Notes 20 largely plagiarized by %khc

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1 1 Notes 20 largely plagiarized by %khc 1 Warig This set of otes covers discrete time. However, i probably wo t be able to talk about everythig here; istead i will highlight importat properties or give radom examples. 1 You are advised to cosult your lecture otes ad your textbook, sice you are resposible for everythig. 2 Discrete Time We begi our discussio of discrete time by attemptig to draw some aalogies betwee cotiuous time (CT) ad discrete time (DT). I DT, our basic sigal is []. This fuctio is ozero at 0 oly, as you would expect from a compariso with (t). The major differece is that [] has a height of 1 at 0, whereas (t) has ifiite height ad area 1. Our ext most basic sigal is u[], which ca be costructed from [] by takig the ruig sum from k,1 to k : Compare this to CT: u[] X k,1 [k] t u(t) ( )d,1 This leads us to the coclusio that the local equivalet of itegratio i DT is the ruig sum. Differetiatio i CT the becomes a fiite differece, the differece betwee two or more successive samples. 2 Sometimes, CT differetial equatios are approximated by fiite differeces to covert them to DT differece equatios. Oe more thig before we move o: i DT, complex expoetials ad siusoids exhibitig a somewhat weird behavior. Cosider two differet complex expoetials i CT: e j!1t ad e j!2t.if! 1 ad! 2 are ot equal, the there is o way for the two complex expoetials to look the same as a fuctio of time. Now, cosider the same two complex expoetials i DT. We ll do this by takig t T, effectively samplig the CT sigal with samplig period T. 3 For the particular choice of! 2! T : e j!2t e j(!1+ 2 T )T e j!1t e j 2 T )T e j!1t e j2 e j!1t We ca explicitly surpress the depedece o the samplig period T by defiig a ew variable Ω!T. We refer to! as uormalized frequecy ad to Ω as ormalized frequecy. Ayway, i uormalized or ormalized frequecy, we still ed up with the same basic result i uormalized frequecy, you caot tell the differece betwee a DT complex expoetial at frequecy! 1 ad aother DT complex expoetial! 2! T. Exercise Show that i ormalized frequecy, the DT complex expoetial Ω 2 ad Ω 2 Ω are the same value at iteger values of. 1 Oce agai, radomess is good. 2 The most basic fiite differece is first order: x[], x[, 1]. More o this if you take math128a ad math128b, umerical aalysis parts I ad II, which is highly recommeded if you are goig to ed up codig a lot. 3 More o samplig later. For ow, let s thik about samplig as closig a switch at multiples of time t T ad recordig the resultig values.

2 EE120: Sigals ad Systems; v DT Covolutio The DT equivalet of the siftig itegral is: x[] k,1 x[k][, k] This says that the etire sequece x[] ca be obtaied oe sample at a time, by summig a ifiite sequece of uit samples, appropriately scaled. If we have a liear system H, ad we take this x[] as iput ito our system, we ed up with the output y[]: y[] H[x[]] H[ k,1 k,1 x[k][, k]] x[k]h[[, k]] This is oly from assumig liearity. As with cotiuous time, we ca defie the impulse respose g[; k] H[[, k]] as a fuctio of two variables ad k,where ca be iterpreted as the curret idex, ad k ca be thought of as whe the impulse occurred. Now, if the system is time-ivariat, we do t really care too much whe that impulse happes, oly the differece betwee whe that impulse happeed ad whatever idex we re curretly at. So g[; k] reduces from a fuctio of two variables back dow to a sigle variable: g[; k] h[, k]. A more elegat way of gettig to the same result is to cosider the defiitio of shift-ivariace, which is the local equivalet of time ivariace i DT. 4 Shift ivariace says that if i shift a sigal ad put it ito a system, the output is goig to be the same as if i put the ushifted sigal ito the system first ad shifted it later. I math, the delay ad system operators commute: H[D k [x[]]] D k [H[x[]]] Applyig this defiitio of shift ivariace, we ote that: H[[, k]] H[D k [[]]] D k [H[[]]] D k [h[]] h[, k] wherewehavedefiedh[] H[[]], the impulse respose. Note that i m goig to refer to shift ivariace ad time ivariace iterchageably. 5 The fial form of the covolutio sum becomes: y[] k,1 x[k]h[, k] What are we actually doig here? Well, we ca iterpret our iput as the sum of a buch of scaled ad shifted uit samples. If we kow that our system is LTI ad that it has some impulse respose h[], the if we put i a fuctio at, k, the respose of the system will be h[, k]. Now, if we put i [, k], scaled by x[k], we will get x[k]h[, P k]. If we have a whole buch of fuctios for various values of k, we ca, by liearity, sum up the whole 1 output as,1 x[k]h[, k]. This process is illustrated i Figure 1. 4 The reaso why the ame is differet is that the idex could represet some other variable such as positio istead of time. For a cocrete example, thik of the positio of a pixel i a image. Now, to warp yourbrai, cosidera 3D image chagigas a fuctio of time 5 Sorry, premature seility. Maybe i should take up skydivig ow before i forget to pull the ripcord.

