191 8.6 Excwises e A blurred image horn a camera car1 be ixiterpre siiiothirig system Thr irnpiilw rCspCJlmc is t hc blurred spot Lhat would be a sharp poiiil on 12ic original image A radio cliannel with multi-path propagation can be clc.cribed by iui impulse Chwter 8.4.43 ) response that consists of multiple time-shift cd echos ( The inertia of ine~~stiring instruments has a smoothing effect on the measiir~cfsigrialb aiitl srnooths sharp stcps into slowly rising slopes The irrqulsc ~ n cmi s offeii be rnotlrl1t.d by ixit RCkircuit with a largc If the i~iipulwrespoiise h ( t ) is known tor sucli a sybtcrn, then it is porsible atterript to invert the iirfliiencc of I ) ( i ) by convolution with i t second J,TI-sy.;tcin g ( Iii the icleal case the result is agaiii a delta irrrpulsc: S(t) A h j t ) = ( t ) (8.77) The systcm function G ( s ) = L ( g ( t ) }of the second sy5tern c'ari b t i cxpiessed by the systeixr function H ( s ) = L { h ( t ) } ,using (lie cmwolution theorem: (8.78) 1he effect of a systnn with system function G(s) is called deconvoZut/on It is rarely possible t o carry out in the ideal form ot (8.78), because the mcasurcii signals usually have additional iioisc interkrence Jri aclditioa, sigrial compoireiits that weie suppressed completely by thc first sytitein (ZP of thc syst,erii h c tiou), (*annotbe ~ccoveredfroni the second system The filter that rcpresents the best coriipronrisc betwern decoiivolitioii and noise reduction will be introduced in ptcr I8 as the Wiencr filter A sy rri with systcin ftirrction G ( s ) (8.78) could also be unstable This problem will be dealt with in Scctioii 16.3 Exorcise 8.1 Show that the response of an RC-cirmiL to a short rectangle impnlsc (8.7) turns into the irnpiilse response (8.4) for To -+ U Exercise 8.2 Using the calculation rules for thr delta impulse, determine: 3L a) f, = J' e - ' h ( ~ ) c l t -T