Discrete-time signals and systems See Oppenheim and Schafer, Second Edition pages 8 93, or First Edition pages 8 79.

Transcription

1 Discrete-time sigals ad systems See Oppeheim ad Schafer, Secod Editio pages 93, or First Editio pages 79. Discrete-time sigals A discrete-time sigal is represeted as a sequece of umbers: x D fxœg; < < : Here is a iteger, ad xœ is the th sample i the sequece. Discrete-time sigals are ofte obtaied by samplig cotiuous-time sigals. I this case the th sample of the sequece is equal to the value of the aalogue sigal x a.t/ at time t D T : xœ D x a.t /; < < : The samplig period is the equal to T, ad the samplig frequecy is f s D =T x[] x a.t / For this reaso, although xœ is strictly the th umber i the sequece, we ofte refer to it as the th sample. We also ofte refer to the sequece xœ whe we mea the etire sequece. Discrete-time sigals are ofte depicted graphically as follows: t x[ 3] x[ ] x[ 4] x[ ] x[4]... x[] x[] x[3] x[] (This ca be plotted usig the MATLAB fuctio stem.) The value xœ is udefied for oiteger values of. Sequeces ca be maipulated i several ways. The sum ad product of two sequeces xœ ad yœ are defied as the sample-by-sample sum ad product respectively. Multiplicatio of xœ by a is defied as the multiplicatio of each sample value by a. A sequece yœ is a delayed or shifted versio of xœ if with a iteger. The uit sample sequece is defied as yœ D xœ ; < ıœ D : D : This sequece is ofte referred to as a discrete-time impulse, or just impulse. It plays the same role for discrete-time sigals as the Dirac delta fuctio does for cotiuous-time sigals. However, there are o mathematical

2 complicatios i its defiitio. A importat aspect of the impulse sequece is that a arbitrary sequece ca be represeted as a sum of scaled, delayed impulses. For example, the sequece ca be represeted as a 4 a 3 a a a xœ D a 4 ıœ C 4 C a 3 ıœ C 3 C a ıœ C C a ıœ C C a ıœ a I geeral, ay sequece ca be expressed as The uit step sequece a Ca ıœ C a ıœ C a 3 ıœ 3 C a 4 ıœ 4: xœ D X kd xœkıœ k: a 3 a 4 The uit step is related to the impulse by uœ D Alteratively, this ca be expressed as X kd ıœk: uœ D ıœ C ıœ C ıœ C D X ıœ Coversely, the uit sample sequece ca be expressed as the first backward differece of the uit step sequece ıœ D uœ uœ : kd Expoetial sequeces are importat for aalysig ad represetig discrete-time systems. The geeral form is xœ D A : If A ad are real umbers the the sequece is real. If < < ad A is positive, the the sequece values are positive ad decrease with icreasig :... For < < the sequece alterates i sig, but decreases i magitude. For j j > the sequece grows i magitude as icreases. A siusoidal sequece... k: is defied as < uœ D : < :

3 has the form xœ D A cos.! C / for all ; with A ad real costats. The expoetial sequece A with complex D j je j! ad A D jaje j ca be expressed as xœ D A D jaje j j j e j! D jajj j e j.! C/ D jajj j cos.! C / C j jajj j si.! C /; so the real ad imagiary parts are expoetially weighted siusoids. Whe j j D the sequece is called the complex expoetial sequece: xœ D jaje j.! C/ D jaj cos.! C / C j jaj si.! C /: The frequecy of this complex siusoid is!, ad is measured i radias per sample. The phase of the sigal is. The idex is always a iteger. This leads to some importat differeces betwee the properties of discrete-time ad cotiuous-time complex expoetials: Cosider the complex expoetial with frequecy.! C /: xœ D Ae j.! C/ D Ae j! e j D Ae j! : Thus the discrete-time siusoid is oly periodic if which requires that A cos.! C / D A cos.! C! N C /;! N D k for k a iteger: The same coditio is required for the complex expoetial sequece Ce j! to be periodic. The two factors just described ca be combied to reach the coclusio that there are oly N distiguishable frequecies for which the correspodig sequeces are periodic with period N. Oe such set is! k D k ; k D ; ; : : : ; N : N Additioally, for discrete-time sequeces the iterpretatio of high ad low frequecies has to be modified: the discrete-time siusoidal sequece xœ D A cos.! C / oscillates more rapidly as! icreases from to, but the oscillatios become slower as it icreases further from to. Thus the sequece for the complex expoetial with frequecy! is exactly the same as that for the complex expoetial with frequecy.! C /. More geerally, complex expoetial sequeces with frequecies.! C r/, where r is a iteger, are idistiguishable from oe aother. Similarly, for siusoidal sequeces xœ D A cosœ.! C r/ C D A cos.! C /: I the cotiuous-time case, siusoidal ad complex expoetial sequeces are always periodic. Discrete-time sequeces are periodic (with period N) if xœ D xœ C N for all : 5 6

