Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio : Digital sigals are discrete i both time (the idepedet variable) ad amplitude (the depedet variable). Sigals that are discrete i time but cotiuous i amplitude are referred to as discrete-time sigals. Z Trasform is a powerful tool for aalysis ad desig of Discrete Time sigals ad systems. The Z Trasform differs from Fourier Trasform because it covers a broader class of DT sigals ad systems which may or may ot be stable. Fourier Trasformoly exists for sigals which ca absolutely itegrated ad have a fiite eergy.z-trasforms is a geeralizatio of Discrete-time Fourier trasform by cosiderig a broader class of sigals. Descriptio: Discrete Time Sigal represetatio Discrete-time sigals are data sequeces. A sequece of data is deoted {x[]} or simply x [] whe the meaig is clear. The elemets of the sequece are called samples. The idex associated with each sample is a iteger. If appropriate, the rage of will be specified. Quite ofte, we are iterested i idetifyig the sample where = 0. This is doe by puttig a arrow uder that sample. For istace, The arrow is ofte omitted if it is clear from the cotext which sample is x[0]. Samplevalues ca either be real or complex. The time iterval betwee samples is ot explicitly show. It ca be assumed to beormalized to uit of time. So, the correspodig ormalized samplig frequecy is Hz. If the actual samplig iterval is T secods, the the samplig frequecy is give by fs=/t. Differece betwee cotiuous ad discrete time sigals A cotiuous time sigal is defied cotiuously with respect to time. A discrete time sigal is defied oly at specific or regular time istats
The example of cotiuous sigal show i the figure above is x(t)=e- at. Note that this sigal is cotiuous fuctio of time. The figure also shows a discrete time versio of expoetial sigal. It is defied as x() = e a T s Thus the discrete time sigal has values oly at 0,T s,2t s,3t s, It is ot defied over cotiuous time. Sigificace i) Aalog circuits process cotiuous time sigals. Such circuits are op-amps, filters, amplifiers, etc. ii) Digital circuits process discrete time sigals. Such circuits are microprocessors, couters, flipflops etc. Discrete-time complex expoetial ad siusoidal sigals Whe the expoet is purely imagiary, the the sigal is said to be complex expoetial sigal. a) Siusoidal Sequeces A siusoidal sequece has the form x() = Acos(ω o + ) This fuctio ca also be decomposed ito its i-phase x i [] ad quadrature x q [] compoets. x[] = Acosøcos ω o +Asiøsiω o = x i [] + x q [] This is a commo practice i commuicatios ad sigal processig. b) Complex Expoetial Sequeces Complex expoetial sequeces are essetially complex siusoids. x()=a e j(ωo+ø) =A cos(ωo+ø)+jasi(ωo+ø) Bilateralz-Trasform Cosider applyig a complex expoetial iput x()=z to a LTI system with impulse respose h(). The system output is give by Where H z = y = h x = h k x k k= = z h k z k = z H(z) k= = h k z k k= k= h k z k or equivaletly H z = = h z H(z) is kow as the trasfer fuctio of the LTI system. We kow that a sigal for which the system output is a costat times, the iput is referred to as a eige fuctio of the system ad the amplitude factor is referred to as the system s eige value. Hece, we idetify z as a eige fuctio of the LTI system ad H(z) is referred to as the Bilateral z-trasform or simply z-trasform of the impulse respose h().
The trasform relatioship betwee x() ad X(z) is i geeral idicated as x Z X(z) Existece of z Trasform I geeral, X z = x z = The ROC cosists of those values of z (i.e., those poits i the z-plae) for which X(z) coverges i.e., value of z for which = x z < Sice z = re jω the coditio for existece is = x r e jω < Sice e jω = Therefore, the coditio for which z-trasform exists ad coverges is = x r Thus, ROC of the z trasform of a x() cosists of all values of z for which x r is absolutely summable. < Relatio betwee Z ad Discrete Time Fourier trasform Whe z = e jω, X e jω = = x e jω correspods to the Discrete Time Fourier trasform (DTFT) of x(), i.e., X z z = e jω = F{x }. Thez trasform also bears a straight forward relatioship to the DTFT whe the complex variable z = re jω. To see this relatioship, cosider X(z) with z = re jω. X z = re jω = x r e jω = or X z = re jω = [x r ]e jω = = F{x r } Uilateralz Trasformhave cosiderable value i aalyzig causal systems ad particularly, systems specified by liear costat coefficiet differece equatios with o-zero iitial coditios( i.e., systems that are ot iitially at rest). The Uilateral z- trasform of a discrete time sigal x() is defied as X z = x z
