Consider the signal x(t) = cos 27T't. Since x(t) is periodic with a fundamental period

A continuous-time periodic signal x(t) is real valued and has a fundamental period T = 8. The nonzero Fourier series coefficients for x(t) are Express x(t) in the form a1 = a_ 1 = 2,a3 = a*_ 3 = 4j. x(t) = L Ak cos(wkt + cPk).

A discrete-time periodic signal x[n] is real valued and has a fundamental period N = 5. The nonzero Fourier series coefficients for x[n] are Express x[n] in the form x[n] = Ao + ~ Ak sin(wkn + cfJk).

For the continuous-time periodic signal (27T ) . (57T ) x(t) = 2 +cos 3 t + 4sm 3 t , 251 determine the fundamental frequency w0 and the Fourier series coefficients ak such that x(t) = ~ akeJkwot k~-x

Use the Fourier series analysis equation (3.39) to calculate the coefficients ak for the continuous-time periodic signal x(t) = { 1.5, -1.5, with fundamental frequency w0 = 1T. Ost

Let x1 (t) be a continuous-time periodic signal with fundamental frequency w 1 and Fourier coefficients ak. Given that X2(t) = X1(1- t) + X1(t- 1), how is the fundamental frequency w2 of x 2(t) related tow 1? Also, find a relationship between the Fourier series coefficients bk of x 2(t) and the coefficients ak. You may use the properties listed in Table 3.1.

Consider three continuous-time periodic signals whose Fourier series representations are as follows: Use Fourier series properties to help answer the following questions: (a) Which of the three signals is/are real valued? (b) Which of the three signals is/are even?

Suppose the periodic signal x(t) has fundamental period T and Fourier coefficients ak. In a variety of situations, it is easier to calculate the Fourier series coefficients bk for g(t) = dx(t)/dt, as opposed to calculating ak directly. Given that JT I: x(t)dt = 2, Chap.3 find an expression for ak in terms of bk and T. You may use any of the properties listed in Table 3.1 to help find the expression.

Suppose we are given the following information about a signal x(t): 1. x(t) is real and odd. 2. x(t) is periodic with period T = 2 and has Fourier coefficients ak. 3. ak = 0 for lkl > I. 4. Ho2 ix(t)j 2 dt = 1. Specify two different signals that satisfy these conditions.

Use the analysis equation (3.95) to evaluate the numerical values of one period of the Fourier series coefficients of the periodic signal x[n] = L {48[n- 4m] + 88[n- 1 - 4m]}.

Suppose we are given the following information about a signal x[n]: 1. x[ n] is a real and even signal. 2. x[n] has period N = 10 and Fourier coefficients ak. 3. a11 = 5. 9 4. Yo 2:: jx[nJI2 = 50. n=O Show that x[ n] = A cos(Bn + C), and specify numerical values for the constants A, B, and C

Each of the two sequences Xt [n] and x2[n] has a period N = 4, and the corresponding Fourier series coefficients are specified as Xt [n] ~ ah x2[n] ~ bh where 1 ao = a3 = -a, 2 Using the multiplication property in Table 3.1, determine the Fourier series coefficients ck for the signal g[n] = Xt [n]x2[n].

Consider a continuous-time LTI system whose frequency response is I x sin(4w) H(jw) = -x h(t)e-jwtdt = w If the input to this system is a periodic signal x(t) = { ~ l, O:s:t<4 4:St<8 with period T = 8, determine the corresponding system output y(t).

When the impulse train x[n] = L o[n - 4k] k=-X is the input to a particular LTI system with frequency response H(eiw), the output of the system is found to be (57T 7T) y[n] = cos 2 n + 4 . Determine the values of H(eikrr/2 ) fork = 0, 1, 2, and 3.

Consider a continuous-time ideallowpass filterS whose frequency response is H(jw) = { 1, 0, lwl :s: 100 lwl > 100 When the input to this filter is a signal x(t) with fundamental period T = 1r/6 and Fourier series coefficients ak. it is found that s x(t) ~ y(t) = x(t). For what values of k is it guaranteed that ak = 0?

Determine the output of the filter shown in Figure P3.16 for the following periodic inputs: (a) x 1[n] = (-1)" (b) x2[n] = 1 + sin(3; n + 'i-) (c) x3[n] = ~~=-oc(~r- ku[n- 4k]

Consider three continuous-time systems S1, S2, and S3 whose responses to a complex exponential input ei51 are specified as sl : ej5t --7 tej5t, S2 : ej5t ----7 ejS(t-1), S3 : ei51 ----7 cos(St). For each system, determine whether the given information is sufficient to conclude that the system is definitely not LTI.