4 ! D! D Discrete-time systems A discrete-time system is defied as a trasformatio or mappig operator that maps a iput sigal xœ to a output sigal yœ. This ca be deoted as yœ D T fxœg:! D 4 x[] Example: Ideal delay T fg y[]! D yœ D xœ d W! D 7 4! D x[] The sequece correspodig to! D is idistiguishable from that with! D. I geeral, ay frequecies i the viciity of! D k for iteger k are typically referred to as low frequecies, ad those i the viciity of! D. C k/ are high frequecies. y[]=x[ ] This operatio shifts iput sequece later by d samples. 7

5 Example: Movig average yœ D M C M C For M D ad M D, the iput sequece XM kd M xœ y[3] k yœ D xœ d is ot uless d D.. Liear systems A system is liear if the priciple of superpositio applies. Thus if y Œ is the respose of the system to the iput x Œ, ad y Œ the respose to x Œ, the liearity implies yields a output with y[] : yœ D.xŒ C xœ C xœ3/ 3 yœ3 D.xŒ C xœ3 C xœ4/ 3 I geeral, systems ca be classified by placig costraits o the trasformatio T fg.. Memoryless systems : A system is memoryless if the output yœ depeds oly o xœ at the same. For example, yœ D.xŒ/ is memoryless, but the ideal delay x[] Additivity: Scalig: T fx Œ C x Œg D T fx Œg C T fx Œg D y Œ C y Œ T fax Œg D at fx Œg D ay Œ: These properties combie to form the geeral priciple of superpositio T fax Œ C bx Œg D at fx Œg C bt fx Œg D ay Œ C by Œ: I all cases a ad b are arbitrary costats. This property geeralises to may iputs, so the respose of a liear system to xœ D P k a kx k Œ will be yœ D P k a ky k Œ..3 Time-ivariat systems A system is time ivariat if a time shift or delay of the iput sequece causes a correspodig shift i the output sequece. That is, if yœ is the respose to xœ, the yœ is the respose to xœ. For example, the accumulator system yœ D X kd xœk 9

6 is time ivariat, but the compressor system yœ D xœm for M a positive iteger (which selects every M th sample from a sequece) is ot..4 Causality A system is causal if the output at depeds oly o the iput at ad earlier iputs. For example, the backward differece system yœ D xœ xœ is causal, but the forward differece system is ot..5 Stability yœ D xœ C xœ A system is stable if every bouded iput sequece produces a bouded output sequece: Bouded iput: jxœj B x < Bouded output: jyœj B y <. For example, the accumulator yœ D X kd xœ is a example of a ubouded system, sice its respose to the uit step uœ is X < < yœ D uœ D : C ; kd which has o fiite upper boud. 3 Liear time-ivariat systems If the liearity property is combied with the represetatio of a geeral sequece as a liear combiatio of delayed impulses, the it follows that a liear time-ivariat (LTI) system ca be completely characterised by its impulse respose. Suppose h k Œ is the respose of a liear system to the impulse ıœ D k. Sice ( ) X yœ D T xœkıœ k ; kd the priciple of superpositio meas that X yœ D xœkt fıœ kg D kd X kd xœkh k Œ: If the system is additioally time ivariat, the the respose to ıœ hœ k. The previous equatio the becomes X yœ D xœkhœ k: kd k at k is This expressio is called the covolutio sum. Therefore, a LTI system has the property that give hœ, we ca fid yœ for ay iput xœ. Alteratively, yœ is the covolutio of xœ with hœ, deoted as follows: yœ D xœ hœ:

