Chapter 1 Answers1.6 (a).NoBecause when t<0, )(1t x =0.(b).NoBecause only if n=0, ][2n x has valuable.(c).Yes Because ∑∞-∞=--+--+=+k k m n k m n m n x ]}414[]44[{]4[δδ ∑∞-∞=------=k m k n m k n )]}(41[)](4[{δδ ∑∞-∞=----=k k n k n ]}41[]4[{δδ N=4.1.9 (a). T=π/5Because 0w =10, T=2π/10=π/5.(b). Not periodic.Because jt t e e t x --=)(2, while t e -is not periodic, )(2t x is not periodic.(c). N=2Because 0w =7π, N=(2π/0w )*m, and m=7.(d). N=10Because n j j e e n x )5/3(10/343)(ππ=, that is 0w =3π/5, N=(2π/0w )*m, and m=3.(e). Not periodic. Because 0w =3/5, N=(2π/0w )*m=10πm/3 , it ’s not a rational number.1.14 A1=3, t1=0, A2=-3, t2=1 or -1dtt dx )( isSolution: x(t) isBecause ∑∞-∞=-=k k t t g )2()(δ, dt t dx )(=3g(t)-3g(t-1) or dtt dx )(=3g(t)-3g(t+1) 1.15. (a). y[n]=2x[n-2]+5x[n-3]+2x[n-4]Solution:]3[21]2[][222-+-=n x n x n y ]3[21]2[11-+-=n y n y ]}4[4]3[2{21]}3[4]2[2{1111-+-+-+-=n x n x n x n x ]4[2]3[5]2[2111-+-+-=n x n x n xThen, ]4[2]3[5]2[2][-+-+-=n x n x n x n y(b).No. For it ’s linearity.the relationship between ][1n y and ][2n x is the same in-out relationship with (a). you can have a try.1.16. (a). No.For example, when n=0, y[0]=x[0]x[-2]. So the system is memory. (b). y[n]=0.When the input is ][n A δ,then, ]2[][][2-=n n A n y δδ, so y[n]=0. (c). No.For example, when x[n]=0, y[n]=0; when x[n]=][n A δ, y[n]=0. So the system is not invertible.1.17. (a). No.For example, )0()(x y =-π. So it ’s not causal.(b). Yes.Because : ))(sin()(11t x t y = , ))(sin()(22t x t y =))(sin())(sin()()(2121t bx t ax t by t ay +=+1.21. Solution:We have known:(a).(b).(c).(d).1.22. Solution:We have known:(a).(b).(e).(g)1.23. Solution:For )]()([21)}({t x t x t x E v -+= )]()([21)}({t x t x t x O d --= then,(a).(b).(c).1.24.For: ])[][(21]}[{n x n x n x E v -+= ])[][(21]}[{n x n x n x O d --=then,(a).(b).1.25. (a). Periodic. T=π/2.Solution: T=2π/4=π/2.(b). Periodic. T=2.Solution: T=2π/π=2.(d). Periodic. T=0.5. Solution: )}()4{cos()(t u t E t x v π=)}())(4cos()()4{cos(21t u t t u t --+=ππ )}()(){4cos(21t u t u t -+=π )4cos(21t π= So, T=2π/4π=0.51.26. (a). Periodic. N=7Solution: N=m *7/62ππ=7, m=3.(b). Aperriodic.Solution: N=ππm m 16*8/12=, it ’s not rational number.(e). Periodic. N=16 Solution as follow:)62cos(2)8sin()4cos(2][ππππ+-+=n n n n x in this equation,)4cos(2n π, it ’s period is N=2π*m/(π/4)=8, m=1.)8sin(n π, it ’s period is N=2π*m/(π/8)=16, m=1.)62cos(2ππ+-n , it ’s period is N=2π*m/(π/2)=4, m=1. So, the fundamental period of ][n x is N=(8,16,4)=16.1.31. SolutionBecause )()1()(),2()()(113112t x t x t x t x t x t x ++=--=. According to LTI property ,)()1()(),2()()(113112t y t y t y t y t y t y ++=--=Extra problems:Sketch ⎰∞-=t dt t x t y )()(. 