Consider three discrete-time systems S1, S2, and S3 whose respective responses to a complex exponential input ei""12 are specified as sl : ej7rnl2 ----7 ejmz/2u[n], s2 : ej7rn/2 ----7 ej37rnl2, s3 : ej7rn/2 --7 2ej57rn/2. For each system, determine whether the given information is sufficient to conclude that the system is definitely not LTI.

Consider a causal LTI system implemented as the RL circuit shown in Figure P3.19. A current source produces an input current x(t), and the system output is considered to be the current y(t) flowing through the inductor. 1f1 Figure P3. 19 (a) Find the differential equation relating x(t) and y(t). (b) Determine the frequency response of this system by considering the output of the system to inputs of the form x(t) = eiwt. (c) Determine the output y(t) if x(t) = cos(t).

A continuous-time periodic signal x(t) is real valued and has a fundamental period T = 8. The nonzero Fourier series coefficients for x(t) are specified as Express x(t) in the form a,= a*_ 1 = j,as =a-s= 2. x(t) = L Ak cos(wkt + cf>k).

Determine the Fourier series representations for the following signals: (a) Each x(t) illustrated in Figure P3.22(a)-(f). (b) x(t) periodic with period 2 and x(t) = e -r for - 1 < t < 1 (c) x(t) periodic with period 4 and x(t) = { sin 7rt, 0,

In each of the following, we specify the Fourier series coefficients of a continuoustime signal that is periodic with period 4. Determine the signal x(t) in each case. { 0 k = 0 (a) ak = ( ')k sin hr/4 otherwise J k1r ' (b) ak = (-l)ksi;~;/8, ao = /6 ( ) - { jk, lkl < 3 c ak - 0, otherwise (d) ak = ' { 1 k even 2, k odd

Let x(t) = { i-t, be a periodic signal with fundamental period T = 2 and Fourier coefficients ak. (a) Determine the value of ao. (b) Determine the Fourier series representation of dx(t)ldt. (c) Use the result of part (b) and the differentiation property of the continuous-tim{ Fourier series to help determine the Fourier series coefficients of x(t)

Consider the following three continuous-time signals with a fundamental period of T = 112: x(t) = cos( 41Tt), y(t) = sin(47Tt), z(t) = x(t)y(t). (a) Determine the Fourier series coefficients of x(t). (b) Determine the Fourier series coefficients of y(t). (c) Use the results of parts (a) and (b), along with the multiplication property of the continuous-time Fourier series, to determine the Fourier series coefficients of z(t) = x(t)y(t). (d) Determine the Fourier series coefficients of z(t) through direct expansion of z(t) in trigonometric form, and compare your result with that of part (c).

Let x(t) be a periodic signal whose Fourier series coefficients are k = 0 otherwise Use Fourier series properties to answer the following questions: (a) Is x(t) real? (b) Is x(t) even? (c) Is dx(t)ldt even?

A discrete-time periodic signal x[n] is real valued and has a fundamental period N = 5. The nonzero Fourier series coefficients for x[n] are \(a_{0}=2, a_{2}=a_{-2}^{*}=2 e^{j \pi / 6}, \quad a_{4}=a_{-4}^{*}=e^{j \frac{\pi}{3}} .\) Express x[n] in the form \(x[n]=A_{0}+\sum_{k=1}^{\infty} A_{k} \sin \left(\omega_{k} n+\phi_{k}\right)\)

Determine the Fourier series coefficients for each of the following discrete-time periodic signals. Plot the magnitude and phase of each set of coefficients ak (a) Each x[n] depicted in Figure P3.28(a)-(c) (b) x[n] = sin(27Tn/3)cos(7Tn/2) (c) x[n] periodic with period 4 and x[n] = 1 - sin :n for 0 :5 n :5 3 (d) x[n] periodic with period 12 and x n [ ] = - 1 sm 1rn f 0 11 4 or :5 n :5

In each of the following, we specify the Fourier series coefficients of a signal that is periodic with period 8. Determine the signal x[n] in each case. (a) ak = cos(k;) + sin( 3!7T) (b) ak = { ~~n(k:;7), ~:; ~ 6 (c) ak as in Figure P3.29(a) (d) ak as in Figure P3.29(b)

Let x[n] = { ~: be a periodic signal with fundamental period N = 10 and Fourier series coefficients ak. Also, let g[n] = x[n] - x[n - 1]. (a) Show that g[n] has a fundamental period of 10. (b) Determine the Fourier series coefficients of g[n]. (c) Using the Fourier series coefficients of g[n] and the First-Difference property in Table 3.2, determine ak for k -:/= 0.

