Introduction to Systems Engineering
Prof. Matthew Campisi
Department of Electrical and Computer Engineering
NYU-Poly
The System Development Process
Modern engineered systems come into being in response to
societal needs or because of new opportunities of
Digital Signal Processing I
Sample Final Exam
Spring 2016
1. A Linear phase FIR filter has the impulse response below:
n
h(n)
0
2
1
1
2
0
3
1
4
2
a) The filter frequency response can be expressed in the form
real function of . Find A( ), and .
H (e j ) A(
EL6113
Formula Sheet for Final
Linear Phase Filter Types
Type
N
h(n)
Symmetry
A( )
Period
Fall 2016
Symmetry
About
Symmetry
About
Form of
H ( e j )
=
=0
1
Odd
Even
2
Even
Even
2
Even
Even
4
A( ) e j
Even
Odd
3
Odd
Odd
2
A( ) e j
Odd
Odd
4
Even
Odd
4
j
% Notch filtering of tonal noise
% Null out tonal noise in a noisy speech signal.
% Compare second-order FIR and IIR filters.
% Load data
clear
[s, Fs] = wavread('clean.wav');
% speech signal
[x, Fs] = wavread('noisy.wav');
% speech signal with tonal nois
% Filter delay demo
% Illustratation of delay induced by an LTI system.
% Start
clear
% close all
% Define system
% The system is defined by the difference equation
%
% y(n) = 0.1 x(n) - 0.1 x(n-1) + 0.1 x(n-2) + 1.5 y(n-1) - 0.8 * y(n-2)
b = 0.1*[1 -1
1]
% Computing the impulse response of a system with complex poles (Example 1)
% This example shows three different ways to compute the impulse response.
% Define system coefficients
% Difference equation: y(n) = x(n) - 2.5 x(n-1) + y(n-1) - 0.7 y(n-2)
b = [
% Computing the impulse response of a system with complex poles (Example 2)
% This example shows three different ways to compute the impulse response.
% Define system coefficients
% Difference equation:
%
y(n) = x(n) - 2.5 x(n-1) + x(n-2) + y(n-1) - 0.7 y
1.6.1 A causal discrete-time system is described by the dierence equation,
y(n) = x(n) + 3 x(n
1) + 2 x(n
4)
(a) What is the transfer function of the system?
(b) Sketch the impulse response of the system.
1.6.1) Solution
H(z) = 1 + 3 z
1
+ 2z
4
h(n)
4
3
2
10.2 You should do this problem by hand (no computer).
in the z-plane. Similarly, poles in the RHS of the imaginary axis in the
s-plane must go to the outside of the unit circle in the z-plane.
(a) Suppose you have a stable causal analog filter. Prove why
1.13 Consider the following 8-point signals, 0 n 7.
(a) [1, 1, 1, 0, 0, 0, 1, 1]
(b) [1, 1, 0, 0, 0, 0,
1,
1]
(c) [0, 1, 1, 0, 0, 0,
1,
1]
(d) [0, 1, 1, 0, 0, 0, 1, 1]
Which of these signals have a real-valued 8-point DFT? Which of these
signals have a im
l, HoMEW/k b
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Note: For problem 2.1 the systems are:
(a) T cfw_x(n) = g (n) x(n)
(b) T cfw_x(n) =
n
x(k )
k = n0
(c) T cfw_x(n) =
n + n0
x(k )
k = n n0
(d) T cfw_x(n) = x(n n0 )
(e)
(f)
(g)
(h)
T cfw_x(n) = e x ( n )
T cfw_x(n) = a x(n) + b
T cfw_x(n) = x(n)
T cfw_x(
2
3 n
1. Design a simple real causal FIR lter that
(a) annihilates the signal cos
(b) annihilates the signal ( 1)n
(c) has unity dc gain
For the system you design:
(a) Find and sketch the impulse response h(n)
(b) Find the dierence equation
(c) Show the p
EL 6113 Final exam
Fall 2013
Provided formulas
DFT:
N 1
2
x(n) ej N nk ,
X(k) =
Name:
k = 0, . . . , N 1
n=0
Inverse DFT
ID Number:
x(n) =
1. Closed book, closed notes.
1
N
N 1
2
X(k) ej N nk ,
n = 0, . . . , N 1
k=0
WN := ej2/N
2. Non-programmable calcul
EL 611 Exam 3
Fall 2013
Provided formulas
DFT:
Name:
X(k) =
N 1
X
x(n) e
j 2 nk
N
,
k = 0, . . . , N
1
n=0
ID Number:
Inverse DFT
x(n) =
1. Closed book, closed notes.
N 1
2
1 X
X(k) ej N nk ,
N
n = 0, . . . , N
1
k=0
2. Non-programmable calculators permit
EL 611 Exam 2
Fall 2013
Provided formulas
DFT:
N 1
2
x(n) ej N nk ,
X(k) =
1. Closed book, closed notes.
k = 0, . . . , N 1
n=0
2. Non-programmable calculators permitted.
Inverse DFT
3. Show your work. Simplify your answers.
4. No cell phones. Put cell ph
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 ) the energy is defined as
Ef =
2
f (t ) dt .
To understand
2004 Polytechnic University
THE SAMPLING THEOREM
A function, f (t ) is called - bandlimited if F ( ) = 0 for
below illustrates the situation.
> . The spectrum
F ( )
In this diagram both the magnitude ( triangle) and phase (smooth curve) of F ( ) are
show
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 over all integers from to , as
. , x(2), x(1), x(0), x(1)
Also, Gf (!) = G(ej ! ) = H( ej ! ) = H(ej ej ! ) = H(ej (!+) ) = H f (! +
). so
3.6 Filter Transformations. An FIR filter with impulse response h(n) is
illustrated below. Define a new FIR filter with impulse response g(n) =
( 1)n h(n). Sketch the,
Gf (!)
3.25 Filter specifications. An analog signal, bandlimited to 20 Hz is corrupted
by high-frequency noise. The spectrum of the noise is from 25 Hz to 35 Hz.
The noisy analog signal is sampled at 60 Hz. A digital lowpass filter is to be
designed so as to rem