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
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 a
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 i
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.
EL-GY 6113/ BE-GY 6403 Digital Signal Processing I
Fall 2017, Yao Wang
HW 1
From Selesnicks Signals Exercises packet
http:/eeweb.poly.edu/iselesni/EL6113/Signals_Exercises.pdf
Reading: Oppenheim Sec.
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 se
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 se
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
2004 Polytechnic University
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 ov
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
2004 Polytechnic University
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 ) th
2004 Polytechnic University
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
2004 Polytechnic University
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
2004 Polytechnic University
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
2004 Polytechnic University
FREQUENCY RESPONSE
We now focus on sinusoidal inputs to our discrete time systems to investigate the
frequency response characteristics. Therefore, consider a discrete time
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 ou
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
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 sh
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 answ
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
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
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 c
EL611
Z Transform Solutions
Fall 2008
1.)
H ( z) =
z (2 z 2 + 15 / 2 z + 15 / 4)
( z 3)( z 1 / 2) 2
.
a) H ( z ) has zeros at z = 0 , z = 4.197 and z = 0.447 . It has a pole at z = 3 and a
double pole