7 The previous derivatio suggests the iterpretatio that the iput sample at D k, represeted by xœkıœ k, is trasformed by the system ito a output sequece xœkhœ k. For each k, these sequeces are superimposed to yield the overall output sequece: 5 h[k] k xœ hœ Reflect h[ k] 5 k xœ ıœ C xœıœ xœ hœ C xœhœ yœ D xœ hœ C C xœhœ A slightly differet iterpretatio, however, leads to a coveiet computatioal form: the th value of the output, amely yœ, is obtaied by multiplyig the iput sequece (expressed as a fuctio of k) by the sequece with values hœ k, ad the summig all the values of the products xœkhœ k. The key to this method is i uderstadig how to form the sequece hœ k for all values of of iterest. To this ed, ote that hœ k D hœ.k /. The sequece hœ k is see to be equivalet to the sequece hœk reflected aroud the origi: Shift The sequece hœ to k D. 5 + h[ k] k is the obtaied by shiftig the origi of the sequece To implemet discrete-time covolutio, the sequeces xœk ad hœ k are multiplied together for < k <, ad the products summed to obtai the value of the output sample yœ. To obtai aother output sample, the procedure is repeated with the origi shifted to the ew sample positio. Example: aalytical evaluatio of the covolutio sum Cosider the output of a system with impulse respose < N hœ D : otherwise to the iput xœ D a uœ. To fid the output at, we must form the sum over all k of the product xœkhœ k. k 3 4

8 x[] h[ k] (N ) 5 k k 4 Properties of LTI systems All LTI systems are described by the covolutio sum X yœ D xœkhœ k: kd Some properties of LTI systems ca therefore be foud by cosiderig the properties of the covolutio operatio: Commutative: xœ hœ D hœ xœ Distributive over additio: Cascade coectio: xœ.h Œ C h Œ/ D xœ h Œ C xœ h Œ: Sice the sequeces are o-overlappig for all egative, the output must be zero: yœ D ; < : For N the product terms i the sum are xœkhœ k D a k, so it follows that X yœ D a k ; N : kd Fially, for > N the product terms are xœkhœ k D a k as before, but the lower limit o the sum is ow N C. Therefore x[] h Œ h Œ y[] x[] h Œ h Œ y[] yœ D hœ xœ D h Œ h Œ xœ D h Œ h Œ xœ. Parallel coectio: h Œ X yœ D a k ; > N : kd N C x[] h Œ yœ D.h Œ C h Œ/ xœ D h p Œ xœ. Additioal importat properties are: y[] A LTI system is stable if ad oly if S D P kd jhœkj <. The ideal 5 6

9 delay system hœ D ıœ average system d is stable sice S D < ; the movig XM hœ D ıœ k M C M C kd M < M D CM M C M : otherwise; the forward differece system hœ D ıœ C ıœ, ad the backward differece system hœ D ıœ ıœ are stable sice S is the sum of a fiite umber of fiite samples, ad is therefore less tha ; the accumulator system hœ D X kd ıœk < D : < D uœ is ustable sice S D P D uœ D. A LTI system is causal if ad oly if hœ D for <. The ideal delay system is causal if d ; the movig average system is causal if M ad M ; the accumulator ad backward differece systems are causal; the forward differece system is ocausal. Systems with oly a fiite umber of ozero values i hœ are called Fiite duratio impulse respose (FIR) systems. FIR systems are stable if each impulse respose value is fiite. The ideal delay, the movig average, ad the forward ad backward differece described above fall ito this class. Ifiite impulse respose (IIR) systems, such as the accumulator system, are more difficult to aalyse. For example, the accumulator system is ustable, but the IIR system is stable sice hœ D a uœ; jaj < X X S D ja j D D (it is the sum of a ifiite geometric series). Cosider the system which has Forward differece jaj D jaj < Oe sample delay hœ D.ıŒ C ıœ/ ıœ D ıœ ıœ C ıœ ıœ D ıœ ıœ : This is the impulse respose of a backward differece system: Oe sample delay Backward differece Forward differece Here a o-causal system has bee coverted to a causal oe by cascadig with a delay. I geeral, ay o-causal FIR system ca be made causal by cascadig with a sufficietly log delay. Cosider the system cosistig of a accumulator followed by a backward differece: 7

