![PSDC ENGINEERING DIPLOMA ASSESSMENT
Signals And Systems ¨C Mid-Term Test 2 (B16 CE/B17 EEA&EEB)
7th January 2009, Wednesday 2:00pm ¨C 5:00pm (3 hours)
Question 1 (Signals and Systems)
(a) Determine whether the following signals are periodic or aperiodic.
If it is a periodic signal, determine its power.
(i) x[n] = 2cos(?¦Ðn) + sin(?¦Ðn)
(ii) x(t) = Odd{cos(2¦Ðt)}
(b) If x(t) = (t2 ¨C 1) [u(t) ¨C u(t¨C4)],
(i) Sketch x(t)
(ii) Sketch x(t ¨C 2)
(iii) Sketch x(¨C3t ¨C 2)
(iv) Sketch Even{x(t ¨C 2)}
(c) Determine the following systems are casual, linear and time-invariant.
(i) y[n] = 3x2[¨C2n]
(ii) y(t) = cos (t+ ? ¦Ð) x(t)
Question 2 (Convolution)
(a) Evaluate the convolution sum for y[n] = x[n]?h[n], where x[n] and h[n] are
shown in Fig. Q2 (a).
Fig. Q2 (a)
(b) Define the convolution of two signals x(t) and h(t).
Then, compute the convolution integral for y(t) = x(t)?h(t) of the following
signals:
h(t) = e¨C3tu(t) and x(t) = e¨C3tu(t¨C1).
1](https://image.slidesharecdn.com/snsmid-term-test2solution-110406043236-phpapp01/85/Sns-mid-term-test2-solution-1-320.jpg)
![PSDC ENGINEERING DIPLOMA ASSESSMENT
Question 3 (Laplace Transform)
(a) Define Bilateral (two-sided) Laplace Transform.
(b) Consider the signal, x(t) = e¨C3tu(t). Find the Laplace transform of x(t) with
the associated region of convergence (ROC).
(c) Consider a continuous-time LTI system described by
d 2 y( t ) dy( t ) dx( t )
2
?3 ? 2 y( t ) = ?3 + x( t )
dt dt dt
Using the shifting property, find the unit impulse response h(t).
Question 4 (z-Transform)
(a) Define Bilateral (two-sided) z-Transform.
(b) Consider the signal, x[k] = aku[k]. Find the z-transform of x[k] with the
associated region of convergence (ROC).
(c) Consider a discrete-time LTI system described by
y[k] ¨C y[k¨C1] ¨C 2 y[k¨C2] = x[k] + 2 x[k¨C1] + 2 x[k¨C2]
Using the shifting property, find the unit impulse response h[k].
"Learning without thought is useless; thought without learning is dangerous."
ѧ¶ø²»Ë¼ÔòØè,˼¶ø²»Ñ§Ôò´ù.
2](https://image.slidesharecdn.com/snsmid-term-test2solution-110406043236-phpapp01/85/Sns-mid-term-test2-solution-2-320.jpg)
Sns mid term-test2-solution
- 1. PSDC ENGINEERING DIPLOMA ASSESSMENT Signals And Systems ¨C Mid-Term Test 2 (B16 CE/B17 EEA&EEB) 7th January 2009, Wednesday 2:00pm ¨C 5:00pm (3 hours) Question 1 (Signals and Systems) (a) Determine whether the following signals are periodic or aperiodic. If it is a periodic signal, determine its power. (i) x[n] = 2cos(?¦Ðn) + sin(?¦Ðn) (ii) x(t) = Odd{cos(2¦Ðt)} (b) If x(t) = (t2 ¨C 1) [u(t) ¨C u(t¨C4)], (i) Sketch x(t) (ii) Sketch x(t ¨C 2) (iii) Sketch x(¨C3t ¨C 2) (iv) Sketch Even{x(t ¨C 2)} (c) Determine the following systems are casual, linear and time-invariant. (i) y[n] = 3x2[¨C2n] (ii) y(t) = cos (t+ ? ¦Ð) x(t) Question 2 (Convolution) (a) Evaluate the convolution sum for y[n] = x[n]?h[n], where x[n] and h[n] are shown in Fig. Q2 (a). Fig. Q2 (a) (b) Define the convolution of two signals x(t) and h(t). Then, compute the convolution integral for y(t) = x(t)?h(t) of the following signals: h(t) = e¨C3tu(t) and x(t) = e¨C3tu(t¨C1). 1
- 2. PSDC ENGINEERING DIPLOMA ASSESSMENT Question 3 (Laplace Transform) (a) Define Bilateral (two-sided) Laplace Transform. (b) Consider the signal, x(t) = e¨C3tu(t). Find the Laplace transform of x(t) with the associated region of convergence (ROC). (c) Consider a continuous-time LTI system described by d 2 y( t ) dy( t ) dx( t ) 2 ?3 ? 2 y( t ) = ?3 + x( t ) dt dt dt Using the shifting property, find the unit impulse response h(t). Question 4 (z-Transform) (a) Define Bilateral (two-sided) z-Transform. (b) Consider the signal, x[k] = aku[k]. Find the z-transform of x[k] with the associated region of convergence (ROC). (c) Consider a discrete-time LTI system described by y[k] ¨C y[k¨C1] ¨C 2 y[k¨C2] = x[k] + 2 x[k¨C1] + 2 x[k¨C2] Using the shifting property, find the unit impulse response h[k]. "Learning without thought is useless; thought without learning is dangerous." ѧ¶ø²»Ë¼ÔòØè,˼¶ø²»Ñ§Ôò´ù. 2