If is a real function of t, show that the inversion

a) Find the Fourier transform of the function shown in Fig. P17.1. b) Find when . c) Sketch versus when and . Hint: Remember that is an even function of . Figure P17.1 f(t) A t t/2 0 t/2 v

The Fourier transform of is shown in Fig. P17.2. a) Find . b) Evaluate . c) Sketch for when and . Hint: Remember that is even. Figure P17.2

Use the defining integral to find the Fourier trans- f(t) form of the following functions:

Derive F{sin wot}.

Find the Fourier transform of each of the following functions. In all of the functions, a is a positive real constant and .

If is a real function of t, show that the inversion integral reduces to f(t) = 1 2p L q -q [A(v) cos vt - B(v) sin vt] dv.

If is a real, odd function of t, show that the inversion integral reduces to f(t) = -2p L q-q B(v) sin vt dv.

Use the inversion integral (Eq. 17.9) to show that f-1 52>jv6 = sgn(t)Hint: Use Problem 17.7.

Find by using the approximating function f(t) = e- P|t| cos v0t

Show that if is an odd function, B(v) = -2 L q 0 f(t) sin vt dt.

Show that if is an even function, A(v) = 2 Lq 0 f(t) cos vt dt,

a) Show that where Hint: Use the defining integral and integrate by parts. b) What is the restriction on if the result given in (a) is valid? c) Show that where F(v) = f5f(t)6

a) Show that fb L t - q f(x)dx r = F(v) jv , where Hint: Use the defining integral and integrate by parts. b) What is the restriction on if the result given in (a) is valid? c) If , can the operational transform in (a) be used? Explain.

a) Show that f5f(at)6 = 1 a F a va b, a 7b) Given that for , sketch for , and 2.0. Do your sketches reflect the observation that compression in the time domain corresponds to stretching in the frequency domain?

Derive each of the following operational transforms:

Given show that where and Hint: Use the defining integral to write f5y(t)6 = L q - q B L q - q x(l)h(t - l)dlRe-jvt dt.Next, reverse the order of integration and then make a change in the variable of integration; that is, let .

Given , show that . Hint: First, use the defining integral to express as Second, use the inversion integral to write Third, substitute the expression for into the defining integral and then interchange the order of integration

a) Show that b) Use the result of (a) to find each of the following Fourier transforms (assuming )

Suppose that , where a) Use convolution in the frequency domain to find . b) What happens to as the width of increases so that includes more and more cycles on ?

a) Use the Fourier transform method to find io(t) in the circuit shown in Fig. P17.20 if vg = 60 sgn(t) V b) Does your solution make sense in terms of known circuit behavior? Explain.

Repeat Problem 17.20 except replace io(t) with vo(t) .

a) Use the Fourier transform method to find vo(t) in the circuit shown in Fig. P17.22. The initial value of vo(t) is zero, and the source voltage is 50u(t) V b) vo(t) Sketch versus t. Figure P17.22

Repeat Problem 17.22 if the input voltage is (vg) changed to 25 sgn(t)

a) Use the Fourier transform to find in the circuit in ig = 40 sgn(t) mA Fig. P17.24 if b) Does your solution make sense in terms of known circuit behavior? Explain. Figure P17.24

Repeat Problem 17.24 except io replace with vo.

The voltage source in the circuit in Fig. P17.26 is given by the expression vg = 8 sgn(t) V. a) Find vo(t) . b) What is the value of vo(0-) ? c) What is the value of vo(0+)? d) Use the Laplace transform method to find vo(t for )t > 0+ . e) Does the solution obtained in (d) agree with vo(t) for t > 0+ from (a)?

Repeat Problem 17.26 except replace with vo(t) from io(t).

a) Use the Fourier transform to find vo in the circuit in Fig. P17.28 if equals 2e-100|t. b) Find vo(0-) . c) Find vo(0+). d) Use the Laplace transform method to find vo for t 0 . e) Does the solution obtained in (d) agree with vo for t > 0+ from (a)?

a) Use the Fourier transform to find in the circuit in Fig. P17.28 if equals . b) Find io(0-) . c) Find io(0+) . d) Use the Laplace transform method to find for t 0 . e) Does the solution obtained in (d) agree with for from t > 0+(a)?