# 0.3 Modelling corruption  (Page 8/11)

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This integral defines the convolution operator $*$ and provides a way of finding the output $y\left(t\right)$ of any linear system, given its impulse response $h\left(t\right)$ and the input $x\left(t\right)$ .

M atlab has several functions that simplify the numerical evaluation of convolutions. The most obvious of these is conv , which is used in convolex.m to calculate the convolution of an input x (consisting of two delta functions at times $t=1$ and $t=3$ ) and a system with impulse response h that is an exponential pulse. The convolution gives the output of the system.

Ts=1/100; time=10;             % sampling interval and total time t=0:Ts:time;                   % create time vectorh=exp(-t);                     % define impulse response x=zeros(size(t));              % input is sum of two delta functions...x(1/Ts)=3; x(3/Ts)=2;          % ...at times t=1 and t=3 y=conv(h,x);                   % do convolution and plotsubplot(3,1,1), plot(t,x) subplot(3,1,2), plot(t,h)subplot(3,1,3), plot(t,y(1:length(t))) convolex.m example of numerical convolution (download file) 

[link] shows the input to the system in the top plot,the impulse response in the middle plot, and the output of the system in the bottom plot. Nothing happens before time $t=1$ , and the output is zero. When the first spike occurs,the system responds by jumping to 3 and then decaying slowly at a rate dictated by the shape of $h\left(t\right)$ . The decay continues smoothly until time $t=3$ , when the second spike enters. At this point, the output jumps up by 2, andis the sum of the response to the second spike, plus the remainder of the response to the first spike. Since there are no more inputs,the output slowly dies away.

Suppose that the impulse response $h\left(t\right)$ of a linear system is the exponential pulse

$h\left(t\right)=\left\{\begin{array}{cc}{e}^{-t}\hfill & t\ge 0\hfill \\ 0\hfill & t<0\hfill \end{array},.\right)$

Suppose that the input to the system is $3\delta \left(t-1\right)+2\delta \left(t-3\right)$ . Use the definition of convolution [link] to show that the output is $3h\left(t-1\right)+2h\left(t-3\right)$ , where

$h\left(t-{t}_{0}\right)=\left[\begin{array}{cc}{e}^{-t+{t}_{0}}& t\ge {t}_{0}\\ 0& t<{t}_{0}\end{array}\right].$

Suppose that a system has an impulse response that is an exponential pulse. Mimic the code in convolex.m to find its output when the input is a white noise (recall specnoise.m ).

Mimic the code in convolex.m to find the output of a system when the input is an exponential pulse and theimpulse response is a sum of two delta functions at times $t=1$ and $t=3$ .

The next two problems show that linear filters commute with differentiation, and with each other.

Use the definition to show that convolution is commutative (i.e., that ${w}_{1}\left(t\right)*{w}_{2}\left(t\right)={w}_{2}\left(t\right)*{w}_{1}\left(t\right)$ ). Hint: Apply the change of variables $\tau =t-\lambda$ in [link] .

Suppose a filter has impulse response $h\left(t\right)$ . When the input is $x\left(t\right)$ , the output is $y\left(t\right)$ . If the input is ${x}_{d}\left(t\right)=\frac{\partial x\left(t\right)}{\partial t}$ , the output is ${y}_{d}\left(t\right)$ . Show that ${y}_{d}\left(t\right)$ is the derivative of $y\left(t\right)$ . Hint: Use [link] and the result of Exercise  [link] .

Let $w\left(t\right)=\Pi \left(t\right)$ be the rectangular pulse of [link] . What is $w\left(t\right)*w\left(t\right)$ ? Hint: A pulse shaped like a triangle.

Redo Exercise  [link] numerically by suitably modifying convolex.m . Let $T=1.5$ seconds.

#### Questions & Answers

can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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