Digital Signal Processing: A Computer-Based Approach

Transcript

1 SOLUTIOS AUAL to accopay Digital Sigal Processig: A Coputer-Based Approac Tird Editio Sait K itra Prepared by Cowdary Adsuilli, Jo Berger, arco Carli, Hsi-Ha Ho, Raeev Gadi, Ci Kaye Ko, Luca Luccese, ad ylee Queiroz de Farias ot for sale

2 Capter (a) 8, 996, 8, 868, 79, (b) 8,, µ Hece,, <, µ Tus,, < µ µ, (a) Cosider te sequece defied by If <, te is ot icluded i te su ad ece, for < O te oter ad, for i te su, ad as a result, for Terefore,,, µ, <,, is icluded (b) Sice,, µ it follows tat,, µ Hece,, <,, <,, µ µ,, Recall µ µ Hece, ( µ µ ) ( µ µ ) ( µ µ ) ( µ µ ) µ µ µ 6µ µ (a) c { }, d y { 7 (b) 8 6 }, (c) e w { }, (d) u y { 8 7 }, v w { s y w { r y { 8 7 (e) }, (f) }, (g) } 6 (a), y 6 8 7, w 7 8, (b) Recall µ µ Hece, ( µ µ ) ( µ µ ) ( µ µ ) ( µ µ ) ( µ µ ) ( µ µ ) ot for sale

3 µ 9µ µ µ µ µ µ µ, 7 (a) z - z - y Fro te above figure it follows tat y (b) z z z β β w β w w z β y Fro te above figure we get w ( β β ) ad y w βw βw aig use of te first equatio i te secod we arrive at y ( β β ) β ( β β ) β ( β β ) ( ( β β ) ( β ββ β ) ( β β β β ) β β ) (c) Figure P(c) is a cascade of a first-order sectio ad a secod-order sectio Te iput-output relatio reais ucaged if te orderig of te two sectios is itercaged as sow below 6 w u y 8 z w z w z y ot for sale

4 Te secod-order sectio ca be redraw as sow below witout cagig its iputoutput relatio 6 w u y 8 z w z z y z w z Te secod-order sectio ca be see to be cascade of two sectios Itercagig teir orderig we fially arrive at te structure sow below: z 6 s 8 z u u y z y z z u Aalyzig te above structure we arrive at s 6, u s 8u u, y u y Fro u y y Substitutig tis i te secod equatio we get after soe algebra y s y 8y 8y aig use of te first equatio i tis equatio we fially arrive at te desired iput-output relatio y y 8y y 6 (d) Figure P9(d) is a parallel coectio of a first-order sectio ad a secod-order sectio Te secod-order sectio ca be redraw as a cascade of two sectios as idicated below: w 8 z w z y z w z ot for sale

5 Itercagig te order of te two sectios we arrive at a equivalet structure sow below: q z y z 8 z y z y Aalyzig te above structure we get q, y q 8y y Substitutig te first equatio i te secod we ave y 8y y (-) Aalyzig te first-order sectio of Figure P(d) give below u 6 z u y we get u u, y 6u Solvig te above two equatios we ave y y 6 (-) Te output y of te structure of Figure P9(d) is give by y y y (-) Fro Eq (-) we get 8y y 8 ad y y Addig te last two equatios to Eq (-) we arrive at y y 8y y 6 8 (-) Siilarly, fro Eq (-) we get y y y 8 Addig tis equatio to Eq (-) we arrive at y y 8y y 8 8 (-) Addig Eqs (-) ad (-), ad aig use of Eq (-) we fially arrive at te iputoutput relatio of Figure P(d) as: y y 8y y 9 6 ot for sale

6 8 (a) }, 7 { * Terefore } 7 { * ( ) }, { *, cs ( ) } { *, ca (b) Hece, ad tus, Terefore, / e / * e / * e ( ), / *, e cs ad ( ) *, ca (c) Hece, ad tus, Terefore, / e / * e / * e ( ), *, cs ad ( ) / *, ca e 9 (a) } { Hece, } { Terefore, } { ) ( ev } { ad } 6 6 { ) ( od } { (b) } { y Hece, } { y Terefore, } { ) ( y y y ev ad } { ) ( y y y od (c) } { w Hece, } { w Terefore ) ( w w w ev ot for sale 6

7 { } ad w od ( w w ) { } (a) µ Hece, µ Terefore, /,,, ev ( µ µ ),, ad /,, /,,, od ( µ µ ),, /, (b) µ Hece, µ Terefore,,,, ev ( µ µ ),, ad,,, od ( µ µ ),,,,, (c) µ Hece, µ Terefore,, ev, od ( µ ( ) µ ) ( µ ( ) µ ) ad (d) Hece,, ev ( ) ( ), od ( ) ( ) Terefore, ad ( ) ev Tus, ( ) a eve sequece Liewise, od ( ) ev od ( ) Hece, ev is Tus, Hece, od is a odd sequece od ev ot for sale 7