Consider the signal x[ n] depicted in Figure P3.32. This signal is periodic with period N = 4. The signal can be expressed in terms of a discrete-time Fourier series as 3 x[n] = L akeik(27T/4)n. (P3.32-1) k=O x[n] f12 1-8 1-4 _Jo 14 18 112 116 n Figure P3.32 As mentioned in the text, one way to determine the Fourier series coefficients is to treat eq. (P3.32-1) as a set of four linear equations (for n = 0, 1, 2, 3) in four unknowns (ao, a1, a2, and a3). (a) Write out these four equations explicitly, and solve them directly using any standard technique for solving four equations in four unknowns. (Be sure first to reduce the foregoing complex exponentials to the simplest form.) (b) Check your answer by calculating the ak directly, using the discrete-time Fourier series analysis equation 3 ak = ~ L x[n]e- jk(21T/4)n.

Consider a causal continuous-time LTI system whose input x(t) and output y(t) are related by the following differential equation: d dty(t) + 4y(t) = x(t). Find the Fourier series representation of the output y(t) for each of the following inputs: (a) x(t) = cos 27rt (b) x(t) = sin 47rt + cos( 67rt + 7r/4)

Consider a continuous-time LTI system with impulse response l h(t) = e-4lrl. Find the Fourier series representation of the output y(t) for each of the following inputs: (a) x(t) = L::_xo(t - n) (b) x(t) = L:: _x(- l)no(t - n) (c) x(t) is the periodic wave depicted in Figure P3.34.

Consider a continuous-time LTI system S whose frequency response is H(jw) = { I, 0, lwl 2: 250 otherwise When the input to this system is a signal x(t) with fundamental period T = 7r/7 and Fourier series coefficients ah it is found that the output y(t) is identical to x(t). For what values of k is it guaranteed that ak = O?

Consider a causal discrete-time LTI system whose input x[n] and output y[n] are related by the following difference equation: I y[n] - 4y[n - I] = x[nl Find the Fourier series representation of the output y[n] for each of the following inputs: (a) x[n] = sinc3; n) (b) x[n] = cos(*n) + 2cos(n)

Consider a discrete-time LTI system with impulse response ( l ~nl h[n] = 2) Find the Fourier series representation of the output y[n] for each of the following inputs: (a) x[n] = L~= -xD[n - 4k] (b) x[ n] is periodic with period 6 and x[n] = { b: n = 0, :tl n = :::2, :::3

Consider a discrete-time LTI system with impulse response \(h[n]=\left\{\begin{array}{ll}1, & 0 \leq n \leq 2 \\-1, & -2 \leq n \leq-1 \\0, & \text { otherwise }\end{array}\right.\) Given that the input to this system is \(x[n]=\sum_{k=-\infty}^{+\infty} \delta[n-4 k]\) determine the Fourier series coefficients of the output y[n].

Consider a discrete-time LTI system S whose frequency response is H(elw) . [ I = ' 0, lwl :5 ~ < lwl <'1T'' Show that if the input x[n] to this system has a period N = 3, the output y[n] has only one nonzero Fourier series coefficient per period.

Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients ak: 1. ak = ak+2 2. ak = a-k 3. J ~a5 x(t)dt = 1. 4. f x(t)dt = 2. Determine x(t).

Let x(t) be a real-valued signal with fundamental period T and Fourier series coefficients ak (a) Show that ak = a*_k and a0 must be real. (b) Show that if x(t) is even, then its Fourier series coefficients must be real and even. (c) Show that if x(t) is odd, then its Fourier series coefficients are imaginary and odd and ao = 0. (d) Show that the Fourier coefficients of the even part of x(t) are equal to ffi-e{ak}. (e) Show that the Fourier coefficients of the odd part of x(t) are equal to jdm{ak}

(a) A continuous-time periodic signal x(t) with period Tis said to be odd hannonic if, in its Fourier series representation +x x(t) = L akejk(27TIT)t, k= -CG ak = 0 for every non -zero even integer k. (i) Show that if x(t) is odd harmonic, then x(t) = -x~ + ~). (ii) Show that if x(t) satisfies eq. (P3.43-2), then it is odd harmonic. (P3.43-1) (P3.43-2) (b) Suppose that x(t) is an odd-harmonic periodic signal with period 2 such that x(t) = t for 0 < t < 1. Sketch x(t) and find its Fourier series coefficients. (c) Analogously, to an odd-harmonic signal, we could define an even-harmonic signal as a signal for which ak = 0 fork odd in the representation in eq. (P3.43- 1). Could T be the fundamental period for such a signal? Explain your answer. (d) More generally, show that Tis the fundamental period of x(t) in eq. (P3.43-1) if one of two things happens: (1) Either a 1 or a-1 is nonzero; or (2) There are two integers k and l that have no common factors and are such that both ak and a, are nonzero.

