EECE 233: Signals and Systems

Final Exam: Part 2 (Dec. 2008)

Time allowed: 4 hours 20 minutes

Student #:

Name:

본 시험 문제지는 답안지와 함께 제출하세요.

Problem 1: / 40

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)TrueFalse

(k) (l) (m) (n) (o) (p) (q) (r) (s) (t)TrueFalse

(u) (v) (w) (x) (y) (z) (aa) (ab) (ac) (ad)TrueFalse

(ae) (af) (ag) (ah) (ai) (aj) (ak) (al) (am) (an)TrueFalse

Problem 2: / 20

Problem 3: / 15

Problem 4: / 15

Problem 5: / 15

Problem 6: / 15

Problem 7: / 20

Problem 8: / 15

Problem 9: / 25

Problem 10: / 30

Total: / 210

Terms, definitions, etc.:

• You don’t need to prove a result already proved in the lecture notes, unless the questionis to prove the very result.

• j ,√−1.

• The arguments of sinusoidal functions are all in radian.

• CT: continuous-time. DT: discrete-time

• δ(t), δ[n]: unit impulse function. u[n], u(t): unit step function

• BIBO: bounded-input bounded-output

• R: the set of all real numbers. C: the set of all complex numbers

• ω and ωi are radian frequencies unless otherwise specified.

• FS: Fourier Series, FT: Fourier transform

• CTFS: Continuous-Time Fourier Series, DTFS: Discrete-Time Fourier Series

• CTFT: Continuous-Time Fourier Transform, DTFT: Discrete-Time Fourier Transform

• In this exam, the CTFT of x(t) is defined as

X(jω) =∫ ∞

−∞x(t)e−jωtdt

or X(j2πf).

• In this exam, the DTFT of x[n] is defined as

X(ejω) =∞∑

n=−∞x[n]e−jωn

or X(ej2πf ).

• “Laplace transform” and “z-transform” means bilateral transforms, unless otherwise spec-ified.

• LT: Laplace transform, ULT: unilateral Laplace transform

• ZT: z-transform, UZT: unilateral z-transform

• ROC: region of convergence

• In this exam, it is defined that the LT of a CT signal does not exist iff the ROC is empty.

• LCCDE: linear constant coefficient differential equation for CT signals and systems, linearconstant coefficient difference equation for DT signals and systems

2

Problem 1. (40 points: 1 point each) Say True or False to the following statements. Mark youranswers in the table on the cover page. (Correct answer= 1, No answer= 0, Incorrectanswer= −1)

(a) A digital video signal can be modeled as a one-dimensional signal x : Z→ R3.

(b) Any square matrix can be written as the sum of a Hermitian matrix and a Hermitiananti-symmetric matrix.

(c) A linear combination of two CT periodic signals is always periodic.

(d) The DTFT of the product of two DT signals is the periodic convolution of the DTFTs ofthe signals.

(e) If x(t) is absolutely integrable, then its CTFT always exists.

(f) The inverse Fourier transform of the frequency response of a DT ideal LPF is a sinc pulsein the time domain.

(g) The DTFT of a DT periodic signal is always a sum of countably infinite Dirac deltafunctions.

(h) For many digital filtering applications, it is not necessary that the unit sample responseof the filter be zero for n < 0 if the processing is not to be carried out in real time.

(i) If the DTFT of x[n] is X(ejω), then the DTFT of (−1)nx[n] is X(ej(ω−π)).

(j) If the frequency ω1 is 10 dB higher than the frequency ω2, then ω1 = 10ω2.

(k) The output of any RLC circuit in initial rest to a sinusoidal input is a sinusoidal signalwith the same frequency as the input signal and, possibly, a phase shift.

(l) When the system function H(s) of an LTI system is rational and has poles at s = pi,for i = 1, 2, ...,M , a necessary condition for this system to be causal and BIBO stable is<(pi) < 0, ∀i.

(m) When L{x(t)} = X(s), L{x(−t)∗} = X(1/s∗)∗.

(n) If a CT signal is right-sided, then the ROC of its LT contains the unit circle of s-plane.

(o) If a CT signal is two-sided, then the ROC of its LT is a strip in the s-plane.

(p) The LT of a causal CT signal is always a rational function of s.

(q) The LT of a CT signal as a function of s does not converge on the poles.

(r) The LT of the sum of two CT signals has the ROC that equals the intersection of theROCs of the LTs of the two signals.

(s) Multiplication of esT to the LT of a signal leads to time-delay −T in t-domain.

(t) A system whose input-output relation is described by an LCCDE is a linear time-invariantsystem.

3

(u) If an LTI system has its input-output relation described by an LCCDE, then its impulseresponse is always uniquely determined.

(v) An LTI system whose input-output relation is described by an LCCDE and is in the initialrest is always causal.

(w) If the system function H(s) of a CT LTI system is rational, then the stability of thesystem implies that the ROC is to the right of the rightmost pole.

(x) Two different causal CT signals x1(t) and x2(t) can have the same unilateral LT.

(y) When a CT linear system is described by an LCCDE, the zero-input response is alwayszero.

