EECS 451
PROBLEM SET #4
ASSIGNED: Sep. 30, 2010. READ: Sects. 4.2 & 4.3 (skip 4.2.7-4.2.8: Z + much easier).
DUE DATE: Oct. 07, 2010. TOPICS: Dierence equations and transfer functions.
Please box your answers. Show your work. Turn in all Matlab plots and
Recitation 2 -EECS 451, Winter 2010
Jan 20, 2010
OUTLINE
Review of Sampling
Practice problems
Concepts
1. Sampling
Sampling Theorem
Ideal Sampling and reconstruction
Practical issues
Interpretation of aliasing in both frequency and time domain
Problems
1.
Recitation 1 -EECS 451, Winter 2010
Jan 13, 2010
OUTLINE
Review of Fourier Transform
Practice problems
Concepts
1. Fourier Transforms
Definition
Basic Properties
Conjugate symmetry
Problems
1. Determine the Fourier transform of the signal x(t )
1 t
Divide large problem into smaller;
Conquer by solving smaller ones.
1000-point problem: (1000)2 =1 million ops.
100 10-point problems: 100(10)2 =10,000 ops.
Great! But how can we do this?
Why: Aliased: see HW. The sign: ej 2(n+(N/2)/N = ej 2n/N .
That is:
9/6/2009
Introduction to EECS 451
TOPICS FOR TODAYS LECTURE:
So what can you DO with DSP?
1. You can filter noisy data:
1. Go over features of the course web site at
www.eecs.umich.edu/~aey/eecs451.html
Includes: Copies of lecture presentations;
Condensed
d3 x
(t)
dt3
h[n] =
d2 x
dt2 (t)
dx
dx
dt (t) dt (t T ) /T . Substitute in itself.
11
1
1
T ( T cfw_1, 1 T cfw_0, 1, 1)= T 2 (cfw_1, 2, 1).
2
2
dx
x
(t) d 2 (t T ) /T . h[n]: gets complicated.
dt2
dt
1
1
Backward y [n]=(x[n]x[n1])/T . h[n]=cfw_ T , T .
1
Means: Maximum frequency is B Hertz.
Means: X ( ) = Fcfw_x(t)=0 for | | 2B .
RADIAN
Means: Bandwidth=2B Hertz=4B SECOND .
3. Ideal Reconstruction from Samples
4. Determining Signal Bandwidths
5. Finite Pulse Width Sampling
1
0
0.5
0
0.5
1
1
Thats: Ban
%
% State Space Representation
% x' = Ax + Bu
% y = Cx + Du
%
% Problem 1 -% Check Controllability and Observability of a 2nd order System
% Given -MatrixA = [3 1 0;-1 0 1;2 0 0];
MatrixB = 0;
MatrixC = [1 0 0];
MatrixD = 0;
% Objective -% 1) To Find Cont
% start of code
% include constants.m
constant
% includes change_data
changedata
% electrical transfer function
Z = tf([1],[La Ra]);
% dynamic transfer function
I = tf([1],[Jm Bm]);
% closed loop transfer function
T = feedback(Z*Ki*I, Kb)
% bandwidth freq
1
CHEM 3512 Winter 2015: Physical Chemistry
Problem Set 3
Name:
Instructors Name:
2
1. Calculate the energy of n=5 to n=4 transition in Li2+ ion in the units of cm-1
In Bohrs theory, lithium ion is essentially identical to hydrogen atom
Where z=3
En=
hcR
THE MACHINE SHOP
Introduction
The production and technology department deals with a lot of heavy machines which are
automatic. If not handled properly, they could be very dangerous. There is also a lot of cutting,
drilling, rolling and manipulation of met
ELECTRIC WELDING.
1.1 DEFINATION OF WELDING.
Its defined as the process of joining two pieces of metal at faces considered liquid or
plastic by application of heat or pressure.
A filler material is usually used to affect the union.
1.2WELDING PROCESSES.
T
WELDING WORKSHOP
Introduction
Welding is the process by which two metals are joined together using hard soldering.