Suppose we are given the following information about a signal x(t): 1. x(t) is a real signal. 2. x(t) is periodic with period T = 6 and has Fourier coefficients ak. 3. ak = 0 for k = 0 and k > 2. 4. x(t) = - x(t - 3). 5. \(\frac{1}{6} \int_{-3}^{3}|x(t)|^{2} d t=\frac{1}{2}\) 6. a1 is a positive real number Show that x(t) = A cos(Bt + C), and determine the values of the constants A, B, and C.

Let x(t) be a real periodic signal with Fourier series representation given in the sine-cosine form of eq. (3.32); i.e., \ -5 x(t) = ao + 2 L[Bk cos kwot - ck sin kwot]. k=I (P3.45-1) (a) Find the exponential Fourier series representation of the even and odd parts of x(t); that is, find the coefficients ak and f3k in terms of the coefficients in eq. (P3.45-1) so that +co Sv{x(t)} = L akeJkwot, k= -00 +co Od{x(t)} = L f3kejkwot. k= -00 (b) What is the relationship between a k and a-kin part (a)? What is the relationship between f3 k and f3- k? (c) Suppose that the signals x(t) and z(t) shown in Figure P3.45 have the sine-cosine series representations x(t) = a0 + 2 ti ~[ Bk cos (21Tkt) - . (21Tkt)~ 3- - Ck sm - 3- r z(t) = do+ 2 ti ~[ Ek cos (21Tkt) - . (21Tkt)~ 3- - Fk sm - 3- 'J Sketch the signal ~r r l 1 y(t) = 4(a0 + d0 ) + 2 t:J Bk+ ).Ek cos (27Tkt) - . (27Tkt)11 3 - + F, sm - 3 - j

In this problem, we derive two important properties of the continuous-time Fourier series: the multiplication property and Parseval's relation. Let x(t) and y(t) both be continuous-time periodic signals having period T0 and with Fourier series represen- tations given by +x +x x(r) = 2= akejkw"', y(t) = 2= bkejkw"'. k=-x (P3.46-l) (a) Show that the Fourier series coefficients of the signal +oc z(t) = x(t)y(t) = L ckejkwot k= -00 are given by the discrete convolution +oo ck = 2:: anbk-n n= -oo (b) Use the result of part (a) to compute the Fourier series coefficients of the signals x1 (t), x2(t), and x3(t) depicted in Figure P3.46. (c) Suppose that y(t) in eq. (P3.46-1) equals x*(t). Express the bk in the equation in terms of ak. and use the result of part (a) to prove Parseval's relation for periodic signals-that is,

Consider the signal x(t) = cos 27T't. Since x(t) is periodic with a fundamental period of 1, it is also periodic with a period of N, where N is any positive integer. What are the Fourier series coefficients of x(t) if we regard it as a periodic signal with period 3?

Let x[n] be a periodic sequence with period N and Fourier series representation x[n] = L akejk(27r!N)n_ k= (P3.48-1) The Fourier series coefficients for each of the following signals can be expressed in terms of akin eq. (P3.48-l). Derive the expressions. (a) x[n - no] (b) x[n] - x[n - 1] (c) x[n] - x[n - ~] (assume that N is even) (d) x[n] + x[n + ~] (assume that N is even; note that this signal is periodic with period N/2) (e) x*[-n] (f) (-l)n x[n] (assume that N is even) (g) (-l)n x[n] (assume that N is odd; note that this signal is periodic with period 2N) (h) y[n] = { x[n], n even 0, n odd

Let x[n] be a periodic sequence with period N and Fourier series representation x[n] = L akejk(27TIN)n. k= (P3.49-1) (a) Suppose that N is even and that x[n] in eq. (P3.49-1) satisfies x[n] = - x [ n + ~] for all n. Show that ak = 0 for all even integers k. (b) Suppose that N is divisible by 4. Show that if x[n] = -x[n +~]for all n, then ak = 0 for every value of k that is a multiple of 4. (c) More generally, suppose that N is divisible by an integer M. Show that if (N/M)-1 [ N] ~ x n + r M = 0 for all n, then ak = 0 for every value of k that is a multiple of M.

Let x[n] be a periodic signal with period N = 8 and Fourier series coefficients ak = -ak-4 A signal ( 1+(-l)ll) y[n] = 2 x[n - 1] with period N = 8 is generated. Denoting the Fourier series coefficients of y[n] by bk. find a function J[k] such that