3 EE120: Sigals ad Systems; v δ [] h[] (a) a impulse ad the correspodig impulse respose x0[] y0[] x1[] y1[] x2[] y2[] (b) scalig ad shiftig the impulse produces a scaled ad shifted impulse respose x[] y[] (c) superpositio of x0[], x1[], ad x2[] from (b) gives x[]; likewise, superpositio of y0[], y1[], ad y2[] gives y[]; so if x[] is iput ito our system with impulse respose h[], the correspodig output is y[] Figure 1: DT covolutio.

4 EE120: Sigals ad Systems; v This covolutio sum is somewhat easier to perform tha the CT versio. Please see OWY sectio 3.2 ad TF sectio 8-4 for examples. 6 Exercise Recocile what you that about CT covolutio with what you kow about DT covolutio. Exercise i claim that you have already see DT covolutio as polyomial multiplicatio. For proof by example, multiply x 2 + 2x + 1by3x 2 + x + 2 ad the covolve x[] [, 2] +2[, 1] +[] ad y[] 3[, 2] + [, 1] +2[]. Compare your results. 4 DT Covolutio Tricks DT covolutio is commutative, associative, ad distributive over additio. Exercise Prove this. Covolutio with a impulse gives back the origial sigal: [] x[] k,1 x[] x[k][, k] where we have used the fact that [, k] is ozero oly at k, ad that at k it takes o value 1. Covolutio with a uit step gives the ruig sum: u[] x[] k,1 X k,1 x[k]u[, k] x[k] where we have used the fact that u[, k] is ozero oly for, k 0, or k. Covolutio with a shifted impulse shifts the origial sigal: [, 0 ] x[] k,1 x[, 0 ] x[k][, 0, k] where we have used the fact that [, 0, k] is ozero oly at k, 0. 5 Eigefuctios The output y[] of a DT LTI system H is just the covolutio of the iput x[] with the impulse respose h[]. What if we let x[] be of the form e jω,where! is some real costat? y[] x[] h[] k,1 k,1 X 1 e jω x[, k]h[k] e jω(,k) h[k] k,1 e jω H(e jω ) e,jωk h[k] 6 OWY refers to the first editio of your textbook ad TF to textbook used last semester. Be wared the DT sectios i TF preset the material i a sufficietly cofusig fashio that i do t recommed those sectios to you.

