Solved: Find the trigonometric Fourier series coefficients

Find the exponential Fourier series for the periodic signal shown in Fig. P15.1. 0 6 f(t) 1 2 3 4 5 6 7 8 9 10 T0 t Figure P15.1

Find the exponential Fourier series for the signal shown in Fig. P15.2. 0 2 4 6 8 10 +1 1 f(t) t Figure P15.2

Find the exponential Fourier series for the periodic pulse train shown in Fig. P15.3. 0 0.1 1 1.1 10 (t) t Figure P15.3

Find the exponential Fourier series for the signal shown in Fig. P15.4. 0 t 1 1 f(t) 2 3 4 5 6 Figure P15.4

Find the exponential Fourier series for the function f(t) with period 1 where f(t) is defined as f(t) = et 0 t < 1

how that the exponential Fourier series for the periodic function f (t) = et ,1 < t < 1 with period 2, can be expressed as f(t) = n = [ (1)n _______ 1 + jn ] [ e1 e1 _______ 2 ]e jn0t

Find the exponential Fourier series for the voltage waveform shown in Fig. P15.7. 1 (t) V 01 2 4 5 6 t(s) Figure P15.7

If the waveform in Problem 15.7 is time-delayed 1 second, we obtain the waveform shown in Fig. P15.8. Compute the exponential Fourier coefficients for this latter waveform and show that they differ from those for the waveform in Problem 15.7 by n(90). 1 2 (t) V 12 5 0 1 3 6 7 t (s) Figure P15.8

What type of symmetry is exhibited by the two waveforms in Fig. P15.9? 0 1 2 4 5 3 6 f1(t) t (a) (b) f 2(t) 01 2 t 3 4 5 Figure P15.9

Find the trigonometric Fourier series for the waveform shown in Fig. P15.10. (t) 01 2 3 4 2 1 1 2 t Figure P15.10

Compute the exponential Fourier series for the waveform that is the sum of the two waveforms in Fig. P15.11 by computing the exponential Fourier series of the two waveforms and adding them. (a) 2 t 1(t) 5 0 3 1 4 3 2 1 1 23456 (b) 2(t) 01 2 3 4 5 6 t 2 1 5 4 3 2 1 Figure P15.11

Given the waveform in Fig. P15.12, determine the type of symmetry that exists if the origin is selected at (a) l1 and (b) l2. l1 l2 Figure P15.12

Use the trigonometric Fourier series to show that the Fourier series for the signal in Fig. P15.13 is given by the expression (t) = 1 __2 n=1 __1 n sin (2nt) V (t) 01 2 3 t(s) 2 Figure P15.13

Derive the exponential Fourier series for the function (t) = A sin t shown in Fig. P15.14. (t) 0 2 t A Figure P15.14

Sketch the missing section in the periodic function shown in Fig. P15.15 between T2 and T, which will make (a) a0 = 0, an = 0 and (b) bn = 0. f(t) 0 T t 2 T 2 3T Figure P15.15

Determine which of the Fourier coefficients are zero for f(t) shown in Fig. P15.16. f(t) 0 t T 2 Figure P15.16

Find the trigonometric Fourier series for the voltage waveform shown in Fig. P15.17. (t) V 0 2 3 4 t 1 Figure P15.17

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.18. (t) 1 1 345 0 1 t Figure P15.18

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.19. t (t) 0 1 3 1 2 4 5 1 Figure P15.19

Find the trigonometric Fourier series for the periodic waveform shown in Fig. P15.20. f(t) 1 2 t T0 4 T0 2 T0 4 T0 2 Figure P15.20

Given the waveform in Fig. P15.21, show that f (t) = __ A 2 + n=1 ____ A n sin ____ 2n T0 t f(t) 0 2 T0 T0 t A Figure P15.21

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.22. (t) 0 3 1 1 42 t Figure P15.22

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.23. (t) 1 1 0 34 2 1 t Figure P15.23