1. SupposeSolution:2. SupposeSketch:(1). )]1(2)1()3()[(--+++t t t t g δδδ(2). ∑∞-∞=-k k t t g )2()(δ(2).Chapter 22.1 Solution:Because x[n]=(1 2 0 –1)0, h[n]=(2 0 2)1-, then(a).So, ]4[2]2[2]1[2][4]1[2][1---+-+++=n n n n n n y δδδδδ (b). according to the property of convolutioin:]2[][12+=n y n y(c). ]2[][13+=n y n y][*][][n h n x n y =][][k n h k x k -=∑∞-∞= ∑∞-∞=-+--=k k k n u k u ]2[]2[)21(2 ][211)21()21(][)21(12)2(0222n u n u n n k k --==+-++=-∑ ][])21(1[21n u n +-= the figure of the y[n] is:2.5 Solution:We have known: ⎩⎨⎧≤≤=elsewhere n n x ....090....1][，，, ⎩⎨⎧≤≤=elsewhere N n n h ....00....1][，，,(9≤N ) Then, ]10[][][--=n u n u n x , ]1[][][---=N n u n u n h∑∞-∞=-==k k n u k h n h n x n y ][][][*][][ ∑∞-∞=-------=k k n u k n u N k u k u ])10[][])(1[][(So, y[4] ∑∞-∞=-------=k k u k u N k u k u ])6[]4[])(1[][( ⎪⎪⎩⎪⎪⎨⎧≥≤=∑∑==4,...14, (140)0N N k Nk =5, then 4≥N And y[14] ∑∞-∞=------=k k u k u N k u k u ])4[]14[])(1[][(⎪⎪⎩⎪⎪⎨⎧≥≤=∑∑==14,...114, (1145)5N N k Nk =0, then 5<N ∴4=N2.7 Solution:[][][2]k y n x k g n k ∞=-∞=-∑（a ）[][1]x n n δ=-,[][][2][1][2][2]k k y n x k g n k k g n k g n δ∞∞=-∞=-∞=-=--=-∑∑(b) [][2]x n n δ=-,[][][2][2][2][4]k k y n x k g n k k g n k g n δ∞∞=-∞=-∞=-=--=-∑∑ (c) S is not LTI system..(d) [][]x n u n =,0[][][2][][2][2]k k k y n x k g n k u k g n k g n k ∞∞∞=-∞=-∞==-=-=-∑∑∑2.8 Solution: )]1(2)2([*)()(*)()(+++==t t t x t h t x t y δδ )1(2)2(+++=t x t xThen,That is, ⎪⎪⎪⎩⎪⎪⎪⎨⎧≤<-≤<-+-=-<<-+=others t t t t t t t t y ,........010,....2201,.....41..,.........412,.....3)(2.10 Solution:(a). We know:Then,)()()(αδδ--='t t t h)]()([*)()(*)()(αδδ--='='t t t x t h t x t y )()(α--=t x t xthat is,So, ⎪⎪⎩⎪⎪⎨⎧+≤≤-+≤≤≤≤=others t t t t t t y ,.....011,.....11,....0,.....)(ααααα(b). From the figure of )(t y ', only if 1=α, )(t y ' would contain merely therediscontinuities.2.11 Solution:(a). )(*)]5()3([)(*)()(3t u et u t u t h t x t y t----==⎰⎰∞∞---∞∞--------=ττττττττd t u e u d t u eu t t )()5()()3()(3)(3⎰⎰-------=tt t t d e t u d et u 5)(33)(3)5()3(ττττ⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧≥+-=-<≤-=<=---------⎰⎰⎰5,.......353,.....313.........,.........0315395)(33)(3393)(3t e e d e d e t e d e t tt t t t t t t t ττττττ(b). )(*)]5()3([)(*)/)(()(3t u e t t t h dt t dx t g t ----==δδ)5()3()5(3)3(3---=----t u e t u e t t(c). It ’s obvious that dt t dy t g /)()(=.2.12 Solution∑∑∞-∞=-∞-∞=--=-=k tk tk t t u ek t t u e t y )]3(*)([)3(*)()(δδ∑∞-∞=---=k k t k t u e)3()3(Considering for 30<≤t ,we can obtain33311])3([)(---∞=-∞-∞=--==-=∑∑ee e ek t u e e t y tk k tk kt. (Because k must be negetive ,1)3(=-k t u for 30<≤t ).2.19 Solution:(a). We have known:][]1[21][n x n w n w +-=(1) ][]1[][n w n y n y βα+-=(2)from (1), 21)(1-=E EE Hfrom (2), αβ-=E EE H )(2then, 212212)21(1)21)(()()()(--++-=--==E E E E E E H E H E H ααβαβ∴][]2[2]1[)21(][n x n y n y n y βαα=-+-+-but, ][]1[43]2[81][n x n y n y n y +-+--=∴⎪⎩⎪⎨⎧=⎪⎭⎫ ⎝⎛=+=143)21(:....812βααor ∴⎪⎩⎪⎨⎧==141βα(b). from (a), we know )21)(41()()()(221--==E E E E H E H E H21241-+--=E EE E ∴][)41()21(2][n u n h n n ⎥⎦⎤⎢⎣⎡-=2.20 (a). 