5 EE120: Sigals ad Systems; v where H(e jω ) P 1 k,1 e,jωk h[k]. We will see later that H(e jω ) is the discrete-time Fourier trasform (DTFT) of h[]. Note also that e jω is a eigefuctio. What if we let x[] be of the form z,wherez is some complex costat? y[] x[] h[] k,1 k,1 X 1 z k,1 z H(z) x[, k]h[k] z,k h[k] z,k h[k] where H(z) P 1 k,1 z,k h[k]. We will see later that H(z) is the trasform of h[]. Note also that z is a eigefuctio. 6 Siusoidal Steady State What is we put x[] cos Ω 0 ito a DT LTI system H? y[] x[] h[] (cos Ω 0 ) h[] 1 2 (ejω0 + e,jω0 ) h[] 1 2 (H(ejΩ0 )e jω0 + H(e,jW0 )e,jω0 ) If we kow othig about system H, this is all we ca say about the output y[]. However, if we kow that h[] is real, the we ca say somethig about H(e jω ). Cosider H (e jω ): H (e jω ) [ h[]e,jω ],1 h []e jω,1 h[]e,j(,ω),1 H(e,jΩ ) I other words, the DTFT H(e jω ) is cojugate symmetric. If we rewrite this i polar: H(e jω ) jh(e jω )je j 6 H(e jω ) H(e,jΩ ) jh(e,jω )je j 6 H(e,jΩ ) H (e jω ) jh(e jω )je,j 6 H(e jω ) The last two equatios are equivalet, as derived above. So we ca equate their magitudes ad phases. This lets us say that the magitude is goig to be eve ad the phase odd. jh(e,jω )j jh(e jω )j 6 H(e,jΩ ),6 H(e jω )

6 EE120: Sigals ad Systems; v So what? From above, we had the output of a DT system for a siusoidal iput: y[] 1 2 (H(ejΩ0 )e jω0 + H(e,jW0 )e,jω0 ) 1 2 [jh(ejω0 )je jω0+j 6 H(e jω 0) + jh(e,jw 0 )je,jω0+j 6 H(e,jW 0) ] 1 2 [jh(ejω0 )je jω0+j 6 H(e jω 0) + jh(e jw 0 )je,jω0,j 6 H(e jw 0) ] 1 2 jh(ejω0 )j[e jω0+j 6 H(e jω 0) + e,jω 0,j6 H(e jw 0) ] jh(e jω0 )j cos(ω H(e jω0 )) I other words, a siusoid iput gives a siusoid output, phase shifted by the phase of the frequecy respose ad amplitude scaled by the magitude respose. 7 Represetatios We begi our discussio of discrete time by otig that the sequece x[] ad the sequece x(t ) cotai the same iformatio. I fact, we ca defie x s (t) as the cotiuous time represetatio of x(t ) such that: x s (t) x(t )(t, T ),1 which is just a ifiite sum of impulses; this also cotais the same iformatio as the plai old sequece x[] or x(t ), except that it s a cotious fuctio, istead of a discrete oe. This should ot be too surprisig. We saw somethig similar to this whe we represeted the Fourier series coefficiets a k as a fuctio of k [this is just a sequece or list of values, a fuctio of a discrete variable], ad as a fuctio of! [this is just a ifiite sum of impulses, a fuctio of a cotiuous variable]. 8 Discrete-Time Fourier Trasform (DTFT) From the begiig of the semester, we saw the FT. Now, the FT trasforms a cotiuous time sigal x(t) ito a fuctio of cotious!. What happes if we try this o our cotious time represetatio of our sequece x[]? X s (j!) 1 x s (t)e,j!t dt,1 1 [ x(t )(t, T )]e,j!t dt,1,1 1 x(t )[ e,j!t (t, T )]dt,1,1 1 x(t )[ e,j!t (t, T )]dt,1,1 1 x(t )e,j!t [ (t, T )]dt,1,1 x(t )e,j!t,1 I cotious time, we decomposed x(t) ito cotious complex expoetials e j!t. I discrete time, we have decomposed x(t) ito discrete complex expoetials.

