Signal_and_System_A

Signal and System A
Attention: Y ou must answer the following questions in English.
1．Determine whether the following signals are periodic or not, if it is periodic signal,
determine its fundamental period (10 points)
(1). t t x 2sin )(=
(2). n n n x 4
sin
3cos
][π
π
+=
2．Determine whether the system is linear? Time invariant? Stable? Causal? Memory?（10 points ）
]1[][-=n x n y
3．The figure of f (2-t ) is showed as following.(15 points)
(1) Please draw the figure of f (t ).
(2) Determine the Fourier Transform of f (t ).
4．Consider an LTI system whose response to the signal
()t x 1
in Figure a is the signal
()t y 1
illustrated in Figure b. Determine and sketch carefully the response of the
system to the input ()t x 2
depicted in Figure c (10 points).
Figure a Figure b Figure c
5． Find and sketch ()()()t h t x t y *=, where x (t ) and h (t ) are shown as(10 points )
6．Determine the parameter M T ,
so that 70% energy of the signal t
t
t x sin 2)(=is
contained within M M w w w ≤≤-，where M M w T π
2= (10 points).
7. Consider a stable continuous-time LTI system for which the input ()t x and
output ()t y are related by the differential equation (20 points).
)(4)
()(2)(3)(22t x dt
t dx t y dt t dy dt t y d +=++ (1) Determine the system function of the system ()S H ;
(2) Determine the unit impulse response ()t h of this system, is this system Memoryless? Causal?
(3) For the input )(2)(3t u e t x t -=, compute the zero status response ()t y ; (4)Draw a block diagram representation of this system.
8. Compute the Z Transform X(Z) of the following sequences ，draw pole-zero pattern of ()z H and indicate the ROC of ()z H .（15 points ）。

t
h (t ) x (t )
2 1
-1
t
1
1
-1
-1 (1) (2) (3)
Advanced
1．Consider the modulation-demodulation system illustrated in Figure . Suppose the Fourier transform ()jw F of ()t f is shown in Figure, and ()())cos(021t w t a t a ==,
b w w ≥0.(20 points)
(1) compute the spectrum )(1jw Y of )(1t y , and draw its figure; (2) compute the spectrum )(2jw Y of )(2t y , and draw its figure; (3) Sketch and determine a filter ()ωj H so that ()()t f t y =
2． Suppose that the signal x[n] has the following properties （10 points ）: (1) x[n] is a real and even signal; (2)x[n] has period N=8 and Fourier coefficients; (3) 59=a ;
(4)
400][2
70
=∑
=n n x
show that )sin(][C Bn A n x +=, and specify all possible numerical values for the constants A,B and C.
Modulation
Demodulation。