8 (a) g ev ev Tus, g ev ev ev ev g Hece, g is a eve sequece (b) u ev od Tus, u ev od ev od u Hece, u is a odd sequece (c) ( ) v Tus, v ( )( ) od od od od v Hece, v is a eve sequece od od od od (a) Sice is causal,, < Also,, > ow, ( ) Hece, ev ( ) ad ev ev, > Cobiig te two equatios we get Liewise, ( ) od ev, ev,, Hece, ( ) od, > Cobiig te two equatios we get od ad, ev, > <,, >, (b) Sice y is causal, y, < Also, y, > Let y y y, were y re ad y i are real causal sequeces re y i ow, ( y ) y ca ca y i Hece, y ( y y ) ad yca y, > Sice y re is ot ow, y caot be fully recovered fro y ca Liewise, ( y y ) y cs cs y re Hece, y ( y y ) ad ycs y, > Sice y i is ot ow, y caot be fully recovered fro y cs Sice is causal,, < Fro te solutio of Proble we ave ev, >, cos( ωo), >, ev,,, cos( ωo) µ, <,, <, (a) { } { A } were A ad are cople ubers wit < Sice for <, ca becoe arbitrarily large, { } is ot a bouded sequece ot for sale 8

9 (b) y A µ A,, < Here,, bouded sequece < Hece, were A ad are cople ubers wit, y A for all values of Hece, { y } is a (c) { } C β µ were C ad β are cople ubers wit β > Sice for >, β ca becoe arbitrarily large, { } is ot a bouded sequece (d) { g } cos( ω ) Sice g for all values of, { g } is a bouded o sequece (e),, v Sice < for > ad, all values of Tus { v } is a bouded sequece ( ) 6 µ ot absolutely suable v < for for, ( ) ow Hece { } is 7 (a) µ ow < Hece, { } is absolutely suable <, sice (b) µ ow <, sice ( ) < Hece, { } is absolutely suable (c) µ ow ( K K ) ( K) ( K) ot for sale 9

10 ) 7( 6 K K 7 ) ( ) ( ) ( ) ( < Hece, is absolutely suable } { 8 (a) a µ ow < a Hece, is absolutely suable } { a (b) ) )( ( b µ ow ) )( ( b < K Hece, is absolutely suable } { b 9 (a) A sequece is absolutely suable if < By Scwartz iequality we ave < Hece, a absolutely suable sequece is square suable ad as tus fiite eergy ow cosider te sequece µ Te covergece of a ifiite series ca be sow via te itegral test Let ), ( f a were a cotiuous, positive ad decreasig fuctio is for all Te te series ad te itegral bot coverge or bot diverge For a ) ( d f ) (, f a But ) (l d Hece, does ot coverge As a result, µ is ot absolutely suable ot for sale

11 (b) To sow tat { } is square-suable, we observe tat ere a, ad tus, f ( ) ow, d words, µ is square-suable See Proble 9, Part (a) solutio Hece, coverges, or i oter cosω c cosω µ ow, c Sice, cosω, c Terefore is square-suable 6 6 Usig te itegral test (See Proble 9, Part (a) solutio) we ow sow tat ot absolutely suable ow, cosω c cosωc d cosω cosω cosit is te cosie itegral fuctio Sice c diverges Hece, is ot absolutely suable K K K K K li K ev K K ev K K K P li li ( ) ( ) od P P li ( )( ) ev od K K K K K P ev od K K li K K K as P P P ev od ow for te give sequece, c d od ev cos it( ω c diverges, od K K K K 6 6 P li od od li li K K K K K K K 6 6 K li 6 K K Hece, P P ev P od ) were is cosωc also K ot for sale

12 si( / ), ow E si ( / ) S ( cos( / ) ) cos( / ) si( / ) C Hece E Let C cos( / ) ad / e Te C S e Tis iplies / e (a) A A µ Te E a a A (b) b µ Te E b b 9 (a) ( ) Te average power K P li li (K ), ad eergy K K E K K K (b) µ Te average power K K K P li li li, ad eergy K K K E K K K K (c) P µ li K K Te average power K li K K ad eergy E K K li K K( K )(K ), 6 (d) ω Ae Te average power P li K K K K ot for sale

13 , ) ( li li li A K A K A K e A K K K K K K K K ω ad eergy ω A e A E (e) cos φ A ote is a periodic sequece Te average power cos cos φ φ A A P Let φ cos C ad cos φ S Te / / φ φ φ e e e e e e S C Hece Terefore C A A P Sice is a periodic sequece, it as ifiite eergy 6 I eac of te followig parts, deotes te fudaetal period ad r is a positive iteger (a) Here ad ) / cos( ~ r ust satisfy te relatio r Aog all positive solutios for ad r, te sallest values are ad r Hece te average power is give by 8 cos ~ P (b) Here ad ) / cos( ~ r ust satisfy te relatio r Aog all positive solutios for ad r, te sallest values are ad r Hece te average power is give by cos ~ 9 P (c) Here ad 7) / cos( ~ r ust satisfy te relatio 7 r Aog all positive solutios for ad r, te sallest values are ad r Hece te average power is give by cos ~ 7 P ot for sale

14 (d) ~ cos( / ) Here ad r ust satisfy te relatio r Aog all positive solutios for ad r, te sallest values are 6 ad r Hece te average power is give by ~ cos P 8 6 (e) ~ cos( / ) cos( / ) We first deterie te fudaetal period r of cos( / ) of cos( / ) Here ad r ust satisfy te relatio r positive solutios for ad, te sallest values are deterie te fudaetal period relatio r Aog all positive solutios for Aog all ad r We et Here ad r ust satisfy te ad r, te sallest values are ad r Te fudaetal period of ~ is te give by LC (, ) LC(,) Hece te average power is give by ~ cos cos P 6cos 9cos cos cos 8 (f) ~ 6 cos( / ) cos( / ) We first deterie te fudaetal period of cos( / ) Here ad r ust satisfy te relatio r Aog all positive solutios for ad r, te sallest values are 6 ad r We et deterie te fudaetal period of cos( / ) Here ad r ust satisfy te relatio r Aog all positive solutios for ad r, te sallest values are ad r Te fudaetal period of ~ 6 is te give by LC (, ) LC(6,) Hece te average power is give by 9 ~ 6 cos cos 6 P 9 6cos 9 cos 9 cos cos 8 7 ow, fro Eq (8) we ave ~ y Terefore ot for sale