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.24. (t) 1 1 0 34 2 1 t Figure P15.24

Derive the trigonometric Fourier series for the waveform shown in Fig. P15.25. (t) +10 10 0 t Figure P15.25

Find the trigonometric Fourier series for the waveform shown in Fig. P15.26. (t) 0 t 2 3 Figure P15.26

Derive the trigonometric Fourier series for the function shown in Fig. P15.27. (t) t V +V T0 2 T0 2 Figure P15.27

Derive the trigonometric Fourier series of the waveform shown in Fig. P15.28. f(t) 0 2 t A 1 3 Figure P15.28

Derive the trigonometric Fourier series for the function (t) = A|sin t| as shown in Fig. P15.29. (t) 0 2 t A Figure P15.29

The amplitude and phase spectra for a periodic function (t) that has only a small number of terms is shown in Fig. P15.30. Determine the expression for (t) if T0 = 0.1 s. 2 4 6 8 1 234 n Dn n 1 23 4 n +80 45 Figure P15.30

Find the trigonometric Fourier series coefficients for the waveform in Fig. P15.31. (t) 0 34 2 1 2 6 t Figure P15.31

Plot the first four terms of the amplitude and phase spectra for the following signal. f (t) = n = 1 n odd ( ___2 n sin ___ n 2 cos ___ n 2 t + ___2 n (2 cos n) sin ___ n 2 t)

Compute the first four terms of the amplitude and phase spectra for the periodic signal defined in Problem 15.17

For the amplitude and phase spectra shown in Fig. P15.34, express the function as a sum of sinusoidal functions using the Fourier series coefficients an and bn. 0 30 60 90 120 n n 20 40 4 6 8 2 Dn 20 40 60 60 n 0 150 180 210 Figure P15.34

Write a function f(t) in cosine form from the amplitude and phase spectra shown in Problem 15.34.

The rectified ac signal shown in Fig. P15.36a is the input to the low-pass filter shown in Fig. P15.36b. Determine the equation for the output signal o(t). 1 1 t (s) 12 0 2 (a) (t) V

The voltage s(t) shown in Fig. P15.37a is applied to the circuit in Fig. P15.37b. Determine the expression for the steady-state voltage o(t) using the first eight harmonics. (a) t (s) s(t) V 1 10 2 3 + (b) + s(t) o(t) R = 10 k C = 10 F 10 Figure P15.37

The discrete line spectrum for a periodic function f(t) is shown in Fig. P15.38. Determine the expression for f(t). 0 1 2 3 4 5 10 20 30 40 50 f (Hz) Dn n 10 20 30 40 50 f (Hz) 90 Figure P15.38

Plot the first four terms of the amplitude and phase spectra for the signal f (t) = n = 1 n odd (___ 2 n sin ___ n 2 cos n0t + ___6 n sin n0t )

Determine the steady-state response of the current io(t) in the circuit shown in Fig. P15.40 if the input voltage is described by the waveform shown in Problem 15.25. + 1 2 io(t) S(t) 2 H Figure P15.40

Determine the first three terms of the steady-state voltage o(t) in Fig. P15.41 if the input voltage is a periodic signal of the form (t) = __1 2 + n=1 ___1 n (cos n 1) sin nt V + 1 1 i(t) (t) o 1 F (t) 1 H + Figure P15.41

Find the average power absorbed by the network in Fig. P15.42 if (t) = 12 + 6cos (377t 10) + 4cos (754t 60)V i(t) = 0.2 + 0.4cos (377t 150) 0.2cos (754t 80) + 0.1cos (1131t 60) A + (t) i(t) Network Figure P15.42

Determine the steady-state voltage o(t) in the network in Fig. P15.43a if the input current is given in Fig. P15.43b. i(t) 01 2 3 t (s) 12 A o(t) + i(t) i2 i (t) 1(t) 1 1 1 2 1 H 1 F (a) (b) Figure P15.43

