier transform to find X(Ω).

(d) Use the Laplace transform of x(t) to verify the above found Fourier transform.

(e) Use MATLAB’s symbolic function fourierto compute the Fourier transform of x(t). Plot the magnitude spectrum corresponding to x(t).

The Fourier transform of finite support signals, which are absolutely

The Fourier transform of finite support signals, which are absolutely integrable or finite energy, can be obtained from their Laplace transform rather than doing the integral. Consider the following signals

x1(t) = u(t + 0.5) − u(t − 0.5), x2(t) = sin(2π t) [u(t) − u(t − 0.5)]

x3(t) = r(t + 1) − 2r(t) + r(t − 1)

(a) Plot each of the above signals.

(b) Find the Fourier transforms {Xi(Ω)}for i = 1, 2, and 3 using the Laplace transform.

(c) Use MATLAB’s symbolic integration function int to compute the Fourier transform of the given signals. Plot the magnitude spectrum corresponding to each of the signals.

As you know, π is an irrational number that can

As you know, π is an irrational number that can only be approximated by a number with a finite number of decimals. How to compute this value recursively is a problem of theoretical interest. In this problem we show that the Fourier series can provide that formulation.

(a) Consider a train of rectangular pulses x(t), with a period

x1(t) = 2[u(t + 0.25) − u(t − 0.25)] − 1,                      −0.5 ≤ t ≤ 0.5

and period T0 = 1. Plot the periodic signal and find its trigonometric Fourier series.

(b)Use the above Fourier series to find an infinite sum for π.

(c) If πN is an approximation of the infinite sum with N coefficients, and πis the value given by , find the value of N so that πN is 95% of the value of π given by MATLAB.

The Fourier transforms of even and odd functions are very

The Fourier transforms of even and odd functions are very important. Let x(t) = e∣t and y(t) = e−t u(t) − et u(−t).

(a) Plot x(t) and y(t), and determine whether they are even or odd.

(b) Show that the Fourier transform of x(t) is found from

which is real function of Ω­, therefore its computational importance. Show that X(Ω) is an even function of ­. Find X(Ω) from the above equation (called the cosine transform).

(c) Show that the Fourier transform of y(t) is found from

which is imaginary function of ­Ω, thus its computational importance. Show that Y(Ω) is and odd function of Ω­. Find Y(Ω)from the above equation (called the sine transform). Verify that your results are correct by finding the Fourier transform of z(t) = x(t) + y(t) directly and using the above results.

(d) What advantages do you see in using the cosine and sine transforms? How would you use the cosine and the sine transforms to compute the Fourier transform of any signal, not necessarily even or odd? Explain.

To understand the Fourier series consider a more general problem,

To understand the Fourier series consider a more general problem, where a periodic signal x(t), of period T0, is approximated as a finite sum of terms

where {φk (t)} are ortho-normal functions. To pose the problem as an optimization problem, consider the square error

and we look for the coefficients {X̂(k)}that minimize ε.

(a) Assume that x(t) as well as x̂(t) are real valued, and that x(t) is even so that the Fourier series coefficients Xk are real. Show that the error can be expressed as

(b) Compute the derivative of ε with respect to X̂n and set it to zero to minimize the error. Find X̂n.

(c) In the Fourier series the {φk(t)} are the complex exponentials and the {X̂n} coincide with the Fourier series coefficients. To illustrate the above procedure consider the case of the pulse signal x(t), of period T0 = 1 and a period

x1(t) = 2[u(t + 0.25) − u(t − 0.25)]

Use MATLAB to compute and plot the approximation x̂(t) and the error ε for increasing values of N from 1to 100.

(d) Concentrate your plot of x̂(t) around one of the discontinuities, and observe the Gibb’s phenomenon. Does it disappear when N is very large? Plot x̂(t) around the discontinuity for N = 1000.