15 ~ y Substitutig r we get ~ y r ~ y r Hece ~ y is a periodic sequece wit a period 8 (a) ow ~ Te portio of ~ p i te rage is p give by { } { } { } { 7 }, Hece, oe period of ~ p is give by { 7 }, ow ~ y y Te portio of ~ y p i te rage is give by p y y y { 6} { 8 7} { } { 8 }, Hece, oe period of ~ y p is give by { 8 }, ow w ~ w Te portio of w ~ p i te rage is give by p w w w { } { } { } { 7 }, Hece, oe period of w ~ p is give by { 7 }, (b) 7 ow ~ p 7 Te portio of ~ p i te rage 6 is give by 7 7 { } { } { } { }, 6 Hece, oe period of ~ p is give by { }, 6 ow ~ y y 7 Te portio of ~ y p i te rage 6 is give by p 7 7 { 6} { 8 7 } { } { 8 7 6}, 6 Hece, oe period of ~ is give by { 8 7 6}, 6 ow w ~ w 7 Te portio of w ~ p i te rage 6 is give by p w 7 w w 7 { { } { } } y p ot for sale

16 { }, 6 Hece, oe period of ~ is give by { }, 6 9 ~ Acos( ωo φ ) (a) { } Hece A, ωo /, φ / (b) { } Hece A, ω o /, φ / (c) { } Hece A, ω o /, φ / (d) { } Hece A, ω /, φ o Te fudaetal period of a periodic sequece wit a agular frequecy satisfies Eq (7a) wit te sallest value of ad r (a) Here Eq (7a) reduces to r wic is satisfied wit ω o, r (b) ω o 8, r Here Eq (7a) reduces to w p ω o 8 r wic is satisfied wit / (c) We first deterie te fudaetal period of Re{ e } cos() I tis case, Eq (7a) reduces to r wic is satisfied wit, r / We et deterie te fudaetal period of I{ e si() I tis case, Eq (7a) reduces to r wic is satisfied wit, r Hece te fudaetal period of ~ is give by c LC (, ) LC(,) (d) We first deterie te fudaetal period of cos() I tis case, Eq (7a) reduces to r wic is satisfied wit, r We et deterie te fudaetal period of si( ) I tis case, Eq (7a) reduces to r wic is satisfied wit, r Hece te fudaetal period of ~ is give by LC (, ) LC(,) (e) We first deterie te fudaetal period of cos( 7) I tis case, Eq (7a) reduces to r wic is satisfied wit, r We et deterie te fudaetal period of cos(6 ) I tis case, Eq (7a) reduces to 6 r wic is satisfied wit, r We fially deterie te fudaetal period of si( ) I tis case, Eq (7a) reduces to r wic is satisfied wit, r Hece te fudaetal period of ~ is give by LC,, ) LC(,,) ( Te fudaetal period of a periodic sequece wit a agular frequecy ω o satisfies Eq (7a) wit te sallest value of ad r ot for sale 6

17 (a) ω o 6, r Here Eq (7a) reduces to 6 r wic is satisfied wit (b) ω o 8, r 7 (c) ω o, r 9 (d) ω o, r (e) ω o 6, r Here Eq (7a) reduces to Here Eq (7a) reduces to Here Eq (7a) reduces to Here Eq (7a) reduces to 8 r wic is satisfied wit r wic is satisfied wit r wic is satisfied wit 6 r wic is satisfied wit ω o 8 Here Eq (7a) reduces to 8 r wic is satisfied wit, r For a sequece ~ si( ω) wit a fudaetal period of, Eq (7a) reduces to ω r For eaple, for r we ave ω / 6 Aoter sequece wit te sae fudaetal period is obtaied by settig r wic leads to ω 6 / Te correspodig periodic sequeces are terefore si(6 ) ad si( ) Te tree paraeters A, Ωo, ad φ of te cotiuous-tie sigal a (t) ca be deteried fro For eaple a ( T ) Acos( ΩoT φ) by settig distict values of Acos φ, Acos( Ωo T φ) Acos( ΩoT )cosφ Asi( ΩoT )si φ β,, Acos( Ωo T φ) Acos( ΩoT )cosφ Asi( ΩoT )si φ γ Substitutig te first equatio ito te last two equatios ad te addig te we get β γ cos( Ω o T ) wic ca be solved to deterie Ω o et, fro te secod equatio we ave Asi φ β Acos( Ω T )cosφ β cos( Ω T ) Dividig tis o β cos( ΩoT) equatio by te last equatio o te previous page we arrive at ta φ si( ΩoT wic ca be solved to deterie φ Fially, te paraeter is deteried fro te first equatio of te last page o ot for sale 7