x[n] is a real periodic signal with period N and complex Fourier series coefficients ak. Let the Cartesian form for ak be denoted by where bk and ck are both real. (a) Show that a-k = a'k. What is the relation between bk and b-k? What is the relation between ck and c - k? (b) Suppose that N is even. Show that aN12 is real. (c) Show that x[n] can also be expressed as a trigonometric Fourier series of the form (N-l)/2 (2 k ) (2 k ) x[n] = ao + 2 ~ bk cos : n - ck sin : n if N is odd or as (N-2l 12 (2 k ) (2 k ) x[n] = (ao + aN12(-lt) + 2 ~ bk cos : n - ck sin : n if N is even. (d) Show that ifthe polar form of ak is Akejek, then the Fourier series representation for x[n] can also be written as x[n] = ao + 2 L Akcos _!!._!!_+Ok (N-1)/2 ( 2 k ) k=l N if N is odd or as if N is even. (e) Suppose that x[n] and z[n], as depicted in Figure P3.52, have the sine-cosine series representations x[n] = ao + 2 ti ~1 bk cos (27Tkn) - . (27Tkn)j 7 - - ck sm - 7 - , z[n] =do+ 2 ti ~1 dkcos (27Tkn) - . (27Tkn)j 7 - - fksm - 7 - . Sketch the signal y[n] = a0 - do+ 2 t; 3 I dkcos (27Tkn) - . (27Tkn)j 7 - +Uk - ck)sm - 7 - .

Let x[n] be a real periodic signal with period N and Fourier coefficients ak. (a) Show that if N is even, at least two of the Fourier coefficients within one period of ak are real. (b) Show that if N is odd, at least one of the Fourier coefficients within one period of a k is real.

Consider the function N-1 a[k] = L ej(21TIN)kn. n=O (a) Show that a[k] = N for k = 0, N, 2N, 3N, .... (b) Show that a[k] = 0 whenever k is not an integer multiple of N. (Hint: Use the finite sum formula.) (c) Repeat parts (a) and (b) if a[k] = L ej(21TIN)kn.

Let x[n] be a periodic signal with fundamental period N and Fourier series coefficients ak. In this problem, we derive the time-scaling property [ ] _ { x[ .!!. ], n = 0, m, 2m, X(m) n - m 0, elsewhere listed in Table 3.2. (a) Show that X(m)[n] has period of mN. (b) Show that if x[n] = v[n] + w[n], then (c) Assuming that x[n] = ej27rkon!N for some integer k0, verify that l m-1 X(m)[n] = - L ej27r(ko+lN)nl(mN)_ m l=O That is, one complex exponential in x[n] becomes a linear combination of m complex exponentials in X(m)[n]. (d) Using the results of parts (a), (b), and (c), show that if x[n] has the Fourier coefficients ak> then X(m)[n] must have the Fourier coefficients ~ak.

Let x[n] be a periodic signal with period N and Fourier coefficients ak. (a) Express the Fourier coefficients bk of Jx[nJl2 in terms of ak. (b) If the coefficients ak are real, is it guaranteed that the coefficients bk are also real?

(a) Let N-1 x[n] = L akejk(27r!N)n (P3.57-1) k=O and N-1 y[n] = L bkejk(27r/N)n k=O be periodic signals. Show that N-1 x[n]y[n] = L ckejk(2TrlN)n, k=O where N-1 N-1 ck = L azbk-l = L ak-1b1. l=O l=O (b) Generalize the result of part (a) by showing that ck = L azbk-l = L ak-1b1. l= l= ( c) Use the result of part (b) to find the Fourier series representation of the following signals, where x[n] is given by eq. (P3.57-l). (i) x[n]cos( ~n) (ii) x[n]L:::,:'_00 8[n - rN] (iii) x[n] (L:::,:_ 008 [n - '~])(assume thatNis divisible by 3) (d) Find the Fourier series representation for the signal x[n]y[n], where x[n] = cos(rrn/3) and { 1 lnl :5 3 y[n] = o: 4 :5 lnl :5 6 is periodic with period 12. ( e) Use the result of part (b) to show that L x[n]y[n] = N L a1b-1, n= l= and from this expression, derive Parseval's relation for discrete-time periodic signals.

Let x[n] and y[n] be periodic signals with common period N, and let z[n] = L x[r]y[n - r] r= be their periodic convolution. (a) Show that z[n] is also periodic with period N. (b) Verify that if ah bh and ck are the Fourier coefficients of x[n], y[n], and z[n], respectively, then (c) Let x[n] = sin (3~n) and [n] = { 1, 0 :5 n :5 3 y 0, 4 :5 n :5 7 be two signals that are periodic with period 8. Find the Fourier series representation for the periodic convolution of these signals. (d) Repeat part (c) for the following two periodic signals that also have period 8: x[n] = [ sm 4 ' - n - ' . (3'lTn) 0 < < 3 0, 4 :5 n :5 7 y[n] = Gro :5 n :5 7.

(a) Suppose x[n] is a periodic signal with period N. Show that the Fourier series coefficients of the periodic signal 00 g(t) = L x[k] 8(t - kT) k= -00 are periodic with period N. (b) Suppose that x(t) is a periodic signal with period T and Fourier series coefficients ak with period N. Show that there must exist a periodic sequence g[n] such that 00 x(t) = L g[k] 5(t - kTIN). k= -00 (c) Can a continuous periodic signal have periodic Fourier coefficients?

