# 0.2 Discrete-time systems  (Page 2/5)

 Page 2 / 5

These definitions will allow a powerful class of analysis and design methods to be developed and we start with convolution.

## Convolution

The most basic and powerful operation for linear discrete-time system analysis, control, and design is discrete-time convolution. We firstdefine the discrete-time unit impulse, also known as the Kronecker delta function, as

$\delta \left(n\right)=\left\{\begin{array}{cc}1\hfill & \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}n=0\hfill \\ 0\hfill & \phantom{\rule{4.pt}{0ex}}\text{otherwise.}\hfill \end{array}\right)$

If a system is linear and time-invariant, and $\delta \left(n\right)\to h\left(n\right)$ , the output $y\left(n\right)$ can be calculated from its input $x\left(n\right)$ by the operation called convolution denoted and defined by

$y\left(n\right)=h\left(n\right)*x\left(n\right)=\sum _{m=-\infty }^{\infty }h\left(n-m\right)x\left(m\right)$

It is informative to methodically develop this equation from the basic properties of a linear system.

## Derivation of the convolution sum

We first define a complete set of orthogonal basis functions by $\delta \left(n-m\right)$ for $m=0,1,2,\cdots ,\infty$ . The input $x\left(n\right)$ is broken down into a set of inputs by taking an inner product of the inputwith each of the basis functions. This produces a set of input components, each of which is a single impulse weighted by a single valueof the input sequence $\left(x\left(n\right),\delta \left(n-m\right)\right)=x\left(m\right)\delta \left(n-m\right)$ . Using the time invariant property of the system, $\delta \left(n-m\right)\to h\left(n-m\right)$ and using the scaling property of a linear system, this gives an output of $x\left(m\right)\delta \left(n-M\right)\to x\left(m\right)h\left(n-m\right)$ . We now calculate the output due to $x\left(n\right)$ by adding outputs due to each of the resolved inputs using the superposition property of linear systems. This is illustratedby the following diagram:

$x\left(n\right)=\left\{\begin{array}{ccccc}x\left(n\right)\delta \left(n\right)\hfill & =\hfill & x\left(0\right)\delta \left(n\right)\hfill & \to \hfill & x\left(0\right)h\left(n\right)\hfill \\ x\left(n\right)\delta \left(n-1\right)\hfill & =\hfill & x\left(1\right)\delta \left(n-1\right)\hfill & \to \hfill & x\left(1\right)h\left(n-1\right)\hfill \\ x\left(n\right)\delta \left(n-2\right)\hfill & =\hfill & x\left(2\right)\delta \left(n-2\right)\hfill & \to \hfill & x\left(2\right)h\left(n-2\right)\hfill \\ ⋮\hfill & & & & ⋮\hfill \\ x\left(n\right)\delta \left(n-m\right)\hfill & =\hfill & x\left(m\right)\delta \left(n-m\right)\hfill & \to \hfill & x\left(m\right)h\left(n-m\right)\hfill \end{array}\right\}=y\left(n\right)$

or

$y\left(n\right)=\sum _{m=-\infty }^{\infty }x\left(m\right)\phantom{\rule{0.166667em}{0ex}}h\left(n-m\right)$

and changing variables gives

$y\left(n\right)=\sum _{m=-\infty }^{\infty }h\left(n-m\right)\phantom{\rule{0.166667em}{0ex}}x\left(m\right)$

If the system is linear but time varying, we denote the response to an impulse at $n=m$ by $\delta \left(n-m\right)\to h\left(n,m\right)$ . In other words, each impulse response may be different depending on when theimpulse is applied. From the development above, it is easy to see where the time-invariant property was used and to derive aconvolution equation for a time-varying system as

$y\left(n\right)=h\left(n,m\right)*x\left(n\right)=\sum _{m=-\infty }^{\infty }h\left(n,m\right)x\left(m\right).$

Unfortunately, relaxing the linear constraint destroys the basic structure of the convolution sum and does not result in anything of this form thatis useful.

By a change of variables, one can easily show that the convolution sum can also be written

$y\left(n\right)=h\left(n\right)*x\left(n\right)=\sum _{m=-\infty }^{\infty }h\left(m\right)x\left(n-m\right).$

If the system is causal, $h\left(n\right)=0$ for $n<0$ and the upper limit on the summation in Equation 2 from Discrete Time Signals becomes $m=n$ . If the input signal is causal, the lower limit on the summation becomes zero. The form of the convolutionsum for a linear, time-invariant, causal discrete-time system with a causal input is

$y\left(n\right)=h\left(n\right)*x\left(n\right)=\sum _{m=0}^{n}h\left(n-m\right)x\left(m\right)$

or, showing the operations commute

$y\left(n\right)=h\left(n\right)*x\left(n\right)=\sum _{m=0}^{n}h\left(m\right)x\left(n-m\right).$

Convolution is used analytically to analyze linear systems and it can also be used to calculate the output of a system by only knowing its impulseresponse. This is a very powerful tool because it does not require any detailed knowledge of the system itself. It only uses one experimentallyobtainable response. However, this summation cannot only be used to analyze or calculate the response of a given system, it can be an implementation of the system. This summation can be implemented inhardware or programmed on a computer and become the signal processor.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!