EL6113 Final Exam, Fall 2008
Directions: Answer all questions completely in the blue exam book. Clearly identify you
ALL work - partial credit will be given. -
Time: 2 Hours, 30 Minutes ,
1. (Spts) Consider the following LTl system:
h[n] = alnl,where |a|
Some of the work can be done using MATLAB: n=1:15; cn=-4*j./n/pi.*sin(pi*n/6).*sin(n*pi/2).*exp(-j*n*pi/3); n=-15:-1; c_n=-4*j./n/pi.*sin(pi*n/6).*sin(n*pi/2).*exp(-j*n*pi/3); cn=[c_n 0 cn]; n=-15:15; subplot(221),stem(n,abs(cn) title('|c_n|') subplot(222
Fourier Series 1. For the following signal: x(t) 2 -7 -6 -4 -3 -2 -1 23 56 8 9 time
a) Find the Fourier series b) Plot the spectra versus frequency, = n 0 . 2. Repeat problem 1 for the following signal: x(t) cos(t)
-10
10
time
3. Compute the Fourier serie
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Fourier Transform For each signal, find the Fourier transform, X(), and then plot |X()| (note, you may want to use MATLAB for the plot in 3.) 1. 2 0 2 t x(t)
2. 3 -4 -2
x(t)
0
2
4
t
3. 8
x(t)
0
4
t
4. x(t) = cos(200t)p4(t) 5. x(t)=e-3tcos(10t)u(t) 6. Find
Name: g
Polytechnic Institute of NYU
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EL6113 Signals, Systems, and Transforms ag: a
FALL 2008 Midterm Exam - 2Hours
Instructions: Answer all questions and show all work. Pafrtial 6cazrefit will be given as applicable.
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Polytechnic Institute of NYU 0"
EL6113 Signals. Systems, and Transforms
Midterm Exam 2 )5 Hours
instructions: Answer all questions and show all work. Partial credit will be given as applicable.
Problem 1 (15 points):
Consider the following thr
1. f(n)= a^|n-1| |a|<1 (causal LTI)
a. find zeros and poles?
b. stable?
c. sketch is signal?
2. (causal LTI)
f(t)=(1/2)^n (1<= n <= 6)
=0
elsewhere
1) find F(z)?
3. H(z)= 1/3(z + z^-1 +z^-2) (causal LTI)
a) h(n)? 1/3(n) +(n-1) + (n-2)
b) find zeros and po
ignals Chap- 3
ses to a complex _;
ient to conclude ._
ive responses I
ient to concl
in Figure P3111"-
)ut is consi w
Chap. 3 Problems
(3) Find the differential equation relating x(r) and r}.
(b) Determine the frequency response of this, system by consi
'
334 The Continuous-Time Fourier Transform Chap. 4;:-
this chapter we have derived and examined many of these properties. Among them
two that have particular signicance for our study of
EL6113
Sampling Practice
Fall 2008
1. A set of samples, f (nT ) , is given below. All samples that are not shown are zero.
f (nT )
1
2
T
T
t
2
Find the unique function, f (t ) , whose bandwidth satisfies / T that passes
through all of these samples. Your
EL (9H3
Assignment! #h/
Name:
1. Determine the z-transform (including ROC of the following:
a. (1/2)n u(n)
b. 41/2)" u(n-l)
c. (l/2)"u(-n)
d. 5(n)
e. 5(n-1)
t. 5(n+i)
g. (1/2)3(u(n)-u(n-10)
la)
For the : transform
. 32 + 223 +52
th)=~3
(z l)(z + 0.5)
a)
EL611
Real Rational Magnitude Characteristic
1. Given the function
V (cos )
5 4 cos
.
(10 6 cos )(17 8 cos )
a) Is this the magnitude function of any rational H (z ) ?
b) If the answer to part (a) is yes, find any H (z ) that has the magnitude function
EL611
LTI-Differential Eq
Fall 2008
1. Find the System functions (transfer functions) of the systems described by the
following differential equations. Also, assuming the systems are causal, state whether
or not they are stable.
a) 2 y (t ) 8 y (t ) 6 y (
EL611 3
Homework #5
1) Given an LTI system with real impulse response (i.e. the impulse response h(n) has no
imaginary part), show that
a) Any real input gives rise to a real output. Use the convolution sum to show this.
b) If the response to a complex in
EL611 3
Assignment 4a
1) An LTI system is to have the transfer function
H ( z)
z (2 z 2 15 / 2 z 15 / 4)
( z 3)( z 1 / 2) 2
.
a) Sketch the pole-zero diagram for H (z ) .
b) If the system is to implemented by stable recursions, find two recursion equatio
EL6113 Assignment 2
1. The impulse response
of an LTI system is known to be zero, except in the interval
. The input x[n] is known to be zero, except in the interval
result, the output is constrained to be zero, except on some interval
and
in terms of
,
,
EL 611
Assignment 1
1. For the given f(t), evaluate
a. f(t-3)
b. f(2t-3)
c. f(-t-3)
d. f(.5t +1)
f(t)
f(t)
4
2
0
-2
-2
-1
-1
0
1
1
2
2
f(t)
Time
2. Decompose the f(t) indicated above into its odd component,
.
3. For the f[n] indicated
EL6113 Assignment 2
1. The impulse response
of an LTI system is known to be zero, except in the interval
. The input x[n] is known to be zero, except in the interval
.
As a result, the output is constrained to be zero, except on some interval
.
Determine
EL611 3
Assignment 4a
1) An LTI system is to have the transfer function
H ( z) =
z (2 z 2 + 15 / 2 z + 15 / 4)
( z 3)( z 1 / 2) 2
.
a) Sketch the pole-zero diagram for H ( z ) .
b) If the system is to implemented by stable recursions, find two recursion e
EL6113
Sampling Practice
Fall 2008
1. A set of samples, f ( nT ) , is given below. All samples that are not shown are zero.
f ( nT )
1
2
T
T
t
2
Find the unique function, f (t ) , whose bandwidth satisfies / T that passes
through all of these samples. You
EL611 3
Homework #5
1) Given an LTI system with real impulse response (i.e. the impulse response has no imaginary
part), show that
a) Any real input gives rise to a real output. Use the convolution sum to show this.
b) If the response to a complex input,
EL611
LTI-Differential Eq
Fall 2008
1. Find the System functions (transfer functions) of the systems described by the
following differential equations. Also, assuming the systems are causal, state whether
or not they are stable.
a) 2 y (t ) + 8 y (t ) + 6