10 Accumulator Backward differece we obtai the recursio represetatio x[] y[] The impulse respose of this system is hœ D uœ.ıœ ıœ / D uœ uœ D ıœ: The output is therefore equal to the iput because xœ ıœ D xœ. Thus the backward differece exactly compesates for (or iverts) the effect of the accumulator the backward differece system is the iverse system for the accumulator, ad vice versa. We defie this iverse relatioship for all LTI systems: hœ h i Œ D ıœ: 5 Liear costat coefficiet differece equatios Some LTI systems ca be represeted i terms of liear costat coefficiet differece (LCCD) equatios NX a k yœ kd k D MX b m xœ md Example: differece equatio represetatio of the accumulator Take for example the accumulator m: Oe sample delay where at we add the curret iput xœ to the previously accumulated sum yœ. Example: differece equatio represetatio of movig average Cosider ow the movig average system with M D : hœ D The output of the system is M C.uŒ uœ M /: yœ D M C XM xœ which is a LCCDE with N D, a D, ad M D M, b k D =.M C /. Usig the siftig property of ıœ, so hœ D kd k; M C.ıŒ ıœ M / uœ x[] Accumulator y[] Backward differece x[] Here yœ yœ D xœ, which ca be writte i the desired form with N D, a D, a D, M D, ad b D. Rewritig as x[] Atteuator =.M C / +.M C / sample delay x Œ Accumulator y[] yœ D yœ C xœ 9

11 Here x Œ D =.M C /.xœ xœ M / ad for the accumulator yœ yœ D x Œ. Therefore yœ yœ D M C.xŒ xœ M /; which is agai a (differet) LCCD equatio with N D, a D, a D, b D b M C D =.M C /. As for costat coefficiet differetial equatios i the cotiuous case, without additioal iformatio or costraits a LCCDE does ot provide a uique solutio for the output give a iput. Specifically, suppose we have the particular output y p Œ for the iput x p Œ. The same equatio the has the solutio yœ D y p Œ C y h Œ; where y h Œ is ay solutio with xœ D. That is, y h Œ is a homogeeous solutio to the homogeeous equatio NX a k y h Œ k D : kd It ca be show that there are N ozero solutios to this equatio, so a set of N auxiliary coditios are required for a uique specificatio of yœ for a give xœ. If a system is LTI ad causal, the the iitial coditios are iitial rest coditios, ad a uique solutio ca be obtaied. 6 Frequecy-domai represetatio of discrete-time sigals ad systems The Fourier trasform cosidered here is strictly speakig the discrete-time Fourier trasform (DTFT), although Oppeheim ad Schafer call it just the Fourier trasform. Its properties are recapped here (with examples) to show omeclature. Complex expoetials are eigefuctios of LTI systems: Defiig yœ D X kd xœ D e j! ; < < hœke j!. k/ D e j! H.e j! / D X kd X kd hœke j!k hœke j!k! we have that yœ D H.e j! /e j! D H.e j! /xœ. Therefore, e j! is a eigefuctio of the system, ad H.e j! / is the associated eigevalue. The quatity H.e j! / is called the frequecy respose of the system, ad H.e j! / D H R.e j! / C jh I.e j! / D jh.e j! /je j ^H.ej!/ : Example: frequecy respose of ideal delay: Cosider the iput xœ D e j! to the ideal delay system yœ D xœ d : the output is yœ D e j!. d / D e j! d e j! : The frequecy respose is therefore H.e j! / D e j! d : Alteratively, for the ideal delay hœ D ıœ d, H.e j! / D X D ıœ d e j! D e j! d : The real ad imagiary parts of the frequecy respose are :

12 H R.e j! / D cos.! d / ad H I.e j! / D si.! d /, or alteratively jh.e j! /j D ^H.e j! / D! d : The frequecy respose of a LTI system is essetially the same for cotiuous ad discrete time systems. However, a importat distictio is that i the discrete case it is always periodic i frequecy with a period : H.e j.!c/ / D D D X D X D X D hœe j.!c/ hœe j! e j hœe j! D H.e j! /: This last result holds sice e j D for iteger. The reaso for this periodicity is related to the observatio that the sequece e j! ; < < has exactly the same values as the sequece e j.!c/o ; < < : A system will therefore respod i exactly the same way to both sequeces. Example: ideal frequecy selective filters The frequecy respose of a ideal lowpass filter is as follows:! c H lp.e j! /! c Oly required part Due to the periodicity i the respose, it is oly ecessary to cosider oe frequecy cycle, usually chose to be the rage to. Other examples of ideal filters are:! c! b! b! a! a H hp.e j! / H bs.e j! /! a! c! a! b H bp.e j! /! b!!!! Highpass Badstop Badpass I these cases it is implied that the frequecy respose repeats with period outside of the plotted iterval. Example: frequecy respose of the movig-average system 3 4