As we have seen, the techniques of Fourier analysis are of value in examining continuous-time LTI systems because periodic complex exponentials are eigenfunctions for LTI systems. In this problem, we wish to substantiate the following statement: Although some LTI systems may have additional eigenfunctions, the complex exponentials are the only signals that are eigenfunctions of every LTI system. (a) What are the eigenfunctions of the LTI system with unit impulse response \(h(t)=\delta(t)\)? What are the associated eigenvalues? (b) Consider the LTI system with unit impulse response h(t) = \(\delta\)(t - T). Find a signal that is not of the form est, but that is an eigenfunction of the system with eigenvalue 1. Similarly, find the eigenfunctions with eigenvalues 1/2 and 2 that are not complex exponentials. (Hint: You can find impulse trains that meet these requirements.) (c) Consider a stable LTI system with impulse response h(t) that is real and even. Show that cos\(\omega\) t and sin \(\omega\) t are eigenfunctions of this system. (d) ConsidertheLTI system with impulse response h(t) = u(t). Suppose that \(\phi(t)\) is an eigenfunction of this system with eigenvalue \(\pi\). Find the differential equation that \(\phi(t)\) must satisfy, and solve the equation. This result, together with those of parts (a) through (c), should prove the validity of the statement made at the beginning of the problem.

One technique for building a de power supply is to take an ac signal and full-wave rectify it. That is, we put the ac signal x(t) through a system that produces y(t) = lx(t)l as its output. (a) Sketch the input and output waveforms if x(t) = cost. What are the fundamental periods of the input and output? (b) If x(t) = cost, determine the coefficients of the Fourier series for the output y(t). (c) What is the amplitude of the de component of the input signal? What is the amplitude of the de component of the output signal?

Suppose that a continuous-time periodic signal is the input to an LTI system. The signal has a Fourier series representation 00 x(t) = L alklejk(1TI4)r, k= -00 where a is a real number between 0 and 1, and the frequency response of the system is H(jw) = { 1' 0, lwl $ W lwi>W. How large must W be in order for the output of the system to have at least 90% of the average energy per period of x(t)?

As we have seen in this chapter, the concept of an eigenfunction is an extremely important tool in the study ofLTI systems. The same can be said for linear, but timevarying, systems. Specifically, consider such a system with input x(t) and output i y(t). We say that a signal t/J(t) is an eigenfunction of the system if l j ' cf>(t) j ---+ Acf>(t). 1 That is, if x(t) = cf>(t), then y(t) = At/J(t), where the complex constant A is called J the eigenvalue associated with t/J(.t). (a) Suppose that we can represent the input x(t) to our system as a linear combination of eigenfunctions cfJk(t), each of which has a corresponding eigenvalue Ak; that is, x(t) = L ckcfJk(t). k= -00 Express the output y(t) of the system in terms of {ck}, {c/Jk(t)}, and {Ak}. (b) Consider the system characterized by the differential equation () - 2 d2x(t) dx(t) y t - t ----;_jfl + t--;[(. Is this system linear? Is it time invariant? (c) Show that the functions are eigenfunctions of the system in part (b). For each cfJk(t), determine the corresponding eigenvalue Ak. (d) Determine the output of the system if 1 x(t) = lOt- 10 + 3t + 2t4 + 7T.

Two functions u(t) and v(t) are said to be orthogonal over the interval (a,b) if r u(t)v*(t) dt = 0. (P3.65-1) If, in addition, r lu(t)l2 dt = 1 = r lv(t)l2 dt, the functions are said to be normalized and hence are called orthonormal. A set of functions {cfJk(t)} is called an orthogonal (orthonormal) set if each pair of functions in the set is orthogonal (orthonormal). (a) Consider the pairs of signals u(t) and v(t) depicted in Figure P3.65. Determine whether each pair is orthogonal over the interval (0, 4). (b) Are the functions sin mwot and sin nwot orthogonal over the interval (0, T), where T = 27Tiwo? Are they alsd orthonormal? (c) Repeat part (b) for the functions c/Jm(t) and c/Jn(t), where cfJk(t) = Jr [cos kwot + sin kw0t]. (d) Show that the functions cf>t(t) = eikwot are orthogonal over any interval of length T = 27rlwo. Are they orthonormal? (e) Let x(t) be an arbitrary signal, and let X0 (t) and Xe(t) be, respectively, the odd and even parts of x(t). Show that X0 (t) and Xe(t) are orthogonal over the interval ( -T, T) for any T. (f) Show that if {k(t)} is a set of orthogonal signals over the interval (a, b), then the set {(1/ jA~)k(t)}, where Ak = f h lk(t)l2 dt, a is orthonormal. (g) Let {;(t)} be a set of orthonormal signals on the interval (a, b), and consider a signal of the form x(t) = ~ a;;(t), where the a; are complex constants. Show that r lx(t)l 2 dt = ~ la;l 2. l (h) Suppose that 1 (t), ... , N(t) are nonzero only in the time interval 0 ::::: t ::::: T and that they are orthonormal over this time interval. Let L; denote the LTI system with impulse response hJt) = ;(T - t). (P3.65-2) Show that if J(t) is applied to this system, then the output at time T is 1 if i = j and 0 if i =P j. The system with impulse response given by eq. (P3.65-2) was referred to in Problems 2.66 and 2.67 as the matched filter for the signal ;(t).

