From definition (3.8a), find the Fourier transfonns of the signals get) shown in Fig. P3.l-4. (a)

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1 Prblems f get) <=> C(w), then shw that g*(t) <=> C*(-w). Frm definitin (3.8a), find the Furier transfnns f the signals get) shwn in Fig. P3.l-4. T {- T (... d.,mcgra l,newy Figure P3.-4 Frm definitin (3.8a), find the Furier transfnns f the signals shwl) in Fig. P3.l-5. lld.ichaet C 4/ g(t) get)t A --'f L J rransfrm. 2 -t 0 t, Figure P3.-5 Frm definitin (3.8b), find the inverse Furiertransfnns f the spectra shwn in Fig. P3.-6. r)ar_ -r 0 < Figure P3.-6 Frm definitin (3.8b), find the inverse Furier transfnns f the spectra shwn in Fig. P3.-7. G(<O)'j n "2 r W --<0 0 0 <0 0 es," Mathe <0 :hecmpact Figure P3.-7 the trign " Find the inverse Furier transfnn f C(w) fr the spectra shwn in Fig. P3.-8.

2 44 ANALYSS AND TRANSMSSON OF SGNALS G(» G(» 7t 0>_ 7t Figure P3.-8 Hint: G(w) = G(w)lejOg(w). Fr part, G(w) = le- jwt, wl:::: W. Fr part, G(w) _ j { le- "/2 = - j 0 < w :::: W - le j "/2 = j 0 > w 2: -W This prblem illustrates hw different phase spectra (bth with the same amplitude speetru represent entirely different signals Sketch the fllwing functins: reet (t/2); t,(3w/00); reet (t - 0/8); (d) sine (rrw/5) ; (e) sine [(w - Orr)/5. (f) sine (t/5) reet (t/lorr). Hint: gca) is g() right-shifted by a. -2 Frm definitin (3.8a), shw that the Furier transfrm f reet (t - 5) is sine (w/2)e-j5w Y-3 (rrt) e jlor Frm definitin (3.8b), shw that the inverse Furier transfrm f reet [(w - O)/2rr] is s' Using pairs 7 and 2 (Table 3.) shw that u(t) {=} rr8(w) + l/jw Shw that cs (wt + (}) {=} rr[8(w + W)e- jo + 8(w - W)e j ]. Hint: Express cs (wt + in terms f expnentials using Euler's frmula Apply the symmetry prperty t the apprpriate pair in Table 3. t shw that: O.5[8(t) + (j/rrt)] 8(t+T)+8(t-T) {=}2es Tw 8(t + T) - 8(t - T) {=} 2j sin Tw. Hint: g(-t) {=} G(-w) and 8(t) = 8(-t). The Furier transfrm f the triangular pulse g(t) in Fig. P3.3-2a is given as.. G(w) = _(e Jw - jwe JW - ) w 2 Using this infrmatin, and the time-shifting and time-scaling prperties, find the Furier transfrms f the signals shwn in Fig. P3.3-2b, e, d, e, and f. Hint: Time inversin in g(t

3 Prblems 45 00_ results in the pulse gl (t) in Fig. P3.3-2b; cnsequently gl (t) = g(-t). The pulse in Fig. P3.3-2c can be expressed as g( - T) + gl (t - T) [the sum f get) and gl () bth delayed by T]. The pulses in Fig. P3.3-2d and e bth can be expressed as g( - T) + gl (t + T) [the sum f get) delayed by T and gl () advanced by T] fr sme suitable chice f T. The pulse in Fig. P3.3-2f can be btained by time-expanding g (t) by a factr f 2 and then delaying the resulting pulse by 2 secnds [r by first delaying get) by secnd and then time-expanding by a factr f 2]. get) 00_ t t 2 g4().5 g/t) - (d) t- -2 (e) 2 t 2 (f) ctrum )7")/5]: ;w is sine 't + B) Figure P l -T Using nly the time-shifting prperty and Table 3., find the Furier transfrms f the signals shwn in Fig. P Hint: The signal in Fig. P3.3-3a is a sum f tw shifted gate pulses. The signal in Fig. P3.3-3b is sin t [u(t) - u(t - 7")] = sin t u(t) - sin u(t - 7") = sin u(t) + sin (t - 7") U( - 7"). The reader shuld verify that the additin f these tw sinusids indeed results in the pulse in Fig. P3.3-3b. n the same way we can express the signal in Figs. P3.3-3c as cs u(t) + sin (t - 7" /2)u (t - 7" /2) (verify this by sketching these signals). The signal in Fig. P3.3-3d is e- a, [U() - u(t - T)] = e-atu(t) e- at e-a(t-t)u( - T). nl2,- n 'St-- T _ (d) Figure P3.3-3 luner get) 3.3/Using the time-shifting prperty, shw that jfg(t) ::=:::;- G(f.t)), then >Y get + T) + g( - T) <==> 2G(w) cs Tw This is the dual f Eq. (3.35). Using this result and pairs 7 and 9 in Table 3., find the Furier transfrms f the signals shwn in Fig. P3.3-4.

