4 Convolution. Recommended Problems. x2[n] 1 2[n]

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1 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P Inpu x[n] Oupuy [n] III xn] yi[n] 0 0 n 0 n x2[n] 1 2[n] 0 0 n 0 0 n X 3 [n] y3[n].-. n -n Figure P4.1-1 Deermine he response y 4 [n] when he inpu is as shown in Figure P I x4[n] Figure P4.1-2 (a) Express x 4 [n] as a linear combinaion of x 1 [n], x 2 [n], and x 3 [n]. (b) Using he fac ha he sysem is linear, deermine y 4 [n], he response o x 4 [n]. (c) From he inpu-oupu pairs in Figure P4.1-1, deermine wheher he sysem is ime-invarian. P4-1

2 Signals and Sysems P4-2 P4.2 Deermine he discree-ime convoluion of x[n] and h[n] for he following wo cases. (a) x[n] h[n] Fiur 11J P (b) Figure P h[ni x [n ] X Figure P4.2-2 P4.3 Deermine he coninuous-ime convoluion of x() and h() for he following hree cases: (a) x() h () Figure P4.3-1

3 Convoluion / Problems P4-3 (b) x() h () 1-- -(- 1) u ( - 1) u( Figure P4.3-2 (c) x() h () F6(-2) -l Figure P4.3-3 P4.4 Consider a discree-ime, linear, shif-invarian sysem ha has uni sample response h[n] and inpu x[n]. (a) Skech he response of his sysem if x[n] = b[n - h[n] = (i)"u[n]. no], for some no > 0, and (b) Evaluae and skech he oupu of he sysem if h[n] = (I)"u[n] and x[n] = u[n]. (c) Consider reversing he role of he inpu and sysem response in par (b). Tha is, h[n] = u[n], x[n] = (I)"u[n] Evaluae he sysem oupu y[n] and skech. P4.5 (a) Using convoluion, deermine and skech he responses of a linear, ime-invarian sysem wih impulse response h() = e- 2 u() o each of he wo inpus x 1 (), x 2 () shown in Figures P4.5-1 and P Use yi() o denoe he response o x 1 () and use y 2 () o denoe he response o x 2 ().

4 Signals and Sysems P4-4 (i) X 1 () = u() 0 Figure P4.5-1 (ii) x 2 () Figure P4.5-2 (b) x 2 () can be expressed in erms of x,() as x 2 () = 2[x() - xi( - 3)] By aking advanage of he lineariy and ime-invariance properies, deermine how y 2 () can be expressed in erms of yi(). Verify your expression by evaluaing i wih yl() obained in par (a) and comparing i wih y 2 () obained in par (a). Opional Problems P4.6 Graphically deermine he coninuous-ime convoluion of h() and x() for he cases shown in Figures P4.6-1 and P4.6-2.

5 Convoluion / Problems P4-5 (a) h() x() Figure P4.6-1 (b) h () x() Figure P4.6-2 P4.7 Compue he convoluion y[n] = x[n] * h[n] when Assume ha a and # are no equal. x[n] =au[n], O < a< 1, h[n] =#"u[n], 0 < #< 1 P4.8 Suppose ha h() is as shown in Figure P4.8 and x() is an impulse rain, i.e., x() = ( of-kt) k= -o0

6 Signals and Sysems P4-6 (a) Skech x(). (b) Assuming T = 2, deermine and skech y() = x() * h(). P4.9 Deermine if each of he following saemens is rue in general. Provide proofs for hose ha you hink are rue and counerexamples for hose ha you hink are false. (a) x[n] *{h[ng[n]} = {x[n] *h[n]}g[n] (b) If y() = x() * h(), hen y(2) = 2x(2) * h(2). (c) If x() and h() are odd signals, hen y() = x() * h() is an even signal. (d) If y() = x() * h(), hen Ev{y()} = x() * Ev{h()} + Ev{x()} * h(). P4.10 Le 1 1 () and 2 2 () be wo periodic signals wih a common period To. I is no oo difficul o check ha he convoluion of 1 1 () and 2 () does no converge. However, i is someimes useful o consider a form of convoluion for such signals ha is referred o as periodicconvoluion.specifically, we define he periodic convoluion of 1 () and X 2 () as TO g() = T 1 (r)- 2 ( - r) dr = 1 ()* 2 () (P4.10-1) Noe ha we are inegraing over exacly one period. (a) Show ha q() is periodic wih period To. (b) Consider he signal a + T 0 Pa() 1(rF)2( - r) dr, = fa where a is an arbirary real number. Show ha 9() = Ya() Hin: Wrie a = kto - b, where 0 b < To. (c) Compue he periodic convoluion of he signals depiced in Figure P4.10-1, where To = 1.