The purpose of this problem is to show that the representation of an arbitrary periodic signal by a Fourier series or, more generally, as a linear combination of any set of orthogonal functions is computationally efficient and in fact very useful for obtaining good approximations of signals. 12 Specifically, let {Jt)}, i = 0, 1, 2, ... be a set of orthonormal functions on the interval a ::::: t ::::: b, and let x(t) be a given signal. Consider the following approximation of x(t) over the interval a ::::: t ::::: b: +N Xn(t) = ~ a;;(t). (P3.66-1) i~-N Here, the a; are (in general, complex) constants. To measure the deviation between x(t) and the series approximation xN(t), we consider the error eN(t) defined as (P3.66-2) A reasonable and widely used criterion for measuring the quality of the approximation is the energy in the error signal over the interval of interest-that is, the integral 12See Problem 3.65 for the definitions of orthogonal and orthonormal functions. of the square of the magnitude of the error over the interval a ::5 t ::5 b: (P3.66-3) (a) Show that E is minimized by choosing a; = r x(t)c/J;(t)dt. (P3.66-4) [Hint: Use eqs. (P3.66-1)-(P3.66-3) to express E in terms of a;, cl>i(t), and x(t). Then express a; in rectangular coordinates as a; = b; + jc;, and show that the equations aE aE . -b = 0 and - = 0, z = 0, :: 1, ::2, ... , N a i ac; are satisfied by the a; as given by eq. (P3.66-4).] (b) How does the result of part (a) change if and the {cP;(t)} are orthogonal but not orthonormal? (c) Let cPn(t) = ejnwot, and choose any interval of length To = 2nlw0 . Show that the a; that minimize E are as given in eq. (3.50). (d) The set of Walsh functions is an often-used set of orthonormal functions. (See Problem 2.66.) The set of five Walsh functions, cPo(t), cP1 (t), ... , cP4(t), is illustrated in Figure P3.66, where we have scaled time so that the cP;(t) are nonzero and orthonormal over the interval 0 ::::; t ::5 1. Let x(t) = sin 7Tt. Find the approximation of x(t) of the form 4 x(t) = ,L a;cP;(t) i=O such that is minimized. (e) Show that XN(t) in eq. (P3.66-1) and eN(t) in eq. (P3.66-2) are orthogonal if the ai are chosen as in eq. (P3.66-4). The results of parts (a) and (b) are extremely important in that they show that each coefficient ai is independent of all the other aj's, i :;i: j. Thtis, if we add more terms to the approximation [e.g., if we compute the approximation .XN+t(t)], the coefficients of cPi(t), i = 1, .. . ,N, that were previously deterinined will not change. In contrast to this, consider another type of se- ries expansion, the polynomial Taylor series. The infinite Taylor series for et is et = 1 + t + t2/2! + ... , but as we shall show, when we consider a finite polynomial series and the error criterion of eq. (P3.66-3), we get a very different result. Specifically, let cf>o(t) = 1, (g) Consider an approximation of x(t) = e1 over the interval 0 s t s 1 of the form io(t) = aocf>o(t). Find the value of a0 that minimizes the energy in the error signal over the interval. (h) We now wish to approximate e1 by a Taylor series using two terms-i.e., i 1 (t) = a0 + a1 t. Find the optimum values for ao and a1. [Hint: Compute E in terms of a0 and a 1, and then solve the simultaneous equations aE = 0 and aao Note that your answer for a0 has changed from its value in part (g), where there was only one term in the series. Further, as you increase the number of terms in the series, that coefficient and all others will continue to change. We can thus see the advantage to be gained in expanding a function using orthogonal terms.]