4 46 ANALYSS AND TRANSMSSON OF SGNALS J J b 6 J ji. P3: h fllwi, '''0.,. g(t) Sn wt 2j [G(w - w) - G(w + W)] 2j [g(t + T) - g(t - T)] G(w) sin Tw Using the latter result and Table 3., find the Furier transfrm f the signal in Fig. P OJ 3. is T signals in Fig. P3.3-6 are mdulated signals with carrier cs lot. Find the Furier transfm::: Figure P3.3-5 these signals using the apprpriate prperties f the Furier transfrm and Table 3.. Sket the amplitude and phase spectra fr parts and. Hint: These functins can be expressed the frm g (t) cs wat. t... Figure P3.3-6 jj-7 Using the frequency-shifting prperty and Table 3., find the inverse Furier transfrm f spectra shwn in Fig. P J,.) at_-l:--t-- -s (by Figure P3.3 7

5 Prblems 49 gh the tin f inless in fr is /2a.,jii. Verify this result by deriving the energy E g frm G(w) using Parseval's therem. Hint: See pair 22 in Table 3.. Use the fact that i:e x2 dx =.,jii by Shw that 00 sinc 2 (kx) dx = -00 Hint: Recgnize that the integral is the energy f g(t) Parseval's therem. rr k = sinc (kt). Find this energy by using terfer 00 gl(t)g2(t) dt = G (-W)G2(w) dw = G (W)G2(-w) dw -00 2rr -00 2rr -00 7"' g(t) 'Z/eneralize Parseval's therem t shw that fr real, Furier transfrmable signals gl (t) and delay in be delay nsfer Shw that sinc (2rr Bt -00 Hint: Recgnize that mrr) sinc (2rr Bt - nrr) dt = { 2B sinc (2rr Bt krr) = sinc [2rr B (t!--)] <==> 2B Use this fact and the result in Prb t shw that m i: n m =n rect () e-jwk/2b 2B 4rrB 00 2 "B. sinc (2rr Bt mrr) sinc (2rr Bt nrr) dt = - e}[(,,-m/2bwdw -00 8rr B2-2" B The desired result fllws frm this integral. Fr the signal 2a g(t) = t +a determine the essential bandwidth B Hz f g(t) such that the energy cntained in the spectral cmpnents f g(t) f frequencies belw B Hz is 99% f the signal energy E g. Hint: Determine G(w) by applying the symmetry prperty [Eq. (3.24)] t pair 3 f Table A lw-pass signal g(t) is applied t a squaring device. The squarer utput g2(t) is applied t a unity gain ideal lw-pass filter f bandwidth 6.j Hz (Fig. P3.7-6). Shw that if 6.j is very small (6.j 0), the filter utput is a dc signal f amplitude 2E g 6.j, where E g is the energy f g(t), Hint: The utput y(t) is a dc signal because its spectrum Y(w) is cncentrated at w = 0 frm -6.w t 6.w with 6.w 0 (impuls,e at the rigin), f g2(t) <==> A(w), and y(t) <==> Y(w), then Y(w) [4rr A(0)6.f]8(w), 8(/).. 2 ()2 8 () 4>;;SS.,,:()", 2EgD.f Figure P3.7-6