7 EE120: Sigals ad Systems; v If we replace x(t ) by x[], this last quatity is the DTFT: X(e j!t ) x[]e,j!t,1 The otatio X(e j!t ) deotes the DTFT. This otatio is more widely used tha the oe i the textbook. Note that the discrete complex expoetial e j!t is periodic with period 2 T : e j(!+ 2 T )T e j!t e 2 e j!t This meas that X(e j!t ) will be periodic with period 2 T. Now a slightly weird thig happes low frequecies show up at multiples of 2 T ad high frequecies ed up at multiples of T. To further supress the time depedece, the otatio Ω!T is itroduced, i which case the trasform X(e jω ) is periodic with period 2; low frequecies the occur at multiples of 2, ad high frequecies at multiples of. Exercise Give that Ω!T, covice yourself that the uits of Ω are radias per cycle. A slightly more ituitive reaso for why the DTFT is periodic: sice you ca t tell the differece betwee e j!t ad e j(!+k 2 T )T, you should t be able to tell what frequecy at which it appears i the frequecy domai either. There is aother iterpretatio of the DTFT, arisig from the fact that you ca thik of the time sigal as the sampled versio of some cotiuous time sigal x(t). To sample i time, you multiply i time by a impulse trai. This meas that you covolve i frequecy with a impulse trai, which makes your spectrum periodic. T ca the be iterpreted as the samplig period. More o this i the last few lectures. Note that multiplyig i time by a impulse trai is a mathematical trick that we use i order to derive the DTFT. What we ca actually do is sample our cotious time x(t) at periodic itervals t T to obtai x(t ). Repeat: multiplyig i time by a impulse trai is a math trick. The DTFT has a umber of properties which are summarized o pages of OWY; commo DTFT trasform pairs are give o OWY pages Persoally, i prefer to work with the trasform, ad the utilize the relatioship betwee the trasform ad the DTFT to fid a give DTFT. O the other had, if you kow that a certai sigal x[] came from samplig x(t) ad you kow X(!), the you ca quickly fid the DTFT by makig copies of X(!), separated by 2 T ad scaled by 1 T. More o this i the ext set of otes. 9 Fourier Series, Revisited Assume that x(t) is periodic with period T 2. Also, assume that x(t) is sufficietly well-behaved such that we ca represet it by a Fourier series: x(t) X e j!0t,1 The Fourier series coefficiets X ca be foud by multiplyig both sides by e,jm!0t ad itegratig over oe period T : x(t)e,jm!0t dt T X e j!0t e,jm!0t dt T,1 X e j!0t e,jm!0t dt,1 T We recall the orthogoality relatioship for complex expoetials: e j!0t e,jm!0t T if m dt 0 otherwise T

8 EE120: Sigals ad Systems; v This lets us reduce the equatio above to: x(t)e,jm!0t dt T X e j!0t e,jm!0t dt,1 T TX m X m 1 T This last equatio is the Fourier series aalysis itegral. If we take the Fourier trasform of x(t): X(!) F [x(t)] x(t)e,jm!0t dt T F [ X e j!0t ],1 X F [e j!0t ],1 2 X (!, ),1 where the X ca be foud by the Fourier series aalysis itegral above. Note that X is just a sequece of values, idexed o. The time sigal to which it correspods i the time domai is periodic i t. 10 DTFT, Revisited Assume that X(e j!t ) is periodic with period 2 T,whereT is the samplig period. We further assume that X(e j!t ) is sufficietly well-behaved such that we ca represet it by a Fourier series: X(e j!t ) x[]e,j!t,1 Note that we choose to use a egative sig, sice we re i the frequecy domai. The Fourier series coefficiet x[] ca be foud by multiplyig both sides by e jm!t ad itegratig over oe period : X(e j!t )e jm!t d! x[]e,j!t e jm!t d!,1 x[],1 We recall the orthogoality relatioship for complex expoetials: e,j!t e jm!t d! This lets us reduce the equatio above to: X(e j!t )e jm!t d!!0 x[],1 x[m] e,j!t e jm!t d! if m 0 otherwise e,j!t e jm!t d!