As we discussed in the text, the origins of Fourier analysis can be found in problems of mathematical physics. In particular, the work of Fourier was motivated by his investigation of heat diffusion. In this problem, we illustrate how the Fourier series enter into the investigation. 13 Consider the problem of determining the temperature at a given depth beneath the surface of the earth as a function of time, where we assume that the temperature at the surface is a given function of time T(t) that is periodic with period I. (The unit of time is one year.) Let T(x, t) denote the temperature at a depth x below the surface at time t. This function obeys the heat diffusion equation with auxiliary condition aT(x, t) at ~k T(x, t) 2 ax2 T(O, t) = T(t). (P3.67-l) (P3.67-2) Here, k is the heat diffusion constant for the earth (k > 0). Suppose that we expand T(t) in a Fourier series: +x T(t) = L anejnZm. (P3.67-3) n= -:o Similarly, let us expand T(x, t) at any given depth x in a Fourier series in t. We obtain +oo T(x, t) = L bn(x)ejnZm, (P3.67- 4) n= -oc where the Fourier coefficients bn(x} depend upon the depth x. 13The problem has been adapted from A. Sommerfeld, Partial Differential Equations in Physics (Nt York: Academic Press, 1949), pp 68-71. (a) Use eqs. (P3.67-1)-(P3.67-4) to show that h11 (x) satisfies the differential equation with auxiliary condition 47T"jn ---yzr-b"(x) (P3.67-5a) (P3.67-5b) Since eq. (P3.67-5a) is a second-order equation, we need a second auxiliary condition. We argue on physical grounds that, far below the earth's surface, the variations in temperature due to surface fluctuations should disappear. That is, lim T(x, t) = a constant. x~x (b) Show that the solution of eqs. (P3.67-5) is bn(x) = [ an exp[- J27Tinl(l + j)xl k], an exp[- J27Tinl(l - j)xlk], n 2:: 0 n s 0 (P3.67-5c) (c) Thus, the temperature oscillations at depth x are damped and phase-shifted versions of the temperature oscillations at the surface. To see this more clearly, let T(t) = ao + a, sin 2m (so that a0 represents the mean yearly temperature). Sketch T(t) and T(x, t) over a one-year period for a0 = 2, and a 1 = 1. Note that at this depth not only are the temperature oscillations significantly damped, but the phase shift is such that it is warmest in winter and coldest in summer. This is exactly the reason why vegetable cellars are constructed!

Consider the closed contour shown in Figure P3.68. As illustrated, we can view this curve as being traced out by the tip of a rotating vector of varying length. Let r(O) denote the length ofthe vector as a function of the angle (J. Then r( (J) is periodic in (J with period 27T and thus has a Fourier series representation. Let {ad denote the Fourier coefficients of r(O). (a) Consider now the projection x( 0) of the vector r( 0) onto the x-axis, as indicated in the figure. Determine the Fourier coefficients for x(O) in terms of the ak's. (b) Consider the sequence of coefficients bk = akejk7rt4. Sketch the figure in the plane that corresponds to this set of coefficients. (c) Repeat part (b) with (d) Sketch figures in the plane such that r(O) is not constant, but does have each of the following properties: (i) r(O) is even. (ii) The fundamental period of r(O) is 7T. (iii) The fundamental period of r(O) is 7T/2.

In this problem, we consider the discrete-time counterpart of the concepts introduced in Problems 3.65 and 3.66. In analogy with the continuous-time case, two discretetime signals ;[n], where the cf>i[n] are orthogonal over the interval (N1, N 2 ), then N? M ::;2 lx[n]i2 = L lail 2 Ai. i= I (d) Let cf>i[n], i = 0, 1, ... , M, be a set of orthogonal functions over the interval (N1, N2), and let x[n] be a given signal. Suppose that we wish to approximate x[n] as a linear combination of the cf>i[n]; that is, M i[n] = L aicf>i[n], i=O where the ai are constant coefficients. Let e[n] = x[n] - i[n], and show that if we wish to minimize N, E = ::;2 le[n]i2, n=N1 then the ai are given by (P3.69-2) [Hint: As in Problem 3.66, express E in terms of ai, cf>i[n], Ai, and x[n], write ai = bi + }ci, and show that the equations aE - = 0 and abi are satisfied by the ai given by eq. (P3.69-2). Note that applying this result when the cf>i[n] are as in part (b) yields eq. (3.95) for ak.] (e) Apply the result of part (d) when the cf>i[n] are as in part (a) to determine the coefficients ai in terms of x[n].

Consider the mechanical system shown in Figure P3.71. The differential equation relating velocity v(t) and the input force f(t) is given by f(t) v(t) ~ B Bv(t) + K I v(t) dt = j(t). (a) Assuming that the output is f,(t), the compressive force acting on the spring, write the differential equation relating f,(t) and f(t). Obtain the frequency response of the system, and argue that it approximates that of a lowpass filter. (b) Assuming that the output is /J(t), the compressive force acting on the dashpot, write the differential equation relating /J(t) and f(t). Obtain the frequency response of the system, and argue that it approximates that of a highpass filter.