18 ow cosider te case Ω T Ωo I tis case Acos( φ) β ad T Acos( ( ) φ) Acos( φ) β Sice all saple values are equal, te tree paraeters caot be deteried uiquely Fially cosider te case Ω T < Ωo I tis case Acos( Ωo T φ) T Acos( ωo φ) iplyig ω o Ω o T > As eplaied i Sectio, a digital siusoidal sequece wit a agular frequecy ω o greater ta assues te idetity of a siusoidal sequece wit a agular frequecy i te rage ω < Hece, Ω caot be uiquely deteried fro Acos( Ωo T φ) o cos( Ω T ) If is periodic wit a period, te o cos Ω T ΩT cos( Ω ( ) ) T Tis iplies ΩoT r wit r ay ozero positive iteger Hece te saplig rate ust satisfy te relatio T r / Ωo If Ω o, ie, T / 8, te we ust ave r Te 8 sallest value of ad r satisfyig tis relatio are ad r Te fudaetal period is tus (a) For a iput i, i,, te output is yi b i b i b i ayi a yi, i, Te, for a iput A B, te output is y b ( A B ) b ( A B ) b ( A B ) a( Ay By ) a( Ay By ) A( b b b ay a y ) B( b b b a a ) By Ay Hece, te syste of Eq (8) is liear (b) For a iput i, i,, te output is / L,, ± L, ± L, L y i i, oterwise For a iput A B, te output for, ± L, ± L, K is y / L A / L B / L Ay By For all oter values of, y A B Hece te syste of Eq () is liear (c) For a iput i, i,, te output is yi i /, i, Te, for a iput A B, te output is y A / B / Ay By Hece te syste of Eq () is liear ot for sale 8

19 (d) For a iput i, i,, te output is y i i, i, Te, for a iput A B, te output is y ( A B ) A B Ay By Hece te syste of Eq (6) is liear (e) Te first ter o te RHS of Eq (6) is te output of a factor-of- up-sapler Te secod ter o te RHS of Eq (6) is siply te output of a uit delay followed by a factor-of- up-sapler, wereas, te tird ter is te output of a uit advace operator followed by a factor-of- up-sapler We ave sow i Part (b) tat te up-sapler is a liear syste oreover, te uit delay ad te uit advace operator are liear systes A cascade of two liear systes is liear ad te liear cobiatio of liear systes is also liear Hece, te factor-of- iterpolator of Eq (6) is a liear syste (f) Followig te arguets give i Part (e), we ca siilarly sow tat te factor-of- iterpolator of Eq (66) is a liear syste 6 (a) y For a iput i, i,, te output is yi i, i, Te, for a iput A B, te output is y A B Ay By Hece te syste is liear For a iput, te output is te ipulse respose As for <, ad te syste is causal Let for all values of Te ( ) y ad y as Sice a bouded iput results i a ubouded output, te syste is ot BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If te y However, y o ( o ) o (b) o o Sice y y, te syste is ot tie-ivariat o y ( ) For a iput i, i,, te output is y ( ), i, Te, for a iput A B, te output is A ( ) B( ) Hece te syste is oliear i i ( A B ) y For a iput, te output is te ipulse respose ( ) As for <, ad te syste is causal ot for sale 9

20 For a bouded iput B <, te agitude of te output saples are y ( ) B < As te output is also a bouded sequece, te syste is BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If o te y ( ) ( o ) y o Hece, te syste is tie-ivariat (c) y β l wit β a ozero costat For a iput, i,, te output is te output is l y β i i l y β l, i, Te, for a iput A B, l Ay By Hece te syste is oliear ( A l B l ) β A l B l For a iput, te output is te ipulse respose β l As l i l l for <, te syste is ocausal For a bouded iput B <, te agitude of te output saples are y β B < As te output is also a bouded sequece, te syste is BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If te y β l y Hece, te syste is o tie-ivariat l o o (d) y l( ) For a iput i, i,, te output is yi l( i ), i, Te, for a iput A B, te output is y l( A B ) Ay By Hece te syste is oliear For a iput, te output is te ipulse respose l( ) For <, l() Hece, te syste is ocausal For a bouded iput B <, te agitude of te output saples are ( B) < y l As te output is also a bouded sequece, te syste is BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If ot for sale

21 o te y l( o ) y o Hece, te syste is tie-ivariat (e) y, wit a ozero costat For a iput, i,, te output is yi i, i, Te, for a iput A B, te output is y A B Ay By Hece te syste is liear For a iput, te output is te ipulse respose For <, Hece, te syste is causal For a bouded iput B <, te agitude of te output saples are y B < As te output is also a bouded sequece, te syste is BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If o te y ( o ) y o Hece, te syste is tie-ivariat (f) y For a iput i, i,, te output is y, i, i i i A B Te, for a iput A B, te output is y Ay By Hece te syste is liear For a iput, te output is te ipulse respose For <, Hece, te syste is causal For a bouded iput B <, te agitude of te output saples are y B < As te output is also a bouded sequece, te syste is BIBO stable Fially, let y ad y be te outputs for iputs ad, respectively If o te y o y o Hece, te syste is tieivariat 7 Let y ad y be te outputs of a edia filter of legt K for iputs ad, respectively If o, te y ed{ K, K,,,, K, K} ed{ K, K,,,, K, o o o o o y o Hece, te syste is tie-ivariat 8 y For a iput i, i,, te output is yi i i i, i, Te, for a iput A B, te output is y A B A B A B Ay By Hece te syste is liear If o, te y o o o y o Hece, te syste is tie-ivariat K} ot for sale

