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Q.1 (20) A DT signal is given as x1 [n] = u[n + 2] - u[n - 3]. a) (2) Use table 5.2 to find the Fourier transform of x1 [n]. If one fundamental period of another 8-periodic signal x2[n] is exactly the same as xa[n], b) (2) Write down the expression of x2[n] in terms of X1 [n]. 8 sinn Another DT signal is given as x3[n] = c) (2) Use table 5.2 to find the Fourier transform of x3[n]. d) (2) Sketch the frequency spectrum of x3[n] for <31. Show important values. πη = If x4[n] = x3[n] * Em-[n - 8m), e) (2) Use tables to find the Fourier transform of x4[n]. f) (2) Sketch the frequency spectrum of x4[n] for 0 <31. Show important values. g) (2) Is the FS coefficients of x4[n] similar to the periodic signal xz[n] ? If your answer is yes, give the name of the property describing the similarity. If your answer is no, briefly explain. The frequency response of a DT filter is given below. 37 4e-jw : 8 H(@jw) = 0 Os| ws <| w5T WST and H(e)w) is 2n-periodic. 377 0 8 If x4[n] is applied to the DT filter to give the output y4[n], h) (2) Sketch the magnitude spectrum of y4[n] for « | <2n. Show important values. i) (2) Write down the mathematical expression of Y4 (ejw) for | 0 | <n. j) (2) Write down the mathematical expression of y4[n].

TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS Fourier transform TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM Section Property Aperiodic signal x(1) x y(t) Signal Fourier transform Fourier series coefficients (if periodic) X(ja) Y() quellement 2. Z 480w - kwoo) a 298W -W) 4, -1 ax-0, otherwise 4.3.1 4.3.2 4.3.6 4.3.3 4.3.5 ax(jw) + bY (jw) ex(jw) Xw-w)) X'(-a) X(-jw) COS! [8(0-0) + 8( + o)] Q-0, otherwise 4.3.5 Linearity Time Shifting Frequency Shifting Conjugation Time Reversal Time and Frequency Scaling Convolution Multiplication Differentiation in Time ax(t) + by() x(1 - 1) Clux(1) x*) *(-1) x(ar) (t) K) + ) X(t) (1) d sinwe! 180w -- ws) 8( + uro)) -0, otherwise 4.4 4.5 xy X(jw) (jw) 1 2X(Han) • Y(ja) xw jax(jw) x(i) 1 2w 8(w) dola, 0, 0 this is the Fourier series representation for any choice of T>0 4.3.4 di (1) 4.3.4 s x(Old! 1 jeux(jas) + "X(0)8(0) Periodic square wave 1. [ <T; x(i) - 0. Till and x{1+T)x() - { <5} * 2 sin kuni 71 8600 - kwa ? ) " sinc kwot Integration Differentiation in Frequency , 7 sin ko kw d 4.3.6 Tx() da(w) 3 84 - 11) 81 ?" 2016 - 2014 a. - for alle - 4.3.3 Conjugate Symmetry X(t) real for Real Signals x(jw) - X'(-10) Relx(j)) - ReX(-)) Sm{X())---(-)) X(w) - X(-1) X(0) --X(-) X(jw) real and even 2 sin aTi x(0) 1. WT 10. W>T xo 4.3.3 sin W W > الله X() - { 4.3.3 X(jw) purely imaginary and odd 0. > W Symmetry for Real and X(t) real and even Even Signals Symmetry for Real and X(t) real and odd Odd Signals Even-Odd Decompo- x) = $v{x() [x(t) real] sition for Real Sig. xo(t) = Od(x() [x() real] - )} (1] nals 8(7) 1 4.3.3 RelX(jo) mo{X(jw)) MO) 1 jo +8) 8(1-10) eu(t). Oefa) > 0 1 d+ja 4.3.7 Parseval's Relation for Aperiodic Signals 1 27 120d = SixCjwildu teu(t), Re{a} > 0 1 1 (a + jo "(t). # Rela) >0 1 (a + jos)

TABLE 5.1 PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM TABLE 5.2 BASIC DISCRETE-TIME FOURIER TRANSFORM PAIRS Signal Fourier Transform Fourier Series Coefficients (if periodic) Section Property Fourier Transform 27k Sate/2/ k(N) 2-3 4:06. - Porte)" at Aperiodic Signal x[n] y[n] ax[n] + by[n] x[n- ne] efon 2ΗΣ 8(a) - wo - 291) 5.3.2 5.3.3 5.3.3 5.3.4 5.3.6 Linearity Time Shifting Frequency Shifting Conjugation Time Reversal (a) Wo 1, km, m N.m + 2N.... ak 0, otherwise (b) irrational → The signal is aperiodic pjuve x[n] X(el) periodic with Y(el) period 27 aX(eds) + by (el) ejung X (edit) X(ell-wron) X'(el) X(e) X(elle) X(el)Y(em) coswort 5.3.7 Time Expansion **[n] x[-n] x[n/k], if n = multiple of k 0, if n #multiple of k x[n]* y[n] x[n]y[n] (81W - wo - 20/) + 8(w + wo - 25/)} (a) wo = 2 * k = Im Im IN, Im + 2N,... ak otherwise (b) irrational → The signal is aperiodic 5.4 Convolution Multiplication 5,5 2. fe xem93%clamando sinon ΤΣ {$(w - wo -2m) - 8(w + wo - 29/)) (a) WO = 21 karir Nr + 2N.... k = --.-- N.-r + 2N... 0, otherwise (b) irrational → The signal is aperiodic 5.3.5 Differencing in Time x[n] – x[n - 1] (1-7)X(el) 1 5.3.5 Accumulation ΣΑΚΙ x[n] = 1 23 8(w - 2m!) k = 0, UN, +2N.... 0, otherwise -- +7X(eº) E 816 – 2017) dX(el) ke=0 5.3.8 Differentiation in Frequency nx[n] Periodic square wave i, in sN x[n] - 0. N, < SN/2 and x[n+ N] = x[n] 21 * ...) sin[(2nk/N(N + )] N sin[27/2N] k+ , N , +2N,... 2N, +1 Nk = 0, 1N, 121.... 8n-kN] 5.3.4 Conjugate Symmetry for (-)。川 - for all k x[n] real Real Signals X(elu) = X*(eds) Re{X(e))} = Re{X(e-/-)} 978{X(e))} = --In{X(e-ju |X(e) = X(e) X(e) = - *X(ej) X(e) real and even au[n], lal! |-- 5.3.4 xre) 1, In S N 10 In> sin[w(N, + ) sin(w/2) 5.3.4 Symmetry for Real, Even x[n] real an even Signals Symmetry for Real, Odd x{n} real and odd Signals Even-odd Decomposition x[n] = {{x[n]} [x[n] real] of Real Signals xo[n] = Od{x{n}} (x[n] real] Parseval's Relation for Aperiodic Signals We sinc (3 sin W 5.3.4 X(1) purely imaginary and odd Re{X(ed)} jm{X(el)} X(w) = (1, 0 = 0 = W 10 W <3 X(w) periodic with period 2 0 <W <T 8[n] 1 5.3.9 un πδίω - 2nk) [ Himle = r pacemy do 1 -- + =- 8[n-nol (n + 1)a":[n), la <1 1 (I - ae" } } (n+r-1)! n!( ra"[n]. Walt 1 (1 -ae-jwy