(z) If a DT LTI system whose input-output relation is described by an LCCDE has a right-sided impulse response, then the system is always causal.

(aa) If a DT signal has a rational ZT, then all zeros of the ZT exists inside the ROC.

(ab) The ZT of −2nu[−n− 1] is 1/(1− 2z−1) with ROC {z ∈ C : |z| < 2} excluding z = 0.

(ac) If a causal DT LTI system has a rational ZT, then the ROC is outside the outermost poleincluding z = ∞.

(ad) When s ∈ C and t ∈ R, (est)∗ = es∗t.

(ae) The system function H(z) of a DT LTI system with real-valued impulse response alwayssatisfies H(z) =

(H(z∗)

)∗.(af) When a DT LTI system has impulse response h1[n] = (0.5)nu[n], its inverse system has

impulse response h2[n] = δ[n]− 0.5δ[n− 1].

(ag) The Nyquist sampling theorem states that a finite-energy band-limited signal sampled atfaster than or equal to the signal bandwidth can be perfectly reconstructed.

(ah) For the perfect reconstruction of a CT signal from its samples taken at faster thanNyquist’s rate, time-shifted sinc pulses can be used.

(ai) If x[n] has the DTFT X(ejω), then y(t) =∑∞

n=−∞ x[n]p(t− nT ) has the CTFT Y (jω) =P (jω)X(ejω).

(aj) The DC term added in AM must be greater than or equal to the maximum value of themessage signal in order to make the demodulation be performed without phase recovery.

(ak) The reason why the DSB modulation wastes half of the bandwidth is because the CTFTof a real-valued signal is conjugate symmetrical.

(al) From the product of a CT message signal with bandwidth B [Hz] and a CT periodicsignal with period T > 1/(2B) [sec] and a nonzero DC term, we can perfectly recover themessage signal.

(am) N× up-sampling is a DT-to-DT conversion that obtains a DT signal by inserting (N − 1)samples between the samples from the original DT signal.

(an) N× down-sampling is a DT-to-DT conversion that obtains a DT signal by dropping(N − 1) samples in every N samples from the original DT signal.

4

Problem 2. (20 points) Suppose that a linear time-varying system has the impulse response

h(t, τ) = a0 +∞∑

k=1

{ak cos

(2πkt

T

)cos

(2πkτ

T

)+ bk sin

(2πkt

T

)sin

(2πkτ

T

)}

defined for t, τ ∈ [0, T ], and that the admissible input signals are time-limited to t ∈ [0, T ].Answer the following questions.

(a) (5 points) Find the Fourier series representation of

z(t) =∞∑

k=−∞w(t− kT )

when w(t) is given by

w(t) ,{ t

T, for 0 ≤ t ≤ T

0, elsewhere.(1)

(b) (5 points) Find the output y(t) when the input x(t) is w(t) defined in Eq. (1).

(c) (10 points) When the output y(t) is given by

y(t) =

( ∞∑

k=−∞cke

j 2πktT

)(u(t)− u(t− T )

),

find the input x(t) in terms of (ak)k, (bk)k, and (ck)k. (Assume ak 6= 0 and bk 6= 0 for allk.)

Problem 3. (15 points) When x(t) has the CTFT X(jω) and X(jω) = 0 for |ω| ≥ πT , answer

the following questions.

(a) (5 points) Find ∫ πT

− πT

X(jω)ejω(t−nT )dω (2)

in terms of x(t).

(b) (5 points) Find the inverse CTFT of

e−jω(t−nT )(u

(ω +

π

T

)− u

(ω − π

T

)).

(c) (5 points) Applying Plancherel’s theorem to Eq. (2), find g(t) such that

x(nT ) =∫ ∞

−∞x(t)g(t− nT )∗dt.

(Hint. Use the result in (b).)

5

Problem 4. (15 points) Define x(t) as

x(t) =1√

2πσ2e−

t2

2σ2

where σ > 0. Answer the following questions.

(a) (5 points) Show that ∫ ∞

−∞x(t)dt = 1.

(Hint. Consider∫∞−∞

∫∞−∞ x(t)x(s)dtds in the polar coordinate.)

(b) (5 points) Find the CTFT of x(t). (Hint. Assume∫ ∞

−∞

1√2πσ2

e−(t+jωσ2)2

2σ2 dt = 1.

)

(c) (5 points) Show that

x(t) ∗ x(t) ∗ · · · ∗ x(t)︸ ︷︷ ︸N − 1 convolution of Nx(t)’s

=1√N

x

(t√N

).

Problem 5. (15 points) When a DT LTI system has the frequency response H(ejω) = jωT for

|ω| < π, answer the following questions.

(a) (5 points) Sketch the magnitude and the phase responses for |ω| < 4π.

(b) (5 points) Find the impulse response h[n] of this system, and sketch it.

(c) (5 points) Discuss how to implement a CT differentiator whose admissible inputs arebandlimited CT signals with bandwidth ωc [rad/sec] by using the above DT LTI system.