1⎰⎰∞∞-∞∞-===1)0cos()cos()()cos()(0dt t t dt t t u δ(b). 0dt t t )3()2sin(5+⎰δπ has value only on 3-=t , but ]5,0[3∉-∴dt t t )3()2sin(5+⎰δπ=0(c). 0⎰⎰---=-641551)2cos()()2cos()1(dt t t u d u πτπττ⎰-'-=64)2cos()(dt t t πδ0|)2(s co ='=t t π 0|)2sin(20=-==t t ππ∑∞-∞=-==k t h kT t t h t x t y )(*)()(*)()(δ∑∞-∞=-=k kT t h )(∴2.27Solution()y A y t dt ∞-∞=⎰,()xA x t dt ∞-∞=⎰,()hA h t dt ∞-∞=⎰.()()*()()()y t x t h t x x t d τττ∞-∞==-⎰()()()()()()()()()(){()}y x hA y t dt x x t d dtx x t dtd x x t dtd x x d d x d x d A A ττττττττττξξτττξξ∞∞∞-∞-∞-∞∞∞∞∞-∞-∞-∞-∞∞∞∞∞-∞-∞-∞-∞==-=-=-===⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰⎰(a) ()()(2)tt y t e x d τττ---∞=-⎰,Let ()()x t t δ=,then ()()y t h t =. So , 2()(2)(2)()(2)()(2)t t t t t h t ed e d e u t τξδττδξξ---------∞-∞=-==-⎰⎰(b) (2)()()*()[(1)(2)]*(2)t y t x t h t u t u t e u t --==+---(2)(2)(1)(2)(2)(2)t t u eu t d u e u t d ττττττττ∞∞-------∞-∞=+------⎰⎰22(2)(2)12(1)(4)t t t t u t e d u t e d ττττ---------=---⎰⎰(2)2(2)212(1)[]|(4)[]|t t t t u t e e u t ee ττ-------=--- (1)(4)[1](1)[1](4)t t e u t e u t ----=-----2.46 SolutionBecause)]1([2)1(]2[)(33-+-=--t u dtde t u e dt d t x dt d t t )1(2)(3)1(2)(333-+-=-+-=--t e t x t e t x t δδ.From LTI property ,we know)1(2)(3)(3-+-→-t h e t y t x dtdwhere )(t h is the impulse response of the system. So ,following equation can be derived.)()1(223t u e t h e t --=-Finally, )1(21)()1(23+=+-t u e e t h t 2.47 SoliutionAccording to the property of the linear time-invariant system: (a). )(2)(*)(2)(*)()(000t y t h t x t h t x t y ===(b). )(*)]2()([)(*)()(00t h t x t x t h t x t y --==)(*)2()(*)(0000t h t x t h t x --=012y(t)t4)2()(00--=t y t y(c). )1()1(*)(*)2()1(*)2()(*)()(00000-=+-=+-==t y t t h t x t h t x t h t x t y δ(d). The condition is not enough.(e). )(*)()(*)()(00t h t x t h t x t y --==τττd t h x )()(00+--=⎰∞∞-)()()(000t y dm m t h m x -=--=⎰∞∞-(f). )()]([)](*)([)(*)()(*)()(000000t y t y t h t x t h t x t h t x t y "=''='--'=-'-'==Extra problems:1. Solute h(t), h[n](1). )()(6)(5)(22t x t y t y dt dt y dtd =++ (2). ]1[][2]1[2]2[+=++++n x n y n y n y Solution:(1). Because 3121)3)(2(1651)(2+-++=++=++=P P P P P P P Hso )()()()3121()(32t u e e t P P t h t t ---=+-++=δ (2). Because )1)(1(1)1(22)(22i E i E EE E E E E E H -+++=++=++=iE Eii E E i -+-+++=1212 so []][)1()1(2][1212][n u i i i k i E E i i E E i n h n n +----=⎪⎪⎪⎪⎭⎫⎝⎛-+-+++=δChapter 33.1 Solution:Fundamental period 8T =.02/8/4ωππ==00000000033113333()224434cos()8sin()44j kt j t j t j t j tk k j t j t j t j tx t a e a e a e a e a e e e je je t t