Assignment 4Continuous-Time Fourier Transform

ECE 223 Signals and Systems IIVersion 1.01Spring 2006

1. Properties of complex numbers. Let c1 = α1 + jβ1 and c2 = α2 + jβ2 be twocomplex numbers.

a. What is the magnitude of c1? Express your answer in terms of α1 and β1.

b. Show that (c1 + c2)∗ = c∗1 + c∗2.

c. Show that (c1c2)∗ = c∗1 c∗2.

2. Frequency concepts. Explain in your own words why there is no natural interpreta-tion of “high-frequency” in continuous-time. Compare and contrast with the discrete-timecase.

3. Relationship to LTI systems.

a. Solve for the transfer function, H(s), for a first-order highpass filter with a passbandgain of 5 and a cutoff frequency of 500 Hz.

b. Plot the impulse response of this system.

c. What range of frequencies does this filter amplify?

d. What range of frequencies does this filter attenuate?

e. Use MATLAB to create the Bode plot of this system.

f. Suppose a sinusoid x(t) = 10 cos(1000πt + 30◦) is applied to the system. What is theoutput?

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g. Repeat the previous question for x(t) = 10 cos(100πt + 45◦).

h. Repeat the previous question for x(t) = 10 sin(100πt − 135◦).

4. Properties.

a. What properties does the CTFT have for all real signals?

b. Real, even signals?

c. Real, even signals with finite energy?

d. Real, odd signals with finite power?

e. Real, periodic signals?

5. Transform examples. Find the Fourier transforms of the following signals. Whichof these signals have Fourier transforms that converge? Which of these signals have Fouriertransforms that are real? imaginary?

a. x(t) = cos(1000t).

b. x(t) = 3δ(t).

c. x(t) = 13 cos(100t) − 7 sin(500t)

d. x(t) = p50(t).

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