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EE303_Bookproblems

# EE303_Bookproblems - 1.1[Find the fundamental period T of...

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Unformatted text preview: 1.1. [Find the fundamental period T of each of the following signals: / 2 (E3 I) si (111:) 8(11; 1‘) sin (2—11- t) cos (3:: t) cosz,n3,co4,3, 5 1.2. Sketch the following signals: ,/ yrf x(t) = sin (%t + 20°) (b) x(t)=t+e3" 05:52 t+2 tS—Z 925,150): 0 45:52 [—2 Zst (d) x(t)=2exp[-t], OSISI, and x(t+1)=x(t)forallt yi. Let cos (TI-t), sin (2m), cos(31rt), sin (417:), cos (3T- 1), sin (% t), x(t)=—r+1, ~15r<0 L 05r<2 2, 2514.:3 0, otherwise {:0 Sketch rm. 2 \ (I1) Sketch x(r — 2}, x(z + 3},x(— 3! — 2), and x6: + 3 and find the analytical expres- sions for these functions. yl/i’s. Sketch the following signals: g4 xl(r} = “(.0 + Suﬁ -— 1) — Zuﬂ -' 2) )12(!)= rm — rt: _ 1) — no ~ 2) {c} x30) = BXP[-flu(f) (d) x40) = 2u(:) + 80 — 1) 1 l - The probability that a random variable x is less than a is found by integrating the proba- bility density function ﬁx) to Obtain P[x s a} = I“ ﬁnd; Given that f(x) = 0.2M): + 2) + O.38(x} + 0.23m: - 1) + 0.1[u(x - 3} — u(x — 6)] ﬁnd (a) P(x 5 -3) lb} Pix 5 1.5] (c) Pp: E 4) (d) P(x 5 6) 2.]. Determine whether the systems described by the following inputfoutput relationships are linear or nonlinear, causal or noncausal, time invariant or time variant, and memoryless or with memory. (a) y(r) = 30) + 3 (b) ya) = 2x20) + 3x0) (1:) HI) = Aw} (d) y(r) = Arx(r) (h) M!) = If: — 5) (i) N) = exvlﬂ!” (j! N!) = xv) xv _ 2) 1 r+ Tﬂ (k) :90) = is] x(¢)d1- r—le a) ‘3'") d; + 2m = 2me 2.3. Evaluate the following convolutions: ta} teeth — am) Hr 60' — b) (b) rect((,r'a) * mouth!) to) meet [Maj * all!) (d) rect Ufa) * sgn(r) (19) my} * “(U (I) titlﬂ} — 110‘ — 1)] * u(t) (g) rectttfa) * I“) (h) rm * [53:10) + [if—f — 1)} ti) [MI +1) - u(1' - l)]sgn(r) * MU) (1} EU) 3" 5'“) 2.4. Graphically determine the convolution of the pairs of signals shown in Figure P14. .tft) Mr} xtr) M!) (a) (b) x (I) Mr) M [c] (d) Figure P14 2.6. The cross correlation of mo different signals is defined as rr R,_u(1)= Jr? x('r)y('r - 0dr = I - .r(-r + 1):,:('-r)d1- ' or (a) Show that Knit] = I0) * yt-r} (bl Show that the cross correlation does not obey the commutative law. (c) Show that R1311) is symmetric (120(1) = Rﬂ(—r)). 2.10. The input to an LTl system with impulse response Mr) is the complex exponential exp Lieu]. Show that the corresponding output is W) = exp [fin-'1 Htw) where H(m) = I no) exp[—jmt]dr 3.8. The signal shown in Figure P3.8 is created when a cosine voltage or current waveform is rectified by a single diode, a process known as half-wave rectification. Deduce the expo- nential Fourier-series expansion for the half-wave rectified signal. 3.10. The signal shown in Figure P310 is created when a sine voltage or current waveform is rectified by a a circuit with two diodes, a process known as full-wave rectification. Deduce the exponential Fourier-series expansion for the full-wave rectified signal. ﬁr) M M H MI!!! Figure P33 Figure P110 3.12. Find the exponentiai Fourier-series representations of the signals shown in Figure P112. Plot the magnitude and phase spectrum for each case. _4—20246: ‘3-l0124; (g) (b) Figure P3.12 4.1. Find the Fourier transform of the followin g signals in terms of X(m). the Fourier trans— form ofxﬁ). (?L_x(—I) ti» m = Wag—1:92 I —x(—t) {c} x00) : {(13 __ “‘3 Jr"‘(r) (e) Benn” = {[1] + :‘ﬂ 2 I — . ﬂy) (f) [mlxﬂj] Z _ (3} ng-Y I \-—-" 4.5. Let X(w) = rect[(w — ])/2]. Find the transform of the following functions. using the properties of the Fourier transform: {a} xl-I) {b} ler) {c} x(t+l} (d) x{-2t+4) (e) {t — 1)x(r + 1) am) (1') 0.! {h} M: - Heml‘ﬂr] (i) x(I)BXP[—ﬂf] {j} rx(I)CXP[-i2tl 4.6. Let x0) = exp[—2I]u(r) and let yﬁ} = 10' + I) + x(.r ‘ 1). Find Y(w). 