ωωωωωωωωωππ∞----=-∞--==+++=++-=-∑3.2 Solution:for, 10=a , 4/2πj ea --= , 4/2πj ea = , 3/42πj ea --=, 3/42πj ea =n N jk k N k e a n x )/2(][π∑>=<=n j n j n j n j e a e a e a e a a )5/8(4)5/8(4)5/4(2)5/4(20ππππ----++++=n j j n j j n j j n j j e e e e e e e e )5/8(3/)5/8(3/)5/4(4/)5/4(4/221ππππππππ----++++= )358cos(4)454cos(21ππππ++++=n n)6558sin(4)4354sin(21ππππ++++=n n3.3 Solution: for the period of )32cos(t πis 3=T , the period of )35sin(t πis 6=Tso the period of )(t x is 6 , i.e. 3/6/20ππ==w)35sin(4)32cos(2)(t t t x ππ++= )5sin(4)2cos(21200t w t w ++=)(2)(21200005522t w j t w j t w j t w j e e j e e ----++=then, 20=a , 2122==-a a , j a 25=-, j a 25-=3.5 Solution:(1). Because )1()1()(112-+-=t x t x t x , then )(2t x has the same period as )(1t x ,that is 21T T T ==, 12w w =(2). 212111()((1)(1))jkw t jkw tk T T b x t e dt x t x t e dt T--==-+-⎰⎰111111(1)(1)jkw tjkw t T Tx t e dt x t e dt T T --=-+-⎰⎰ 111)(jkw k k jkw k jkw k e a a e a e a -----+=+=3.8 Solution:kt jw k k e a t x 0)(∑∞-∞==while:)(t x is real and odd, then 00=a , k k a a --=2=T , then ππ==2/20wand0=k a for 1>kso kt jw k k e a t x 0)(∑∞-∞==t jw t jw e a e a a 00110++=--)sin(2)(11t a e e a t j t j πππ=-=-for12)(2121212120220==++=-⎰a a a a dt t x∴2/21±=a ∴)sin(2)(t t x π±=3.13 Solution:Fundamental period 8T =.02/8/4ωππ==kt jw k k e a t x 0)(∑∞-∞==∴t jkw k k e jkw H a t y 0)()(0∑∞-∞==0004, 0sin(4)()0, 0k k H jk k k ωωω=⎧==⎨≠⎩ ∴000()()4jkw t k k y t a H jkw e a ∞=-∞==∑Because 48004111()1(1)088T a x t dt dt dt T ==+-=⎰⎰⎰So ()0y t =.kt jw k k e a t x 0)(∑∞-∞==∴t jkw k k e jkw H a t y 0)()(0∑∞-∞== ∴dt e jkw H t y Ta t jkw Tk 0)()(10-⎰=for⎪⎩⎪⎨⎧>≤=100, (0100),.......1)(w w jw H ∴if 0=k a , it needs 1000>kwthat is 12100,........1006/2>>k kππand k is integer, so 8>K3.22 Solution:021)(1110===⎰⎰-tdt dt t x Ta Tdt te dt te dt e t x T a t jk t jk t jkw T k ππ-----⎰⎰⎰===1122112121)(10t jk tde jk ππ--⎰-=1121⎥⎥⎦⎤⎢⎢⎣⎡---=----111121ππππjk e te jk t jk tjk ⎥⎦⎤⎢⎣⎡---+-=--ππππππjk e e e e jk jk jk jk jk )()(21⎥⎦⎤⎢⎣⎡-+-=ππππjk k k jk )sin(2)cos(221[]πππππk jk k j k jk k)1()cos()cos(221-==-=0............≠k404402()()1184416tj tj t t j tt j t H j h t edt ee dte edt e e dtj j ωωωωωωωω∞∞----∞-∞∞----∞===+=+=-++⎰⎰⎰⎰A periodic continous-signal has Fourier Series:. 0()j kt k k x t a e ω∞=-∞=∑T is the fundamental period of ()x t .02/T ωπ=The output of LTI system with inputed ()x t is 00()()jk t k k y t a H jk e ωω∞=-∞=∑Its coefficients of Fourier Series: 0()k k b a H jk ω= (a)()()n x t t n δ∞=-∞=-∑.T=1, 02ωπ=11k a T==. 