6

Problem 6-A. (15 points: Choose Problem 6-A or Problem 6-B, and solve it. If yousolve both of them, only Problem 6-A will be graded.) When the input-output relationof a causal CT system is described by

d2y(t)dt2

+ 3dy(t)dt

+ 2y(t) = 2d2x(t)

dt2+ 4

dx(t)dt

− 6x(t)

and it satisfies the initial rest condition, answer the following questions.

(a) (5 points) Find the response of this system to the impulse input x(t) = δ(t).

(b) (5 points) Draw the block diagram of this system using differentiators, adders, and con-stant multipliers.

(c) (5 points) Draw the block diagram of this system using integrators, adders, and constantmultipliers.

Problem 6-B. (15 points: Choose Problem 6-A or Problem 6-B, and solve it. If yousolve both of them, only Problem 6-A will be graded.) When the input-output relationof aCT system is described by

d2y(t)dt2

+ 3dy(t)dt

+ 2y(t) = x(t)

with y(0−) = 1, y′(0−) = −3, answer the following questions.

(a) (5 points) Find the response y(t) of this system for t ≥ 0− to the input x(t) = 0.

(b) (5 points) Find the response y(t) of this system for t ≥ 0− to the input x(t) = δ(t).

(c) (5 points) Find the response y(t) of this system for t ≥ 0− to the input x(t) = u(t).

7

Problem 9-A. (25 points: Choose Problem 9-A or Problem 9-B, and solve it. Ifyou solve both of them, only Problem 9-A will be graded.) Suppose that a CT signalxc(t) that is bandlimited to |ω| < π

T [rad/sec] is sampled at the rate of 1/T [Hz] to generatexd[n] , xc(nT ). When the CTFT of xc(t) is Xc(jω), answer the following questions.

(a) (5 points) Find the DTFT Y (ejω), in terms of Xc(jω), of the M times upsampled versionyd[n] defined as

yd[n] ,{

xd

[nM

], for n = mM,

0, elsewhere.

(b) (10 points) Suppose that hd[n] is an ideal LPF with cutoff frequency π/M [rad/sec] andfilters the DT signal yd[n] defined in (a). When the DT signal zd[n] , yd[n] ∗ hd[n] isinterpolated by using a first-order hold as

zc(t) ,∞∑

n=−∞zd[n]p

(t− nT

M

),

where

p(t) ,{

1− M |t|T , |t| < T/M,

0, elsewhere,

find the CTFT Zc(jω) of zc(t) in terms of Xc(jω).

(c) (10 points) When the CT signal zc(t) defined in (b) is filtered as zc(t)∗hc(t) by a frequency-selective LPF hc(t) with cutoff frequency πM/T [rad/sec], find the necessary and sufficientcondition on the CTFT Hc(jω) of hc(t) for zc(t) ∗ hc(t) to be equal to xc(t) for any CTsignal xc(t) bandlimited to |ω| < π

T .

9

Problem 9-B. (25 points: Choose Problem 9-A or Problem 9-B, and solve it. If yousolve both of them, only Problem 9-A will be graded.) In a communication system, adata sequence (b[n])n is linearly modulated, transmitted over a channel, and received as

y(t) =∞∑

m=−∞b[m]p(t−mT )

where p(t) is a known waveform. The receiver first matched-filters the received signal as

z(t) , y(t) ∗ p(−t)∗

and samples z(t) at every t = nT, ∀n, to obtain

z[n] , z(nT ).

The objective is to design a DT filter h[n] such that

w[n] , z[n] ∗ h[n] =∞∑

m=−∞b[m]δ[n−m].

Define p̃(t) , p(t) ∗ p(−t)∗ and q[n] , p̃(nT ). Assuming q[n] = 0 for |n| > N with a positiveinteger N , answer the following questions.

(a) (5 points) Find z(t) in terms of (b[n])n and q(t), and also find z[n] in terms of (b[n])n andq[n].

(b) (5 points) Find the DTFT Q(ejω) of q[n] in terms of the CTFT P (jω) of p(t).

(c) (5 points) Show that q[n] = q[−n]∗,∀n, and find the ZT H(z) of h[n] in terms of(q[n])N

n=−N .

(d) (5 points) Show that if H(z) has a pole at rejθ for some r > 0 and θ ∈ (0, 2π] then it alsohas pole at (1/r)ejθ. (Hint. Use the results in (c).)

(e) (5 points) Show that if h[n] is stable then h[n] cannot be causal. (Hint. Use the result in(d).)

10

Problem 10. (30 points) A continuous real-valued signal x(t) satisfies

x(t) =K∑

k=−K

akej 2πk

T0t,∀t,

where T0 > 0. We multiply an impulse train

p(t) =∞∑

n=−∞δ(t− nT1)

to x(t) to obtain q(t), where1T1

<1T0

.

Then, q(t) is lowpass filtered to generate r(t) by an ideal LPF with cutoff frequency ωc thatsatisfies

2π

(K

T0− K

T1

)< ωc < 2π

(−K

T0+

K + 1T1

).

Answer the following questions.

(a) (5 points) Find the CTFS of p(t).

(b) (5 points) Find the CTFT of p(t).

(c) (20 points) Find r(t) in terms of x(t).

11