22 Te ipulse respose of te syste is ow Sice for all values of <, te syste is ocausal 9 y For a iput i, i,, te output is yi i i i, i, Te, for a iput A B, te output is y ( A B ) ( A B )( A B ) Ay By Hece te syste is oliear If o, te y o o o y o Hece, te syste is tieivariat Te ipulse respose of te syste is Sice for all values of <, te syste is causal y y ow for a iput µ, te output y y coverges to soe costat K as Te iput-output relatio of te syste as reduces to K K fro wic we get K or i oter words K K For a iput i, i,, te output is,, i yi yi i Te, yi A for a iput A B, te output is B y y y O te oter ad, A B Ay By Ay y y By y Hece te syste is oliear o If o, te y y y o y Hece, te syste is tie-ivariat y y y For a iput i, i,, te output is yi i yi yi, i, Te, for a iput A B, te output is y A B y y O te oter ad, Ay By ot for sale

23 A Ay Ay B By By y Hece te syste is oliear Let y be te output for a iput, ie, y y y Te, for a iput o te output is give by y o o y o o, or i oter words, te syste is tie-ivariat ow, for a iput µ, te output y coverges to soe costat K as Te differece equatio describig te syste as K, ie, K reduces to K K K, or Te ipulse respose of te factor-of- iterpolator of Eq (66) is te output for a iput ad is give by u { },,,,, ( ) ( ) or equivaletly by { } Te iput-output relatio of a factor-of- L iterpolator is give by L L y u ( u u ) Its ipulse respose is te output for L a iput u or equivaletly by { } ad is tus give by L L L ( ) L L L L {,, K,,,,,, K,, }, L L L L L L Te ipulse respose of a causal discrete-tie syste satisfies te differece equatio a Sice te syste is causal, we ave for < Evaluatig te above differece equatio for, we arrive at a ad tus et, for, we ave a ad tus a Cotiuig we get for, a, ie, a a Assue a wit > Fro te differece equatio we te ave a, ie, a a Sice te last equatio olds for,,, by iductio, it olds for As ad are rigt-sided sequeces, assue for all < ad ad < Hece, y O* for all < ad tus L L L L ot for sale

24 is also a rigt-sided sequece Terefore, y O * y as for all Hece, < y 6 (a) < µ µ µ µ µ,,,, O * µ (b) > µ µ µ µ µ,,, O * 7 ow fro Eq (7) a arbitrary iput ca be epressed as wic ca be rewritte usig Eq (b) as ( ) µ µ µ µ Sice is te respose of a LTI syste for a iput s, µ is te respose for a iput ad s µ s is te respose for a iput Hece, te output for a iput is give by µ µ µ O O * * s s s s y 8 Hece, Tus, is also a periodic sequece wit a period ~ y ~ y y y 9 I tis proble we ae use of te idetity O * r r ot for sale

25 (a) y O* ( ) O* ( ) O* O* 6 O* O* O* O Hece 8 * y (b) y O* ( ) O* ( ) O* 7 O* O* 6 O* O O 7 7 * * 6 (c) y O* ( ) O* ( ) 9 O* O* O* 6 O* O* O 9 6 * (d) y O* ( ) O* ( ) O* O* O* O* O O 8 * * 8 8 (a) u O* y {,,,,,, 66,,, 7,,, }, 8 (b) v O* w {, 7,,, 6,, 8,,, 6,,, }, (c) g w O* y { 8,,,,, 6, 6,, 6,, 6, 9, }, ot for sale

26 y ow, g is defied for Tus, for, is defied for, or equivaletly, for Liewise, for, is defied for, or equivaletly, for For te specified sequeces,,, 6 (a) Te legt of y is 6 ( ) (b) Te rage of for y is i(, ) a(, ), ie, For te specified sequeces te rage of is O* v y y O* ow, v Te Let g O O y O were y O ow * * * * v O* Defie v O* Te fro te results of Proble, v y Hece, y O Terefore, aig use of te results of Proble * agai we get y y O* Substitutig by i tis epressio, we get coutative y O* Let O( ) ( Hece te covolutio operatio is y * ) O* O* covolutio operatio is also distributive Hece te O O O( O ) As O is a ubouded * * * * * sequece, te result of tis covolutio caot be deteried But ot for sale 6

27 O O O( O ) ow O for all values * * * * * of, ad ece te overall result is zero As a result, for te give sequeces O O O O * * * * 6 w O O g Defie y O* ad * * w ( O* )O* g f O* g g Cosider y O* g g ow cosider w O( O g ) * * O* f g Te differece betwee te epressios for w ad w is tat te order of te suatios is caged A) Assuptios: ad g are causal sequeces, ad for < Tis iplies, for <, y Tus, w g y -, for g All sus ave oly a fiite uber of ters Hece, te itercage of te order of te suatios is ustified ad will give correct results B) Assuptios: ad g are stable sequeces, ad is a bouded sequece wit B < Here, y ε, wit ε, ε B I tis case, all sus ave effectively oly a fiite uber of ters ad te error ε ca be reduced by coosig ad, sufficietly large As a result, i tis case te proble is agai effectively reduced to tat of te oe-sided sequeces Tus, te itercage of te order of te suatios is agai ustified ad will give correct results Hece, for te covolutio to be associative, it is sufficiet tat te sequeces be stable ad sigle-sided 7 y Sice is of legt ad defied for, ( ) te covolutio su reduces to y y will be ozero for all tose values of ad for wic satisfies iiu value of ad occurs for lowest at ad aiu value of ad occurs for aiu value of at Tus Hece te total uber of ozero saples ot for sale 7