7 Convoluion / Problems P4-7 e R2 () Figure P (d) Consider he signals x1[n] and x 2 [n] depiced in Figure P These signals are periodic wih period 6. Compue and skech heir periodic convoluion using No = 6. IT I '1 x, [n] I II... I-61 0II T II.. 2 1? 11 X2 [n] Figure P (e) Since hese signals are periodic wih period 6, hey are also periodic wih period 12. Compue he periodic convoluion of xi[n] and x2[n] using No = 12. P4.11 One imporan use of he concep of inverse sysems is o remove disorions of some ype. A good example is he problem of removing echoes from acousic signals. For example, if an audiorium has a percepible echo, hen an iniial acousic impulse is

8 Signals and Sysems P4-8 followed by aenuaed versions of he sound a regularly spaced inervals. Consequenly, a common model for his phenomenon is a linear, ime-invarian sysem wih an impulse response consising of a rain of impulses: h() = [ hkb(-kt) (P4.11-1) k=o Here he echoes occur T s apar, and hk represens he gain facor on he kh echo resuling from an iniial acousic impulse. (a) Suppose ha x() represens he original acousic signal (he music produced by an orchesra, for example) and ha y() = x() * h() is he acual signal ha is heard if no processing is done o remove he echoes. To remove he disorion inroduced by he echoes, assume ha a microphone is used o sense y() and ha he resuling signal is ransduced ino an elecrical signal. We will also use y() o denoe his signal, as i represens he elecrical equivalen of he acousic signal, and we can go from one o he oher via acousic-elecrical conversion sysems. The imporan poin o noe is ha he sysem wih impulse response given in eq. (P4.11-1) is inverible. Therefore, we can find an LTI sysem wih impulse response g() such ha y() *g() = x() and hus, by processing he elecrical signal y() in his fashion and hen convering back o an acousic signal, we can remove he roublesome echoes. The required impulse response g() is also an impulse rain: g() = ( k=o gkao-kt) Deermine he algebraic equaions ha he successive gk mus saisfy and solve for gi, g 2, and g 3 in erms of he hk. [Hin: You may find par (a) of Problem 3.16 of he ex (page 136) useful.] (b) Suppose ha ho = 1, hi = i, and hi = 0 for all i > 2. Wha is g() in his case? (c) A good model for he generaion of echoes is illusraed in Figure P4.11. Each successive echo represens a fedback version of y(), delayed by T s and scaled by a. Typically 0 < a < 1 because successive echoes are aenuaed. x() ± y() Delay T (i) Figure P4.11 Wha is he impulse response of his sysem? (Assume iniial res, i.e., y() = 0 for < 0 if x() = 0 for < 0.) (ii) Show ha he sysem is sable if 0 < a < 1 and unsable if a > 1. (iii) Wha is g() in his case? Consruc a realizaion of his inverse sysem using adders, coefficien mulipliers, and T-s delay elemens.

9 Convoluion / Problems P4-9 Alhough we have phrased his discussion in erms of coninuous-ime sysems because of he applicaion we are considering, he same general ideas hold in discree ime. Tha is, he LTI sysem wih impulse response h[n] = ( hks[n-kn] k=o is inverible and has as is inverse an LTI sysem wih impulse response g[n] = (g [nkn] k=o I is no difficul o check ha he gi saisfy he same algebraic equaions as in par (a). (d) Consider he discree-ime LTI sysem wih impulse response h[n] = ( S[n-kN] k=-m This sysem is no inverible. Find wo inpus ha produce he same oupu. P4.12 Our developmen of he convoluion sum represenaion for discree-ime LTI sysems was based on using he uni sample funcion as a building block for he represenaion of arbirary inpu signals. This represenaion, ogeher wih knowledge of he response o 5[n] and he propery of superposiion, allowed us o represen he sysem response o an arbirary inpu in erms of a convoluion. In his problem we consider he use of oher signals as building blocks for he consrucion of arbirary inpu signals. Consider he following se of signals: $[n] = (i)"u[n], #[n ] = [n - k], k = 0, 1, ±2 3,... (a) Show ha an arbirary signal can be represened in he form + 00 x[n] = ( ak4[n - k] k= by deermining an explici expression for he coefficien ak in erms of he values of he signal x[n]. [Hin:Wha is he represenaion for 6[n]?] (b) Le r[n] be he response of an LTI sysem o he inpu x[n] = #[n]. Find an expression for he response y[n] o an arbirary inpu x[n] in erms of r[n] and x[n]. (c) Show ha y[n] can be wrien as y[n] = 0[n] * x[n] * r[n] by finding he signal 0[n]. (d) Use he resul of par (c) o express he impulse response of he sysem in erms of r[n]. Also, show ha 0[n] *#[n] = b[n]

10 MIT OpenCourseWare hp://ocw.mi.edu Resource: Signals and Sysems Professor Alan V. Oppenheim The following may no correspond o a paricular course on MIT OpenCourseWare, bu has been provided by he auhor as an individual learning resource. For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.