Determine the steady-state voltage o(t) in the circuit shown in Fig. P15.44a if the input signal is shown in Fig. P15.44b. (a) (b) + o (t) s(t) + 1 H 1 F 2 1 s(t) V 0 t (s) 1 1 2 3 2 1 2345 Figure P15.44

Find the average power absorbed by the network in Fig. P15.45 if (t) = 60 + 36 cos (377t + 45) + 24 cos (754t 60) V + 24 30 mH (t) 12 50 F Figure P15.45

Find the average power absorbed by the 12- resistor in the network in Fig. P15.45 if (t) = 50 + 25 cos (377t + 45) + 12.5 cos (754t 45) V

Determine the Fourier transform of the waveform shown in Fig. P15.47. (t) A T t T A Figure P15.47

Find the Fourier transform of the function f (t) = sin 0t.

Derive the following properties of the Fourier transform. (a) [e j0t f(t)] = F( 0) (b) [cos 0tf(t)] = __1 2 [F( 0) F( 0)]

Find the Fourier transform of the function f(t) = teatu(t).

Show that (a) [f(at)] = (1a) F(a) for a > 0 (b) [f(t a)] = ejaF()

Given that F() = _____________ 1 + j 2 + j8 + 12 , find the Fourier transform of (a) f(2t) (c) 8f(t2) (b) f(t 2) (d) f(4t 1)

Find the Fourier transform of the function f(t) = 12e2#t# cos 4t

Derive the Fourier transform for the following functions: (a) f(t) = e 2t cos 4tu(t) (b) f(t) = e 2t sin 4tu(t)

Show that [f1(t) f2 (t)] = ___1 2 F1(x)F2( x) dx

Use the transform technique to find o(t) in the network in Fig. P15.43a if (a) i(t) = 4( e t e 2t )u(t) A and (b) i(t) = 12 cos 4t A.

The input signal to a network is i (t) = e 3t u(t) V. The transfer function of the network is H(j) = 1(j + 4). Find the output of the network o(t) if the initial conditions are zero.

Determine o(t) in the circuit shown in Fig. P15.58 using the Fourier transform if the input signal is is(t) = ( e 2t + cos t)u(t) A. + 1 1 1 H 2 1 F o i (t) S(t) Figure P15.58

Use the Fourier transform to find i(t) in the network in Fig. P15.59 if i (t) = 2 e t u(t). 1 s(t) 1 i(t) + 1 H Figure P15.59

Use the transform technique to find o(t) in the network in Fig. P15.60 if (a) i (t) = 4e t u(t) V and (b) i (t) = 4( e2t + 2 e 4t )u(t) V. 3 1 1 i (t) 6 2 o(t) + 1 F + Figure P15.60

Determine the total 1- energy content of the signal (t) = ( e t e 2t )u(t)

Determine the 1- energy content of the signal in Problem 15.61 in the frequency band from 0 to 1 rad/s.

Determine the relationship between the 1- energy at the input and output of the network shown in Fig. P15.63. o(t) + + 1(t) = 2e2t u(t) V 1 1 F Figure P15.63

The input signal for the network in Fig. P15.64 is i (t) = 10 e 5t u(t) V. Determine the total 1- energy content of the output o(t). o(t) + + i (t) 1 1 F Figure P15.64

Compute the 1- energy content of the signal o(t) in Fig. P15.64 in the frequency range from = 2 to = 4 rad/s.

Determine the 1- energy content of the signal o(t) in Fig. P15.64 in the frequency range from 0 to 1 rad/s.

Compare the 1- energy at both the input and output of the network in Fig. P15.67 for the given input forcing function ii (t) = 2 e 4t u(t) A. o ii (t) (t) + 1 1 1 F Figure P15.67

The waveform shown in Fig. P15.68 demonstrates what is called the duty cycle; that is, D illustrates the fraction of the total period that is occupied by the pulse. Determine the average value of this waveform. t Vp 0 T0 2T0 DT0 DT0 (1 D)T0 (1 D)T0 Figure P15.68