EL6113
Solution Assignment 3
1.
T
n (t ) k (t ) dt
T
=
0
=
1
T
1
T
0
e
jn
2
j (n k )
t
T
e
T
0
2
t
T
1
T
e
jk
2
t
T
1
dt =
T
T
e
j (n k )
(
)
dt
0
1 ; n=k
2 T
j (n k )
t
dt =
1
T
e
;
j (n k )2
0
1 ; n=k
1
=
e j ( n k ) 2 1 ;
j ( n k )2
2
t
T
nk
nk
an
Name: _
Polytechnic Institute of NYU
EL6113 Signals, Systems, and Transforms
SPRING 2011 - Midterm Exam 2.5Hours
Instructions: Answer all questions and show all work. Partial credit will be given as applicable.
Problem 1 (20 points):
Consider the followin
EL6113 Assignment 3
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 ou
THE DISCRETE FOURIER TRANSFORM (DFT)
Let x (n ) be a periodic discrete time signal with period N . The
DFT, X (k ) , of x (n ) is defined by
X (k ) =
N 1
x(n)
e
j
2
kn
N
.
(9.1)
n =0
It is easy to see that the X (k ) sequence calculated in this way is
al
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
1
2
3 Attention a discrete-time signal, n must be an integer. In another word, you wont get any value
For
for f[n] when n is a fraction like 1/2, -1/2.
4.
51)
b.
d.
f.
Causal
Stable
Time-invariant
NOT memoryless
Because the current output y(t) depends on
EL611 3
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 input, x(n) ,
THE DISCRETE FOURIER TRANSFORM (DFT)
Let x (n ) be a periodic discrete time signal with period N . The
DFT, X (k ) , of x (n ) is defined by
X (k ) =
N 1
x(n)
e
j
2
kn
N
.
(9.1)
n =0
It is easy to see that the X (k ) sequence calculated in this way is
al
THE DISCRETE TIME FOURIER TRANSFORM (DTFT)
The Discrete-Time Fourier Transform of a signal, x (n ) , is simply
its z-transform evaluated on the unit circle. Thus, given x (n ) , the
DTFT is given by
j
X (e ) =
x(n) e j n
.
(4.1)
n =
Clearly, the DTFT is
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DISCRETE TIME SIGNALS
A discrete time signal, x(n) , is a sequence of numbers, real or complex. In general the signal
can be bi-infinite with the time index, n , running over all integers from to , as
. , x(2), x(1), x(0), x(1)
7.7 OPTIMUM APPROXIMATIONS 0F FIR FILTERS
The design of FIR ﬁlters by windowing is straightforward and is quite general, ev
though it has a number of limitations as discussed below. However, we often wis
Chapter 7
542
compute. The Kaiser window is defined as
/0[,8(1
w[nl
=
(
- [en - a)/a]2)1/2]
10(fJ)
0:11:
M,
otherwise,
0,
where a = M /2, and 100 represents the zeroth-order modified Bessel function a
first kind. In contrast to the other windows in Eqs. (
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ORTHOGONAL FUNCTIONS AND FOURIER SERIES
THE ENERGY IN A FUNCTION
The energy in a function often plays an important role in signal processing. For a given
function f (t ) the energy is defined as
Ef =
2
f (t ) dt .
To understand
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THE SAMPLING THEOREM
A function, f (t ) is called - bandlimited if F ( ) = 0 for
> . The spectrum
below illustrates the situation.
F ( )
In this diagram both the magnitude ( triangle) and phase (smooth curve) of F ( ) are
show
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MAGNITUDE CHARACTERISTICS FOR REAL RATIONAL TRANSFER
FUNCTIONS
In this section we will restrict ourselves to causal, stable systems H (z ) . Recall that such
systems always possess a frequency response, i.e. the unit circle is
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THE Z TRANSFORM
One of the primary tools for the analysis of discrete time LTI systems is the z transform. For a
given signal, f (n) , the z-transform is a function of the complex variable, z , and is defined as
F ( z) =
f ( n
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FREQUENCY RESPONSE
We now focus on sinusoidal inputs to our discrete time systems to investigate the
frequency response characteristics. Therefore, consider a discrete time LTI system with a
pure complex sinusoid of frequency a
FIR Linear Phase Filters
Consider the ideal low pass filter frequency response
H (e j )
1
c
c
The impulse response of this filter is
1
h( n) =
2
H (e
j
)e
j n
1
d =
2
c
(1) e j n d
c
Which works out to be
h( n) =
sin c n c sin c n
=
.
n
c n
(5.1)
h(n)
EL611
Real Rational Magnitude Characteristic
Fall 2008
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 ma
1)
The output is given by the convolution sum
h( k ) x ( n k ) .
a) y (n) =
k =
If h(n) and x(n) are both real then it is obvious that y ( n) will also be real.
b) Note that, since h(n) is real we can take the conjugate of the convolution sum equation
a
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
EL 611
Assignment 2
1. Given , sketch x(t) and x(-2t-4)
2. Determine if the following systems are:
a. Memoryless
b. Causal
c. Stable
d. Linear
e. Time invariant
e.i.
e.ii.
e.iii.
e.iv.
e.v.
3. You are given an LTI system where the input produces the outpu