14.6 Discrete Corivalutioii and Tnipulsc~Hrsporise 35 Figiire 14.11: Block diagram of t,hr discrrtc systrm in Example 14.3 The impulse response of a system is its reaction to a pulse-shaped input I ‘l’hc impi~lseresponse is obtained by inverse transforlning the system function The output signal of a systern is obt,a.inedin the time-domain by convolution of the input sigiial and the impulse response The dcriviition of blicse properties was not simple, as the pulse-shaped input needed to geiierate itri irnpulse respoiise is not an ordina,ry function The price for the elega7nccof this system descripliori was generalising fiinctions to distributions The delta irriplusr introdiiced in Chapter 8.3 cxmnot be chara.c,t,erisedsimply by it,s vihlues, only by its effects on other funct,ions, in particular, the selective property .scd on thc irnpulse respoiise can be transferred to that, for a discrete system, tlic impulse response has the same fiintlamcnt,al properties as for a contirmous system In one poivrt the description of discrete syskrns becorncs a, lot simpler: the pulse-skiaped input signal requirccl to genera.te tlic impulse rrsponsc is now a very simple sec1uenc:e It i s the unit, impulse iritxoctimd in Chapterl2.2.1, which is a sequence of zeros and a single one Nre can deal with these w,lues without the use of generalised fimctions 14.6.1 ~ a ~ c ~Q€~the a System ~ i ~ n iscrete ~ o ~ i v Q l ~ t ~ ~ n To introduce the impulse response of a discrete system wc consider Figiire 14.12 It shows a discrete system with input signal c[k] arid output signal pylk] T h e response h [ k ] to a ctisc*r& unit impulse (see Chapter 12.2.1) will be called thc impulse response of a discrete system