Relatio betwee Laplace, Fourier ad z- trasforms As we have see the relatio betwee the Laplace ad Cotiuous Time Fourier Trasform ad the relatio betwee the z-trasform ad Discrete Time Fourier Trasform, Lets uderstad the relatio betwee Laplace ad z-trasform. Let x(t) be a cotiuous sigal sampled with a samplig time of T uits. Call this sampled sigal as x s (t). We represet this sampled sigal by x s t = x(kt)δ( kt) k= Applyig the Laplace trasform to x s (t) results L x s t = k= Iterchagig the order of itegratio ad summatio x(kt)δ( kt) e st dt L x s t = x(kt) k= δ( kt)e st dt = x(kt) k= δ( kt)e skt dt = x(kt)e skt k= For uiform samplig x(kt) x(k) δ( kt)dt = x(kt)e skt k= The L x s t = k= x(k)e skt = x k e k= st k Comparig this with the z-trasform formula We get a relatio that z=e st X z = x k z k k=0 Examples: Illustratio Fidig the z-trasform of a) x = a u() X z = a u()z = = a z = az
az For covergece of X(z), we require that <. Cosequetly, the regio of covergece is that rage of values of z for which az <, or equivaletly, z > a ad is show i figure below The X z = az = = az z z a b) x = a u( ) X z = a u( ) z = a z = a z = = = = a z = a z = z = az z a This result coverges oly whe a z <, or equivaletly, z < a. The ROC is show below If we cosider the sigals a u() ad -a u(--), we ote that although the sigals are z differig, their z Trasforms are idetical which is.thus, we coclude that to distiguish z-trasforms uiquely their ROC's must be specified. Summary: z trasforms of elemetary fuctios ad ROC z a
Examples: Solved Problems: Problem :Fid the z-trasform of x() = δ(). What is the regio of covergece? Sice δ = The z trasform of a geeral sigal x() is defied as, = 0 0 0 X z = x z = Z δ = δ z =
Z δ = δ z = z 0 = = SiceZ δ =, which is a costat, the result of z trasform coverges for all values of z, i.e., ROC is etire z-plae Problem 2:Determie the Uilateral z-trasform of the fuctio x = a u( + ) Uilateral z-trasform of x() is give by X z = x z Therefore, X z = a u( + )z = a z = az = az Problem 3: Fid the uilateral z trasform of x = δ + + δ + a +3 u( + ) Uilateral z-trasform of x() is give by X z = x z Therefore, X z = X z = 0 + + δ + z + δ z + a +3 u( + )z a +3 z = + a 3 az a 3 = + az Problem 4:Determie the z-trasform ofx() = 0.2 {u u 4 } Therefore, u u 4 =, 0 3 0, 4, X z = 3 3 0.2 z = 0.2z = Problem 5:Fid the z-trasform of x = 2 u + 3 u( ) We kow that a u Z az ; ROC: z > a AlsoWe kow that a u Z az ; ROC: z < a 0.2z 4 0.2z Therefore,Z x = Z 2 u Z 3 u = = 2z 3z ROC: 2< z <3 z 5z +6z 2 Problem 6: Show that the z-trasform of ay ati-symmetric sequece has a zero at z=
Sice the sigal is ati symmetric x(0) = 0 ad x(-)=-x() We kow that sice x(0)=0 X z = x z = x z + x z = = = x z + x z = = x z x z = = Therefore, X(z)=0 for z= = x z z = Hece the sigal x() if it is ati-symmetric has a zero at z= Problem 7:Fid the z-trasform of a cos π 2 a cos π 2 Z 2 z a z a π cos 2 π cos 2 + z a Problem 8:Fid the two sided z-trasform of the sigal 2 = + z a 2 x() = 3 2 0 Here x() ca be writte as x = 3 u + 2 u( ) Therefore, the z-trasform of x() is ROC: /3 < z < 2 X z = 3 z + 2z
Problem 9:Fid the z-trasform of x() = e 3 u( ) X z = x z = X z = e 3 z = ROC: (e 3 z ) <, i. e., z > e 3 = = (e 3 z ) = = (e 3 z ) + (e 3 z + (e 3 z ) 2 + } = (e 3 z ) (e 3 z ) Problem 0:Fid the z-trasform of x() = δ( m) X z = x z = X z = δ( m)z = Sice δ m is the impulse existig at =m = Therefore, X(z) = z m with ROC etire z-plae except z= Assigmet: Problem : Fid the Uilateral ad Bilateral z-trasform of the sigal x() = a + u( + ) Problem 2: Determie the z-trasform ofx = b ; 0 < b < Problem 3: Fid the z-trasform ad ROC of x = Ar cos ω o + φ u ; 0 < r < Problem 4: Determie the Uilateral z-trasform of x() = 4 u(3 ) Problem 5: If x[] = (/3) (/2) u[], the fid the regio of covergece (ROC) of itsz -trasform i the z -plae. Problem 6: Fid the z-trasform of x = a cos(π/2) Problem 7:Determie the Uilateral z-trasform ofx = {,2,5,4,0,3) where sigal starts at origi Problem 8: Fid the z-trasform of x() = 3 [u u 8 ] Problem 9: Fid the z-trasform of t 2 e at after samplig the sigal at every T= sec Problem 0: Fid the z-trasform ad ROC of x[] = hece plot the frequecy respose. 3 u() 2 u[ ], ad Simulatio: Z-trasform usig MATLAB is performed with the help of the fuctio ztras
Example: Fidig the z trasform of x = 4 u() >>syms >>syms z >>ztras(((/4)^()*heaviside())) as = /(4*z - ) + /2 >> pretty(as) ------- + /2 4 z - Refereces: [] Ala V.Oppeheim, Ala S.Willsky ad S.Hamid Nawab, Sigals & Systems, Secod editio, Pearso Educatio, 8 th Idia Reprit, 2005. [2] M.J.Roberts, Sigals ad Systems, Aalysis usig Trasform methods ad MATLAB, Secod editio,mcgraw-hill Educatio,20 [3] Joh R Buck, Michael M Daiel ad Adrew C.Siger, Computer exploratios i Sigals ad Systems usig MATLAB,Pretice Hall Sigal Processig Series [4] P Ramakrisha rao, Sigals ad Systems, Tata McGraw-Hill, 2008 [5] Taru Kumar Rawat, Sigals ad Systems, Oxford Uiversity Press,20