1—- '1‘ 4.9. Find the energy of the following signals. using Parseval‘s theorem. {a} x03) = expf—2:]u(t) {b} .r(:) = um * u(r — 5) {c} 4’0) = MIN) sin(1r_r) (d) x(t) = — 11'! 4.15. A signal has Fourier transform X( }_ so? +__j4uo + 2" "’ " -m2 +j4m + 3 Find the transforms of each of the following signals: (a) x(—21 + l) ('3) W) Expl—ﬂ] dx (t) (c) d: (d) x(1‘) sin (11!) {8) IO) * 50‘ - 1} (f) I(f)*1(f'"1} 4.26 4.27. 4.28 . As discussed in Section 4.41, AM demodulation consists of multiplying the received sig— nal y(t) by a replica, Acos (not, of the carrier and low-pass filtering the resulting signal 7, (3). Such a scheme is called synchronous demodulation and assumes that the phase of the car- rier is known at the receiver. If the carrier phase is not known, z (r) becomes z(t) = y(t)Acos(mor + 8) where 6 is the assumed phase of the carrier. (a) Assume that the signal x(r) is band limited to mm, and find the output \$0) of the demodulator. (b) How does Ht) compare with the desired output x{t)‘? A single-sideband. amplitude-modulated signal is generated using the system shown in Figure P421 (a) Sketch the spectrum of y(r) for “r = mm. ([1) Write a mathematical expression for hf(t). Is it a realizable ﬁlter? Mtw) aqua) en: an t Figure P427 . Consider the system shown in Figure P4.28{a). The systems 11,0) and i120) respectively have frequency responses lel 010) sinwor (a) Figure P4.28(a) H10») = £11100» — mu) + Holt” + won and —1 HA“) : 2j—[Hn(w " ‘90) _ Helm + 090)] (a) Sketch the Spectrum of y(t). (l1) Repeat part (a) for the How} shown in Figure P4.28(b]. {10(9)} ‘uo - u. “'“o 0 we mo + can to (b) Figure P4.28(b} 4.30. In natural sampling. the signalx (I) is multiplied by a train of rectangular pulses. as shown in Figure P430. (10 Find and sketch the spectrum of 13(1). (I1) Can x(.*) be recovered without any distortion? no) X(w} ﬂr B xi!) X9“) 1 A _23r' _T 0 T IT I _wc 0 one u: p (t) Figure P430 Example 5.5.1 Suppose we want to ﬁnd the Laplace transform of (A + B exp[~br])rr(i) From Table 5.1, we have the transform pair 1 i Mr) H s and exp[—br]u{r) <—) m Thus, using linearity, we obtain the transform pair A B (A + 3).: + Ab A + _ E ,_ _.._. = _____-...._ “(1) Bupl 5‘1”“) s + s + b 5(s + b) The ROC is the intersection of Re is} b -—b and Re M > O, and, hence, is given by Rels] > max(-b,0). Example 5.5.2 Consider the rectangular pulse 350:) = rect[(t — a)/2a). This signal can be written as rect((t — a)/2a) = MU) -— u(t — 2a) Using linearity and time shifting, we find that the Laplace transform of x(t) is 1_ _. X{s) = :- — (mm—255]: = -M, Re{s} >0 It should be clear that the time shifting property holds for a right shift only. For exam- ple. the Laplace transform of x(r + to), for to > 0, cannot be expressed in terms of the Laplace transform of x(t). (Why?) Example 5.5.3 From entry 8 in Table 5.1 and Equation (5.5.3), the Laplace transform of x(t) = A expf—at] cosmnr + 6)u(r) is X{s} = EElA exp[-ar](eoswntcose — sinwor sin8)u(r)l = .‘ElA exp[—ar} cosmotcose r40)! — \$[A exp[-ar] sinwor sine 110)} _ 11(3 1:“) c059 _ "”4030 sine— _ (5 + (I)? + {.03 (5 + “)2 _+-"33 _ A [(5 1.93'9953l05i9 61 _— (s + £02 + m3 Re{s} > —a Example 5.5.13 Suppose that the input x(t) = exp [—2r}u (t) is applied to a relaxed (zero initial conditions) LTI system. The output of the system is ya) = Emmi—:1 + eXP[—211* expi~3r1)u(r) Then 1 X(s)_s+2 and 2 2 2 + 3n + 1) 3(3 + 2) _ 3(s + 3) Using Equation (5.5.14). we conclude that the transfer function H (s) of the system is _g 2(s+2)_2(s+2) H(S)_3+3(s+l) 3(s+3) _ 2(sz+6s+7) _3(s+1)(s+3) 2 1 1 =— +——— 3[1 + J from which it follows that Mr) = £50) + % [exp{—r] + exp[-3t]]u(t) ...
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EE303_Bookproblems - 1.1[Find the fundamental period T of...

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