Types of Welding
There are three most common types of welding, namely:
a) Electric/Arc welding
b) Spot welding
c) Gas welding
ELECTRIC/ARC W
Running head: The Shawshank Redemption
1
The Shawshank Redemption
The Shawshank Redemption is a story about upholding dignity and feelings when living in a
hopeless situation .In most prisons, integrity is missing and the movie illustrates how people can
ABSTRACT
Sheet metalwork involves bending, cutting and joining of such materials as tinplate,
aluminium, brass and galvanized mild steel. Initially, the metal is developed and cut to
shape using sheet metalwork tools. One requires knowledge in graphic com
MATLAB CODES USED TO ANALYSE MOTOR CHARACTERISTICS
Save the codes in separate files and the third one to view bode plot of open loop characteristics
% start of code
% motor parameters
La = 0.0048 % henry
Ra = 0.7454 % ohms
Ki = 0.067 % Nm/A
Jm = 6.87e-005
% any change in data
display = input('do you want to change the values(y/n): ', 's');
if strcmp(display,'y')
La = input('enter
Ra = input('enter
Ki = input('enter
Jm = input('enter
Bm = input('enter
Kb = input('enter
end
% end of code
the
the
the
the
the
2z
1 2 z
z 2 z
z 2
x[n]=2(2)n u[n]2(2)n1 u[n1].
That is: Poles are the roots of denominator=0.
Compute: Poles in Matlab: roots([aN , . . . a0 , 0])
Note: an realpoles in complex conjugate pairs.
POLES OF RATIONAL FUNCTION
Poles: cfw_0, p1 . . . pN are r
Q: How to tell whether a system is linear?
A: If doubling input doubles output,
then the system is likely linear.
This rule works most (not all) of the time.
y [n] = 3x[n 2]; y [n] = sin(n)x[n].
y [n] = x[n + 1] nx[n] + 2x[n 1].
EXAMPLES OF LINEAR SYSTEMS
EECS 451
SOLUTIONS TO PROBLEM SET #2
1a. Y ES is LTI. Read o impulse response h[n]=cfw_. . . 1 , 1 , 1, 0, 1, 1 , 1 . . ..
3
2
2
3
1b. N O not causal since h[n]= 0 for n < 0 (y[n] depends on future inputs).
(c) N OT BIBO stable since
|h[n]|=2(1+ 1 + 1 +.
EECS 451
SOLUTIONS TO PROBLEM SET #3
1a. z 2 + 3z + 4 ; ROC: 0 z < . (1b)
1c.
z z 2
z
z 1 z 2 z 2
3
z 1
3
z 1
3
= z2 5z/3 2 . ROC:
7
z+
3
3
z z 1
2z 2 3z
+ z z2 = z2 3+2
z 2 z 1 z 1 z 2
z
1
3
. ROC: |z | > 2 .
< |z | < 2 .
1d. DOES NOT EXIST since ROC is
EECS 451
PROBLEM SET #9
ASSIGNED: Nov. 18, 2010. READ: Sects. 10.3 & 11.1-11.4.
DUE DATE: Dec. 02, 2010. TOPICS: FIR and IIR lter design.
Please box your answers. Show your work. Turn in all Matlab plots and Matlab code.
[15] 1. IIR lter design of a digit
EECS 451
PROBLEM SET #7
ASSIGNED: Oct. 28, 2010. READ: Sects. 8.1 & 8.2.1-8.2.3.
DUE DATE: Nov. 04, 2010. TOPICS: Discrete Fourier Transform (DFT).
Please box your answers. Show your work. Turn in all Matlab plots and Matlab code.
[40] 1. Compute the circ
EECS 451
PROBLEM SET #6
ASSIGNED: Oct. 21, 2010. READ: Sects. 5.4-5.6. All of Chap. 7.
DUE DATE: Oct. 28, 2010. TOPICS: Frequency response.
Please box your answers. Show your work. Turn in all Matlab plots and Matlab code.
[20] 1. We are given that 3+4cos
EECS 451
PROBLEM SET #5
ASSIGNED: Oct. 14, 2010. READ: Sects. 4.4 & 5.1-5.2.
DUE DATE: Oct. 21, 2010. TOPICS: DTFT and DTFS.
Please box your answers. Show your work. Turn in all Matlab plots and Matlab code.
[20] 1. We are given x[n]=cfw_. . . 18, 12, 6,