28 8 y Te aiu value of y occurs at we all product ters are preset Te aiu value is give by y a a 9 y Te aiu value of y occurs at we all product ters are preset Te aiu value is give by y a b 6 (a) y g O* g ow, ev ev ev ev ev y ev g ev ev ( ) g ev gev O* ev y g Replace by Te te suatio o te left becoes y Hece is a eve sequece ev (b) y g ev O* od od g ev ow, od ev od ev od ev od ev Hece gev O* od is a odd sequece y g g ( ) g g y (c) y g od O* od od g od ow, od od od od od od od od Hece god O* od is a eve sequece y g g ( ) g g y 6 Te ipulse respose of te cascade is give by O were * µ ad β µ Hece, ( β ) µ 6 ow µ Terefore y y Hece, y y Tus te iverse syste is give by y Te ipulse respose of te iverse syste is give by {, }, 6 Fro te results of Proble 6 we ave ( β ) µ ow, ot for sale 8

29 y β µ β Substitutig β r i te last epressio we get β β r r y r r r r r r r β β r r r r r r r K β y Te iverse syste is terefore give by ) ( β β y y y 6 (a) O O O O O * * * * * (b) O O O O * * * * 6 ow O * ) )O( ( O * * O O O O * * * * 6 ad 7 Terefore, (a) Te legt of is 8 Usig y we arrive at },,,, {, } { (b) Te legt of is 7 Usig y we arrive at },,, {, } { (c) Te legt of is 8 Usig y we arrive at },,, {, } { ot for sale 9

30 67 y ay b Hece, y ay b et, ( ay b ) b a y ab b a way we obtaied y a y a b (a) Let y be te output due to a iput Te y a y a b If o, te y ay b Cotiuig furter i a siilar o o r y a y a b o a y r a b r However, r y o a y a b o o a y o o r a b Hece y y o if y, ie, te syste is tie-variat Te syste is tie-ivariat if ad oly if y, as te y y o (b) Let y ad y be te outputs due to iputs ad, respectively Let y be te output due to a iput β However, y βy β a y a y a b β a b, wereas, y a y a b β a b Hece, te syste is oliear if y ad is liear if ad oly if y (c) Geeralizig te above result it ca be sow tat a t order causal discretetie syste is liear ad tie-ivariat if ad oly if y r, r 68 y p p d y leads to y y, wic is te differece equatio caracterizig te iverse syste I oter words, siply solve te equatio for i ters of preset ad past values of y ad 69 s µ,, ad s, < Sice is oegative, s is a ootoically icreasig fuctio of for, ad is ot oscillatory Hece, tere is o oversoot 7 (a) f f f Let f r, te te differece equatio reduces to r r r wic reduces furter to r r resultig i ± r Tus, f As f, ece Also f,, ad ece Solvig for ad, we get Hece, p d p p p r ot for sale

31 f (b) y y y As te syste is LTI, te iitial coditios are equal to zero Let Te y y y Hece, y y y ad y y y For >, te correspodig differece equatio is y y y wit iitial coditios y ad y, wic are te sae as tose for te solutio of te Fiboacci s Tus deotes te ipulse respose of a causal LTI syste described by te differece equatio y y y sequece Hece y 7 y y Deotig y y y, ad a b, we get re i re yi ( a b)( yre yi ) y Equatig te real ad te iagiary parts, ad otig tat is real, we get yre ayre byi, yi byre ayi Fro te a secod equatio we ave yi yi yre Substitutig tis equatio i te top left equatio we arrive at y b b re re a i a re ayre ( a b ) yre yre ayre byi ayre ( a b ) yre ay y y, fro wic we get byi a Substitutig tis equatio i te equatio we arrive at yre a wic is a secod-order differece equatio represetig y re i ters of 7 Te first-order causal LTI syste is caracterized by te differece equatio y p p d y Lettig we obtai te differece Solvig it for,,, we get p, p d p d p, ad d d p ( p d p ) Solvig tese equatios we get p, d, ad p equatio represetatio of its ipulse respose p p d 7 p d y Let Te p d Tus, r b a p d r Sice te syste is assued to be causal, r ot for sale

32 r r for all > r Hece, p d r d 7 For a filter wit a cople-valued ipulse respose, te first part of te proof is te sae as tat for a filter wit a real-valued ipulse respose Fro y we get y Sice te iput is bouded B Terefore y B So if S <, te y BS idicatig tat y is also bouded To prove te coverse we eed to sow tat if a bouded iput is produced by a bouded iput te S < Cosider te followig bouded iput defied by * * Te y S ow sice te output is bouded, S < Tus for a filter wit a cople ipulse respose is BIBO stable if ad oly if < S 7 Te ipulse respose of te cascade is g r r Tus g r r r Sice r r ad are stable, < ad < Hece g < result, a cascade of two stable LTI systes is also stable 76 Te ipulse respose of te parallel structure is g ow, ad as a g Sice ad are stable, < ad < Hece g < parallel coectio of two stable LTI systes is also stable ad as a result, a 77 Cosider a cascade coectio of two passive LTI systes wit a iput ad a output y Let y ad y be te outputs of te two systes for te iput ow y ad y Let y y satisfyig te above iequalities Te y y y ad as a result, > y Hece, te parallel coectio of two passive LTI systes ay ot be passive 78 Cosider a parallel coectio of two passive LTI systes wit a iput ad a output y Let y ad y be te outputs of te two systes for te iput ot for sale

