2e t 1 , t 1
t 1
0,
e 2 t , t 0
0, t 0
and x(t )
Problem 3 (30 pts): h(t )
as shown below. Compute the
convolution y(t) = x(t)*h(t).
x(t)
h(t)
2
e-2t
t
0
t
0
1

A linear time invariant (LTI) system has an impulse
response function
. Find the output of the
system when the input is
.
Calculate Impulse Response Function
Ex1:
dVO (t ) 1
1
Vo (t )
Vi (t )
dt
RC
RC
`
1
t
( t ) 1
Vo (t ) e RC
Vi ( )d
RC
0
R
Vi(t)
C
V

Chapter 5
FOURIER TRANSFORM
Recap
The representation of signals as a linear combination of complex
exponentials leads to a convenient expression for the response of a LTI
system.
Specifically, if the input to a continuous-time LTI system is represented
as

USING FT TO ANALYZE LTI
SYSTESM
Review of Convolution Property of FT
if
x t X ,
h t H
x t h t X H
x (t )
h (t )
y (t )
1
LTI Response to a periodic complex exponential
x (t ) exp[ j (0t )]
h (t )
y(t )
H ( )
y(t )
Response to a sinusoid
x (t ) cos(0t

Refresher
x t
Fourier Series:
ce
k
jk0t
0
k
2
T
1
ck x t e jk0t dt
T T
Fourier Transform:
X x t e jt dt
(Fourier transform)
x t
1
2
0
X e jt d
(Inverse Fourier Transform)
6. Duality
FT and IFT are similar but not identical. This symmetry leads to a

Sampling Theorem
Let
be a band-limited signal with
X
= 0 for | > B.
Then
can be uniquely determined by its
samples
, n=0, 1, 2, . if
2 .
s = 2B is called Nyquist sampling frequency.
Signal Reconstruction
X()
1
0
-B
p(t ) (t nT )
n
x(t)
X
xs(t)=x(t)p(t)

R
R
C
R
C
R
C
C
Convert LPF to Bandpass filter
Multiply the impulse response function of a lowpass
filter with 2 cos 0 t
e jtd
H BP ( )
0
hLP (t )
B
sinc[
B
BL BH
otherwise
(t td )]
hBP (t ) 2hLP (t ) cos[
B
BH BL
2
BH BL
(t td )]
2
1
Exercise
Ideal F

Laplace Transform
Laplace Transform (LT)
X ( s ) x(t )e st dt
0
In the Fourier analysis, we focus on s j .
In the Laplace Transform, we let s j .
Fourier analysis is more focused on steady-state response
We focus on causal signals x(t) = 0 for t<0, so we

load indicator_dilution.mat
%create time vector based on length of input and output
t=[1:length(input)]'*dt;
0etermine where the appropriate times occur in each matrix
x=find(t=10);
y=find(t=30);
z=find(t=100);
figure
%plot the time vs the input
plot(t(1:

File Edit View Dsign Analysis Tools Windmnr Help I File Edit WindmIr Help
x0%|\@s®|k? JQI‘QIEI
oot Locus Editor for Open Loﬂpembp Bode Editor for Open Loop Step Rag ponse
4 I I 10 1-2
0
3 — -E
—10
1
System: Closed Loop r to y
2 _ _ —20 I10: rt) '51
l Peak

BIM 108
Biomedical Signals and Control
Lecture 5
Indicator dilution method for measuring
blood flow and volume
Meier and Zierler 1954, Zierler 2000 (on smartsite)
In-flow orifice P, outflow orifice Q. The system contains a
volume V of fluid which enters

BIM 108
Biomedical Signals and
Control
Jinyi Qi
Biomedical Engineering
What is about
y(t)
System H
Input signal
Output signal
A language for describing signals and systems
An extremely powerful set of tools for analyzing
them
1
Signals
A signal is a va

=
BIM 108
Biomedical Signals and Control
Fall 2012
=
University of California, Davis
Department of Biomedical Engineering
Course Content: Representation and analysis of biomedical signals and linear systems. Elementary
signals. Basic system properties. Ti

STA 131A
Fall 2009
More Sample Questions
1. (Events)
(i) Write down the sample space for the following random experiment: Diodes from a batch are tested
one at a time and marked either 'Defective'(D) or 'Nondefective'(N). This is continued until either tw

STA 131A
Fall 2009
More Sample Questions ( 5 0
I
(~(4
s
1. (Events)
(i) Write down the sample Lpace for the following random experiment: Diodes from a batch are tested
one at a time and marked either 'Defective9(D)or 'Nondefectivel(N).This is continued un

Practice problems for Fourier transform
=
1. Compute the Fourier transform of the following signals using the integral formula
(definition).
t + 1, 1 < t 0
a ) x(t ) = 1 t , 0 < t < 1
0, elsewhere
e2t , 0 t | 1
b ) x( t ) =
0, elsewhere
cos( t ), | t |

=
FT Practice Solution
=
1.
t 1, 1 t 0
a) x(t ) 1 t , 0 t 1
0, elsewhere
1
0
x(t )e jt dt (t 1)e jt dt (1 t )e jt dt
0
1
0
1
1
te jt dt te jt dt e jt dt
0
1
1
X ( )
Note: te at dt
1 at
te e at dt
a
1 jt
1 jt
X ( )
e
te
j
j
1
1 jt
1 jt 1 jt
e
e
t

=
BIM 108
Biomedical Signals and Control
Fall 2012
=
Homework #1 (Due October 10)
Reading: chapter 1 (Kamen&Heck)
1. For each signal x(t) shown below,
i.
Express x(t) using one expression with the aid of the unit-step function u(t). For
example, the signa

ENGR 012 Slide 1
Examples of Convolution
There
Theres no need to fear,
convolution is here!
Review
Impulse response: h(t)
Input: x(t)
Output: y(t)
y(t) = h(t) x(t) = x(t) h(t)
+
=
+
h()x(t )d = x()h(t )d
Given the impulse response of a system, and the in

Recap
The representation of signals as a linear combination of complex
exponentials leads to a convenient expression for the response of a LTI
system.
Specifically, if the input to a continuous-time LTI system is represented
as
x(t ) c k e jk0t
k
then the

Ex:
2
1
2
3
4
SIGNAL CLASSIFICATION AND
PROPERTIES
1
Continuous-time signals
For a continuous-time signal x(t), the independent
variable t is continuous.
We assume that t can take on all real values between -
and +.
The dependent variable x can take a

(x (t ), y(t ) 0
(ax 1(t ) bx 2(t ), ay1(t ) by2(t ) 0
(x (t t0 ),y(t t0 ) 0
Systems described by differential equations
1
System Memory
A system is memoryless if the output depends only on the
present input.
Ideal amplifier
A system with memory has an