Figure P.3.3-3 134 3.3-3 Using only the time-shifting property and Table 3.1, find the Fourier transforms of the signals shown in Fig. P3.3-3. -T 10 1 T (a) 1 cos t e-at (c) 10 π/2 sin (b) 元 (d) Hint: The signal in Fig. P3.3-3a is a sum of two shifted rectangular pulses. The signal in Fig. P3.3-3b is sin t[u(t)u(t)] = sin tu(t) sin tu(t) = sin tu(t) + sin (7) ANALYSIS AND TRANSMISSION OF SIGNALS u(t). The reader should verify that the addition of these two sinusoids indeed results in the pulse in Fig. P3.3-3b. In the same way, we can express the signal in Fig. P3.3-3c as л/2)u(tл/2) (verify this by sketching these signals). The signal in [u(t)u(t − T)] = e¯atu(t) - e-Te-a(1-T)u(t − T). cos tu(t) + sin († Fig. P3.3-3d is e TABLE 3.1 Short Table of Fourier Transforms g(t) G(f) 1 e-alu(t) a > 0 a+jлƒ 2 eatu(-t) 3 e-a 4 te u(t) 5 the-atu(t) 6 8(t) 7 ] 82 for a-2f 2a a² + (2πf)² I 1 (a+j2xƒ)² (a+2)+1 8(√) sf-fo) 0.5 [+fa) +80f-√6)] j0.5 [(f+fo)-8f-fo)] a> 0 2 >0 a > 0 2>0 9 cos 2πfot 10 sin 2дfor 1 ][ u(t) (f)+ 12xf 2 12 sgnt j2xf j2zf 13 cos 2лfot u(t) 186) + 4 +√)] + (2/6)² - (2)² 1 2f0 14 sin 2лfor u(t) [8ƒf — fo) — 5(f+fo)] + 4j (2πfo)² - (2лf)² 2лfo 15 e-at sin 2лfot u(t) a > 0 16 e-at cos 2π fot u(t) 17 П "(4) 18 2B sinc (2л Br) 19 A 20 B sinc? (B) 21 n=— 22 e-1²/202 (a + j2nƒ)² + 4x²ƒ a+j2rf (a+j2nƒ)²+4x²ƒZ r sinc (ft) τ Δ ( 2B sinc² (17) (7) (n) foxx (f-nfo) συ Σπε-2(σπι)2 a0 Π fo =