33 y satisfyig te above iequalities Te y y y ad as a result, > y ow y ad y Let y parallel coectio of two passive LTI systes ay ot be passive Hece, te 79 Let te differece equatio p y d y represets te causal IIR digital filter For a iput, te correspodig output is te y, te ipulse respose of te filter As te uber of coefficiets { p } is ad te uber of coefficiets { d } is, tere are a total of uows To deterie tese coefficiets fro te ipulse respose saples, we copute oly te first ipulse respose saples To illustrate te etod, witout ay loss of geerality, we assue Te, fro te differece equatio we arrive at te followig 7 equatios: p, d p, d d p, d d d p, d d d, d d d, 6 d d d Writig te last tree equatios i atri for we arrive at d d d, ad ece, d 6 d 6 d Substitutig tese values i te first four equatios writte i atri for we get p p d p d p d 8 y y l l y l lµ l y l l y ( ) (a) For y, y (b) For y, y ( ) ( ) ot for sale

34 ( T ( ) T T y T y y( T ) ad ( T ) 8 y T ) y( ( ) T ) ( τ) dτ y( ( ) T ) T ( ( ) ) Terefore, te differece equatio represetatio is give by y were l l l l ( ) y y y 8 y l l, ow y l l,, ie, Tus, te differece equatio represetatio is give by 8 y y µ wit y Te total solutio is give by y yc y p, were y c is te copleetary solutio ad y p is te particular solutio y c is obtaied by solvig yc yc To tis ed we set y c λ, wic yields λ λ resultig i te solutio λ Hece y c () For te particular solutio we coose y p β Substitutig tis solutio i te differece equatio represetig te syste we get β β µ For we get β β, ie, ( ) β ad ece β / 6 8 / Terefore y yc y p (), For, we tus ave 8 8 y () iplyig Te total solutio is tus give by y (), µ 8 y y y wit y ad y Te total solutio is give by y yc y p, were y c is te copleetary solutio ad y p is te particular solutio y c is obtaied by solvig yc yc yc To tis ed we set y λ, wic yields λ λ λ resultig i te solutios c λ or λ Hece y () ( ) For te particular solutio we coose 8 c y p β() differece equatio represetig te syste we get β() β() β β() β() β() µ wic yields For Substitutig tis solutio i te we ave β 66 Terefore ot for sale

35 y yc y p () ( ) 66(), For ad () ( ) 66() we tus ave y () ( ) 66() ad y Solvig tese two equatios we get 89 ad Hece, y 89() ( ) 66(), 8 y y y wit, y ad y Te total solutio is give by y yc y p, were y c is te copleetary solutio ad y p is te particular solutio Fro te solutio of Proble 8, te copleetary solutio is of te for y c () ( ) µ To deterie y p we observe tat te it is give by te su of te particular solutio y p of te differece equatio Proble 8, we ave µ y y y ad te particular solutio y p of te differece equatio y y y µ Fro te solutio of y p β() Hece, y p y p β() Terefore, y p y p y p β() β() µ µ For te above equatio reduces to β β Tus, β Terefore, te total solutio is give by y yc y p () ( ) () (), For ad () ( ) () () we tus ave y () ( ) () () ad y Solvig tese two equatios we get 88 ad Hece, y 88() ( ) () () 86 Te solutio is give by, is te copleetary solutio ad p is te particular solutio If is te ipulse respose, te c p, were c p Fro Proble 8 we ote tat () Tus, Tis iplies Hece, (), 87 Te overall syste ca be regarded as te cascade of two causal LTI systes: S: y y y ad S: c ot for sale

36 Te ipulse respose of te syste S ca be foud by solvig te copleetary solutio of Let te copleetary solutio be c λ, we ave λ λ λ ece λ {, } Terefore, te ipulse respose is give by c A() B( ), Solvig costats A, B, we get A 8 ad B Hece 8() ( ), Te ipulse respose of te syste S is give by Te ipulse respose of te overall syste is ( 8() ( ) ) µ ( 8() ( ) ) µ * 9() µ 8( ) µ 88 ( ) µ, < < Step respose is te give by s O* µ ) µ Oµ ( ) µ µ * ( ),,, < ( ( ),, K, < K A 89 Let A ( λi ) Te λi ow li Sice tere A eists a positive iteger A coverges A λi o suc tat for all > o, < < < Hece A 9 { } {,,,,,, },, { y } {6,,,, 8, 7, },, { w } {,,,,,, }, 8 (a) r l l, 6 l 6 { r l} { 8,,,,,, 9,,,,,, 8}, 6 l 6, ryy l y y l, 6 l 6 { r l} {, 8, 9,,, 7, 6, 7,,, 9, 8, }, 6 l 6, yy r 6 ww l 6 w w l, 6 l 6 { r ww l} {,, 6,, 6,, 7,, 6,, 6,, }, 6 l 6, K ot for sale 6