01/221/21()()1jkw t jk tk T a x t e dt t e dt Tπδ---===⎰⎰ （Note ：If ()()n x t t nT δ∞=-∞=-∑,1k a T=） So 2282(2)16(2)4()k k b a H jk k k πππ===++ (b)()(1)()n n x t t n δ∞=-∞=--∑ .T=2, 0ωπ=,11k a T== 01/23/21/21/2111()()(1)(1)221[1(1)]2jkw t jk tjk t k T k a x t e dt t e dt t e dtT ππδδ----==+--=--⎰⎰⎰So 24[1(1)]()16()k k k b a H jk k ππ--==+, (c) T=1,02ωπ=01/421/4sin()12()jk t jk tk T k a x t e dt e dt Tk ωπππ---===⎰⎰28sin()2()[16(2)]k k k b a H jk k k ππππ==+ 3.35 Solution: T= /7π,02/14T ωπ==.kt jw k k e a t x 0)(∑∞-∞==∴t jkw k k e jkw H a t y 0)()(0∑∞-∞==∴0()k k b a H jkw =for⎩⎨⎧≥=otherwise w jw H ,.......0250,.......1)(,01,. (17)()0,.......k H jkw otherwise ⎧≥⎪=⎨⎪⎩that is 0250250, (14)k k ω<<, and k is integer, so 18....17k or k <≤. Let ()()y t x t =,k k b a =, it needs 0=k a ,for 18....17k or k <≤.3.37 Solution:11()[]()212()21312411511cos 224nj j nj n n n n j nn j nn n j j j H e h n ee ee e e e ωωωωωωωωω∞∞--=-∞=-∞-∞--=-∞=-===+=+=---∑∑∑∑A periodic sequence has Fourier Series:2()[]jk n Nk k N x n a eπ=<>=∑.N is the fundamental period of []x n .The output of LTI system with inputed []x n is 22()[]()jk jk n NNk k N y n a H eeππ=<>=∑.Its coefficients of Fourier Series: 2()jk Nk k b a H eπ=(a)[][4]k x n n k δ∞=-∞=-∑.N=4, 14k a =.So 2314()524cos()44j k Nk k b a H e k ππ==-3165cos()42k b k π=-3.40 Solution: According to the property of fourier series: (a). )2cos(2)cos(20000000t Tka t kw a e a ea a k k t jkw k t jkw k k π==+='- (b). Because 2)()()}({t x t x t x E v -+=}{2k v k k k a E a a a =+='-(c). Because 2)(*)()}({t x t x t x R e +=2*kk k a a a -+='(d). k k k a Tjka jkw a 220)2()(π=='(e). first, the period of )13(-t x is 3T T ='then 3)(1)13(131213120dme m x T dt e t x T a m T jk T t T jk T k +'--'-'-'⎰⎰'=-'='ππTjkk m T jk T T jk T jk m T jk T ea dm e m x T e dm e e m x T πππππ221122211)(1)(1---------=⎥⎦⎤⎢⎣⎡==⎰⎰3.43 (a) Proof:（i ）Because ()x t is odd harmonic ,(2/)()jk T t k k x t a e π∞=-∞=∑,where 0k a = for everynon-zero even k.(2/)()2(2/)(2/)()2T jk T t k k jk jk T tk k jk T tk k T x t a ea e e a e ππππ∞+=-∞∞=-∞∞=-∞+===-∑∑∑It is noticed that k is odd integers or k=0.That means()()2Tx t x t =-+（ii ）Because of ()()2Tx t x t =-+,we get the coefficients of Fourier Series222/200/222(/2)/2/20022/2/200111()()()11()(/2)11()()(1)jk t jk t jk t T T T T T T k T jk t jk t T T T T Tjk t jk t T T k TT a x t e dt x t e dt x t e dtT T T x t e dt x t T e dt T T x t e dt x t e dt T T πππππππ-----+--==+=++=--⎰⎰⎰⎰⎰⎰⎰ 2/21[1(1)]()jk t T kT x t e dt T π-=--⎰It is obvious that 0k a = for every non-zero even k. So ()x t is odd harmonic ,(b)Extra problems:∑∞-∞=-=k kT t t x )()(δ, π=T(1). Consider )(t y , when )(jw H ist(2). Consider )(t y , when )(jw H isSolution:∑∞-∞=-=k kT t t x )()(δ↔π11=T , 220==Tw π(1).kt j k k tjkw k k e k j H a ejkw H a t y 20)2(1)()(0∑∑∞-∞=∞-∞===ππ2=(for k can only has value 0)(2).kt j k k tjkw k k e k j H a e jkw H a t y 20)2(1)()(0∑∑∞-∞=∞-∞===πππte e t j t j 2cos 2)(122=+=- (for k can only has value –1 and 1)。
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