13 The frequecy respose of the movig average system < M hœ D CM M C M : otherwise is give by H.e j! / D e j!.m CM C/= e j!.m CM C/= e j!.m MC/ M C M C e j! D M C M C D M C M C For M D ad M D 4, jh.e j! /j ^H.e j! / e j!.m CM C/= e j!.m CM C/= e j!.m M/ e j!= e j!= siœ!.m C M C /= e j!.m M / : si.!=/ 5 This system atteuates high frequecies (at aroud! D ), ad therefore has the behaviour of a lowpass filter.!! 5 7 Fourier trasforms of discrete sequeces The discrete time Fourier trasform (DTFT) of the sequece xœ is X.e j! / D X D xœe j! : This is also called the forward trasform or aalysis equatio. The iverse Fourier trasform, or sythesis formula, is give by the Fourier itegral xœ D Z X.e j! /e j! d!: The Fourier trasform is geerally a complex-valued fuctio of!: X.e j! / D X R.e j! / C jx I.e j! / D jx.e j! /je j ^X.ej!/ : The quatities jx.e j! /j ad ^X.e j! / are referred to as the magitude ad phase of the Fourier trasform. The Fourier trasform is ofte referred to as the Fourier spectrum. Sice the frequecy respose of a LTI system is give by H.e j! / D X kd hœke j!k ; it is clear that the frequecy respose is equivalet to the Fourier trasform of the impulse respose, ad the impulse respose is hœ D Z H.e j! /e j! d!: A sufficiet coditio for the existece of the Fourier trasform of a sequece xœ is that it be absolutely summable: P D jxœj <. I other words, the Fourier trasform exists if the sum P D jxœj coverges. The Fourier trasform may however exist for sequeces where this is ot true a rigorous mathematical treatmet ca be foud i the theory of geeralised fuctios. 5 6

14 Symmetry properties of the Fourier trasform Ay sequece xœ ca be expressed as xœ D x e Œ C x o Œ; where x e Œ is cojugate symmetric (x e Œ D xe Œ ) ad x oœ is cojugate atisymmetric (x o Œ D xo Œ ). These two compoets of the sequece ca be obtaied as: x e Œ D.xŒ C x Œ / D x e Œ x o Œ D.xŒ x Œ / D x o Œ : If a real sequece is cojugate symmetric, the it is a eve sequece, ad if cojugate atisymmetric, the it is odd. Similarly, the Fourier trasform X.e j! / ca be decomposed ito a sum of cojugate symmetric ad atisymmetric parts: X.e j! / D X e.e j! / C X o.e j! /; Sequece xœ Trasform X.e j! / x Œ X.e j! / x Œ X.e j! / RefxŒg X e.e j! / j ImfxŒg X o.e j! / x e Œ X R.e j! / x o Œ jx I.e j! / Most of these properties ca be proved by substitutig ito the expressio for the Fourier trasform. Additioally, for real xœ the followig also hold: Real sequece xœ Trasform X.e j! / xœ X.e j! / D X.e j! / xœ X R.e j! / D X R.e j! / xœ X I.e j! / D X I.e j! / xœ jx.e j! /j D jx.e j! /j where X e.e j! / D ŒX.ej! / C X.e j! / xœ ^X.e j! / D ^X.e j! / x e Œ X R.e j! / x o Œ jx I.e j! / X o.e j! / D ŒX.ej! / With these defiitios, ad lettig X.e j! /: 9 Fourier trasform theorems X.e j! / D X R.e j! / C jx I.e j! /; the symmetry properties of the Fourier trasform ca be summarised as follows: Let X.e j! / be the Fourier trasform of xœ. The followig theorems the apply: 7

15 Sequeces xœ, yœ Trasforms X.e j! /, Y.e j! / Property axœ C byœ ax.e j! / C by.e j! / Liearity xœ d e j! d X.e j! / Time shift e j! xœ X.e j.!! / / Frequecy shift xœ X.e j! / Time reversal xœ j dx.ej! / Frequecy diff. d! xœ yœ X.e j! /Y.e j! / Covolutio xœyœ R X.ej /Y.e j.! / /d Modulatio Some useful Fourier trasform pairs are: Sequece Fourier trasform ıœ ıœ e j! P. < < / kd ı.! C k/ a uœ.jaj < / uœ. C /a uœ.jaj < / si.! c / < M xœ D : otherwise ae j! C P e j! kd. ae j! / ı.! C k/ < j!j <! X.e j! c / D :! c < j!j siœ!.m C/= e j!m= si.!=/ e j! P kd ı.!! C k/ 9