37 (b) r y l y l, 8 l { r y l} {8, 8,,,,, 68,,,,, 6, }, 8 l, r w l 8 y l, l { r l} {,,, 8,, 7,,, 7,,,, 6}, l, w 9 (a) µ l r µ l l l l µ µ l l l l l, l <,, l, l, l, l <, l ote for l, r, l ad for l <, r l Replacig l wit l ( l) l i te secod epressio we get r l r l l Hece, r is a eve fuctio of l aiu value of r l occurs at l sice is a decayig fuctio for icreasig we < (b),, ow r, were, oterwise l l, for l < ( ), l, for ( ) l,, l l, l Terefore, r, oterwise, for, l l l, for < l,, for l > It follows fro te above tat r is a triagular fuctio of, ad ece is a l l eve fuctio wit a aiu value of at l 9 (a) cos were is a positive iteger Period of is, ad ( l) ece r l l cos cos l l cos cos cos si si l cos cos l l ot for sale 7

38 ow cos cos cos cos Let C cos ad ( / ) S si Te C S e e Tis iplies C Tus cos / e l l Hece, r cos cos l (b) { } 6 {,,,,, }, It is a periodic sequece wit a period 6 Tus, r 6, l 6 l l r is also a l periodic sequece wit a period 6 r ( ), 6 6 r ( ), 6 6 r ( ), r ( ), 6 6 r ( ), 6 6 r ( ) 6 6 (c) ( ) is a periodic sequece wit a period Tus, r l l, l Hece, r ( ), r ( ) r l is also a periodic sequece wit a period (a) Te iput data etered durig te eecutio of Progra are: Type i real epoet -/ Type i iagiary epoet pi/6 Type i gai costat Type i legt of sequece (b) Te iput data etered durig te eecutio of Progra are: Type i real epoet -/ Type i iagiary epoet pi/6 Type i gai costat Type i legt of sequece ot for sale 8

39 Real part Iagiary part Aplitude Aplitude Tie ide ~ a Tie ide (a) e Te plots geerated usig Progra are sow below: Iagiary part Real part Aplitude - - Tie ide Aplitude - - Tie ide (b) Te code fraget used to geerate ~ b si(8 8 ) is as follows: si(8*pi* 8*pi); Te plot of te periodic sequece is give below: Aplitude - - (c) Te code fraget used to geerate ~ / / Re( e ) I( e ) c is as follows: real(ep(i*pi*/) iag(ep(i*pi*/); Te plot of te periodic sequece is give below: ot for sale 9

40 Aplitude - - (d) Te code fraget used to geerate ~ d cos() si( ) is as follows: *cos(*pi*)-*si(*pi**pi); Te plot of te periodic sequece is give below: Aplitude - - (e) ~ Te code fraget used to geerate e cos( 7) cos(6) si( ) is as follows: *cos(*pi*7*pi)*cos(6*pi*)-si(*pi*); Te plot of te periodic sequece is give below: Aplitude - - (a) L iput('desired legt '); A iput('aplitude '); oega iput('agular frequecy '); pi iput('pase '); :L-; ot for sale

41 A*cos(oega* pi); ste(,); label('tie Ide'); ylabel('aplitude'); title('\oega{o} ',ustr(oega/pi),'\pi'); (b) ω o 6 ω o 8 Aplitude - Aplitude Aplitude - Tie Ide ω o Tie Ide Aplitude - Tie Ide ω o - ω o Tie Ide Aplitude Tie Ide t ::; fo iput('frequecy of siusoid i Hz '); FT iput('saplig frequecy i Hz '); g cos(*pi*fo*t); plot(t,g,'-'); label('tie'); ylabel('aplitude'); old ::FT; gs cos(*pi*fo*/ft); plot(/ft,gs,'o'); old off t ::8; ot for sale

42 g cos(6*pi*t); g cos(*pi*t); g cos(6*pi*t); plot(t/8,g,'-', t/8, g, '--', t/8, g,':'); label('tie'); ylabel('aplitude'); old ::8; gs cos(6*pi*); plot(/8,gs,'o'); old off 6 As te legt of te ovig average filter is icreased, te output of te filter gets ore sooter However, te delay betwee te iput ad te output sequeces also icreases (Tis ca be see fro te plots geerated by Progra for various values of te filter legt) 7 alpa iput('alpa '); y ; y *(y (alpa/y)); wile abs(y-y)> y *(y(alpa/y)); y y; y y; ed disp('squre root of alpa is'); disp(y); 8 forat log alpa iput('alpa '); y ; y zeros(,6); L legt(y) - ; y() alpa - y*y y; ; wile abs(y(-) - y) > y alpa - y(-)*y(-) y(-); y y(-); y() y; ; ed disp('square root of alpa is');disp(y(-)); :-; err y(:-) - sqrt(alpa); ste(,err); ais( - i(err) a(err)); label('tie ide '); ylabel('error'); title('\alpa ',ustr(alpa)); Te displayed out is Square root of alpa is 8789 ot for sale

43 786 Error Tie ide 9 { } {,,,,,, },, { y } {6,,,, 8, 7, },, { w } {,,,,,, }, 8 { r } { 8,,,,,, 9,,,,,, 8}, 6 6 { } {, 8, 9,,, 7, 6, 7,,, 9, 8, }, 6 6 r yy { r ww } {,, 6,, 6,, 7,, 6,, 6,, }, r r yy Aplitude Aplitude Lag ide Lag ide Aplitude 6 r ww Lag ide Aplitude r y - Lag ide ot for sale

44 r w 6 r yw Aplitude Aplitude Lag ide Lag ide iput('legt of sequece '); :-; ep(-8*); y rad(,)-; legt()-; r cov(y,fliplr(y)); (-):; ste(,r); label('lagide'); ylabel('aplitude'); Aplitude Lag ide ot for sale