356 14 Discrete-Time LTI-Syst,cnis For a = this system represents ari nc.c*imiulatorthat sunib all incoming input values The irrrpnlse response of tlie accurnulator is the discrete unit Figure 14.14 for a = 1) We calculated the F*-tr;ansform of the discrete writ step iir Chapter 12.3.3.3aiid the corresponding z-transform in Example 13.2 The folIotviiig tcrnis will ) i ~i s t ~ i : Systems with finite impulse responses are also called non-rccurszve or FIRsystems (FIR - Finite I m p u l s ~Response) Systems with iiifinit,e impulse respoiwe are alsct called rem~r'szueor TIRsystems (IIR Infinite Iinynlse Response)& 14.6.4 Discrete C o ~ v o l ~ t i o r i practical irnportnnce of discrete convolution comes about because [lie expression iyI :/[x.] ~[x.I ~ r / ? [ ~ j [pi] h[k ICI (14.38) K=-m c m be implcmeiitetl irnrrrcdiately as a computcr progr i>iYj In the usual programming languages, two FOR-loops are required to calculate (14.38), wlierf the outer loop runs via tlie iiidex X and the irriier loop via the index IC Tliere are, liowevw, processors with ii special archit ecd i i r e (digital signal procclssors) which cmi carry n with K iri (14.38) as lie scalai protlnct of t W J vectors very khtx preparation illid testing of such programs it is vital to have convolution on paper We will therefore dea1 with it, iii even greater detail Lhan tlie calculation of the convolution integral in Chapter 8.4.3 To begin with we show a rnetJiod similar to thc convolutioii iritegral in Chapter 8.4.3: Draw 44 and h [ ~with ] I Sliift h f - ~ ,by ] k positions to the right: h[-r;] -+ h [ k - kj 4 Miiltiplicatiou of = [ ~with ] h [ k - and summation of the product for all values ol I(: yields orw vdiic\ of y[k], Repeat steps and fc eps (*ailalso be carried ollt es Tile c;dciiIat,ion of cliscretc cymvoliiiion iisirrg tllese in sonw examples iise of the comniutativity of convolution, th