- All Question Numbers Shown Below Refer To The Numbering From The Course Textbook Example 5 2 A Periodic Rectangular Wa 1 (20.46 KiB) Viewed 26 times
- All Question Numbers Shown Below Refer To The Numbering From The Course Textbook Example 5 2 A Periodic Rectangular Wa 2 (20.46 KiB) Viewed 26 times
- All Question Numbers Shown Below Refer To The Numbering From The Course Textbook Example 5 2 A Periodic Rectangular Wa 3 (47 KiB) Viewed 26 times
- All Question Numbers Shown Below Refer To The Numbering From The Course Textbook Example 5 2 A Periodic Rectangular Wa 4 (41.78 KiB) Viewed 26 times
- All Question Numbers Shown Below Refer To The Numbering From The Course Textbook Example 5 2 A Periodic Rectangular Wa 5 (20.64 KiB) Viewed 26 times
EXAMPLE 5.4: Sketch the magnitude and phase spectra for the periodic signal X(t) considered in Example 5.2 with A = 10,7 = 8 ms, and a = 3 ms. EXAMPLE 5.8: We determine the Fourier Transform of the decaying exponential signal X(t) = e-fu(t) where a is real and positive. EXAMPLE 5.9: We determine the Fourier Transform of the pulse signal *(t) = Apa(e) = 1 [u(t+) -u(-23) Note that this is a single pulse centered at the origin with amplitude A and duration a, as shown in Figure 5.19a EXAMPLE 5.10: From X(t) = 1 + X(w) = 216(w), we determine the transform of (a) x (t) = e/?, (b) X(t) = sin(t), and (c) X(t) = cos(xt). EXAMPLE 5.12: Given that cos wol [8we) + 8(w + 4)], we want to obtain the transform of sint. EXAMPLE 5.24: We determine the transfer function () and the frequency-response behaviour for the circuit shown in Figure 5.37, where v; () = x(t) is the input and 1(t) = y(t) is the output. From H(w), we obtain the impulse response h(t), and the response of the circuit for (a) v. (t) = x(t) = 2 and (b) v (t) = *(t) = 3sin (200xt + 45"). R=100 -21 = 20 L=0.2H FIGURE 5.37 Circuit for Example 5.24 5.2 Express the following periodic signals in exponential Fourier series. (a) X(t) = A + A,cos (Wot +60") (b) Xp(t) = A - B sin (art + 30") (@)X(t) = 5 + 2c0s (wt + 60') - 3sin (wt - 60') 5.4. Sketch the amplitude and phase spectra for each of the periodic signals shown in Problem 5.2.
5.10. In the Fourier series expansion of the periodic signal xy(t) shown in Figures P5.10a and b. (a) what is the average value? (b) what are the amplitudes of the fundamental component in complex exponential form, i.e.. I for k = -1 and +1 ? (e) what is the amplitude of the fundamental component in combined cosine form? (d) what is C? (e) what is G? () what are the values of A and B in Acos (6000nt + )? (g) what are the values of A and in Acos (1500mt + 8) ? (h) what are the values of A and 8 in Acos (5000xt+)? 5.17 Using the defining integral, determine the Fourier transform of the following functions. (a) x(t) = eat (b)x(t) = e-ul!cos (bt) (e) x(t) = te fu(t) (d) etsin (bt)u(t),ab > 0 5.19 Evaluate the Fourier transform of g(t) = e-fu(t) and h(t) = e-bu(t). Now evaluate y(t) = 9(t) • h(t) and find the Fourier transform of y(t). Verify your answer using the convolution property 5.31 For the circuit shown in Figure P5.31 with R = 104 and C = 1aF, determine (a) The differential equation model relating the input x (voltage v, ) to output y voltage P()).. (b) The frequency response function, H(w). R t) FIGURE P5.31
5.42 Sketch the frequency response magnitude and phase for a system given by 3w HW) 100+ jw Determine the response of the system to (a) X(t) = 5. (b)x(t) = 3 cos (50mt). (c)X(t) = 3cos (100mt), and (d) X(t) = 3 cos (1000mt). What is the 3 - dB frequency of the system?