Manipulations involving the signal amplitude (dependent variable).
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1 Oulie Maipulaio of discree ime sigals: Maipulaios ivolvig he idepede variable : Shifed i ime Operaios. Foldig, reflecio or ime reversal. Time Scalig. Maipulaios ivolvig he sigal ampliude (depede variable). Maipulaio of discree ime sigals Maipulaios ivolvig he idepede variable. Maipulaios ivolvig he sigal ampliude (depede variable). Maipulaio ivolvig he idepede variable (ime):. Shifed i ime Operaios: Give a D-T sigal x [ ad a posiive ieger p, he y[ p] is he p -sep righ shif of x [ ha resuls i a delay of he sigal by p uis of ime (replacig by p ). y[ p] is he p -sep lef shif of x [ ha resuls i a advace of he sigal by p uis of ime (replacig by p ). Examples: a) P [ ] : Three-sep righ shif of D-T recagular pulse P [ ] (see figure -8). b) P [ ] : Three-sep lef shif of D-T recagular pulse P [ ] (see figure -8). P [ ] P [ ] Figure _8: Shifed D-T Recagular Sigals c) The D-T sigal, x [, 0 0 ad
2 Fid he ime-shifed sigal y [ ] Aswer:, x [ 5, 6 0, 6 ad. Foldig, reflecio or ime reversal: Le x [ be he origial sequece, ad y [ be refleced sequece, he y [ is defied by y[, his meas ha we replace he idepede variable by ; he resul of his operaio is a foldig or reflecio of he sigal abou he ime origi 0. I is impora o oe ha he operaio of foldig ad ime delayig (or advacig ) a sigal are o commuaive: if SO (shifed operaio, for example Time-Delay) ad FO (foldig operaio), we ca wrie: SO k {} = -k], k>0. FO{}=-. Now: SO k {FO{}}= SO k { -}= -+k] where as FO{SO k {}}=FO{-k}= --k] so : SO k {FO{}} FO{SO k {}} Examples: a) Show he graphical represeaio of he sigals ad x [ ], where x [ is he sigal illusraed i figure Aswer: (see figure -0) Figure _9
3 - - - ] Figure _0 A simple way o verify ha he resul is correc is o compue samples, such as: y[ ] ], y[ ] y[ ] y[0] y[] y[] ], ], ] 0, 0], ], y[] ] b) The D-T sigal 0 oherwise Fid he composie sigal y[ Aswer: y [ 0; for all ieger values of. I able -, he precedece rules for he ime shifig ad he ime foldig operaios are explaied. Table -: Precedece rules for ime shifig ad ime foldig Order of shifig ad foldig Oupu sigal operaios FO SOR. Foldig Shif o he righ ( p)] p] SO FO. Shif o he lef Foldig L p] p]. Foldig Shif o he lef FO SO L ( p)] p] SO FO. Shif o he righfoldig R p] p]
4 . Time Scalig Le x [ deoe a D-T sigal, he he sigal y [ obaied by scalig he idepede variable, ime, by a facor a is defied by y [ a, a 0. If a, he sigal is a compressed versio of x [ ad some values of he discree ime sigal y[ are los. if 0 a, he he sigal y [ is a expaded versio of x [. Example: a) For a ; i x [, he samples x [ for,, 5,... are los. for odd b) 0 oherwise Deermie y[ Aswer: y[ 0 for all Precedece Rules for ime shifig ad ime scalig: C-T case: Suppose ha y( x( c s), his relaio bewee y ( ad x ( saisfies he codiios: s y( 0) x( s) ; ad y( ) x(0) c which provide useful checks o y ( i erms of correspodig values of x (. The correc order for ime shifig ad scalig operaios: a) The ime shifig operaio is performed firs o x (, we ge a iermediae sigal v( x( s) ; he ime shif has replaced by s. b) The ime scalig operaio is performed o v (, replacig by c ad he resul y( v( c x( c s). Examples:. Voice sigal recorded o a ape recorder: Compressio: if he ape is played back a a rae faser ha he origial recordig rae. Expasio: if he rae is slower ha he origial.. Cosider he recagular pulse x ( of ui ampliude ad a duraio of uis, depiced i figure -. Fid y ( x( ). Soluio: s c, s y(0) x( ) 0; y( ) y( ) x(0), he graphical soluio is c represeed i figure -.
5 v(. Shif o he righ v( x( ) x( y(. Scalig y( v( Figure _: Recagular Sigal D-T case: The same rules are used i he case of D-T sigals, i he followig example, hese rules are explaied. Example: x [,, 0,,. Fid y [ ] Suppose ha Soluio: s c, s y[0] ] 0; y[ ] y[ ] 0] 0, c The graphical soluio is represeed i figure -. To ge y[ v[, we calculae he followig pois: y [ 0], y[] ad y [] y[0] v[0] y[] v[] y[] v[6] Figure _ v[. Shif o he righ: v[ ] v[. Scalig: y[ v[ Figure _ 5
6 Maipulaio ivolvig he sigal ampliude (depede variable): Trasformaios performed o ampliude (depede variable) are show i able -. Table -: Trasformaio performed o ampliude Operaio D-T sigals C-T sigals Physical device. Ampliude scalig y[ c y( cx( Elecroic c - scalig facor amplifier. Addiio y[ y( x( x( Audio mixer. Muliplicaio y[ y( x( x( Modulaor. Differeiaio Differece equaio x( y( d d Iducor 5. Iegraio Summaio y( x( ) d Capacior 6
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