9 EE120: Sigals ad Systems; v This is the iverse DTFT. Takig the iverse Fourier trasform of X(e j!t ): x[m] 1 X(e j!t )e jm!t d! T 2 x(t) F,1 [X(e j!t )] 2 T X(e j!t )e jm!t d! F,1 [ x[]e,j!t ],1,1 x[]f,1 [e,j!t ] x[](t, T ),1 This is the DTFT. Note that x[] is just a sequece of values, idexed o. Its Fourier trasform is periodic i!. x(t) is just the cotiuous time represetatio of the discrete time sigal x[] as discussed above. NowwehavetheetireDTFTpair: X(e j!t ) x[] T 2 I ormalized frequecy, the DTFT pair becomes: X(e jω ) x[]e,j!t0,1 x[] T X(e j!t )e j!t0 d! x[]e,jω,1 X(e jω )e jω d! 2 I other words, periodic i oe domai implies discrete i the other. Here, periodic i frequecy [X(e jω )] implies discrete i time [the x[]], idexed o. From the previous sectio, periodic i time [x(t)] implies discrete i frequecy [the X ], idexed o. Exercise Does t this look familiar? i lifted it from ps7 solutios. Take aother look ad make sure that you ca see the duality relatioship betwee the FS ad the DTFT. 11 -Trasform (T) As with the DTFT, if we start out i cotiuous time ad the take the Laplace trasform of x s (t): X s (s) 1 x s (t)e,st dt,1 1 [ x(t )(t, T )]e,st dt,1,1 1 x(t )[ e,st (t, T )]dt,1,1

10 EE120: Sigals ad Systems; v x(t )[,1 1 x(t )e,st [,1 x(t )e,st,1 x[]z,,1 e,st (t, T )]dt,1 1 (t, T )]dt,1 where we have made the chage of otatio x(t )x[] ad z e st. This developmet is similar to that of the DTFT. So why do we bother studyig the trasform? For the same reaso as why we study the Laplace trasform there are some sigals for which the DTFT does ot exist. What is z e st? It s a coformal mappig betwee the s-plae ad the z-plae, takig vertical lies i the s-plae ad makig them ito circles i the z-plae [cosider s + j! ad ote that e T is the radius of a circle ad e j!t is the agle]. I particular, the j! axis eds up as the uit circle, the left half plae eds up iside the uit circle, ad the right half plae eds up outside. This also suggests that, for a system to be stable, its trasform should have poles iside the uit circle. Exercise Verify the cotets of this paragraph. For some values of z, the sum will ot coverge. Those values for which the sum does coverge comprise the regio of covergece (ROC) [compare to the ROC of the Laplace trasform]. The ROC is a circle cetered o the origi, with its radius the distace from the origi to the outermost pole. Because we cosider oly the uilateral trasform, the ROCs all go outwards. The properties are summarized o OWY page 654, ad are scattered throughout sectio 8-3 of TF; commo trasforms are available o OWY page 655, TF page Relatioships The relatioships betwee the five trasforms that you have studied ad the DTFS/DFS/DFT that you will see agai i ee123 are summarized i the followig table: R P 1 LT:X(s),1 x(t)e,st ze dt $ st 1 T:X(z) k,1 x[k]z,k s j! # z e j!t # R P 1 FT:X(j!),1 x(t)e,j!t discretize:tt dt! DTFT:X(e j!t 1 ),1 x[]e,j!t periodicize # periodicize # R FS:a k T x(t)e,jk!0t discretize:tt dt! DFS/DFT Give that you have the Laplace trasform of some time sigal x(t), if you evaluate the trasform o the j! axis, you ed up with the Fourier trasform. If you make the time sigal periodic, you ed up with a discrete spectrum [Fourier series]. O the other had, we could start with a cotious time sigal x(t) ad make it discrete. If we take its Laplace trasform ad utilize the coformal mappig z e st,wheret is the samplig period, we ed up with the trasform. Evaluatig the traform o the uit circle gives you the the DTFT. To get from the cotiuous time trasforms to the discrete time trasforms, we ca thik of takig our CT x(t) ad multiplyig it by a impulse trai to get x(t ). This is just a math trick though sice we oly care about the x(t ), we ca get this by samplig x(t) at itervals t T [perhaps by usig a sample-ad-hold, followed by a A/D coverter]. Exercise Follow the liks betwee the six trasforms ad try to get everythig straight i your mid. There s a lot of material summarized i that oe table, so make sure that it makes sese. Drop by durig office hours if ot.