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

δ ( n ) = 1 for n = 0 0 otherwise.

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

y ( n ) = h ( n ) * x ( n ) = m = - h ( n - m ) x ( m )

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 δ ( n - m ) for m = 0 , 1 , 2 , , . The input x ( n ) 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 ( x ( n ) , δ ( n - m ) ) = x ( m ) δ ( n - m ) . Using the time invariant property of the system, δ ( n - m ) h ( n - m ) and using the scaling property of a linear system, this gives an output of x ( m ) δ ( n - M ) x ( m ) h ( n - m ) . We now calculate the output due to x ( n ) by adding outputs due to each of the resolved inputs using the superposition property of linear systems. This is illustratedby the following diagram:

x ( n ) = x ( n ) δ ( n ) = x ( 0 ) δ ( n ) x ( 0 ) h ( n ) x ( n ) δ ( n - 1 ) = x ( 1 ) δ ( n - 1 ) x ( 1 ) h ( n - 1 ) x ( n ) δ ( n - 2 ) = x ( 2 ) δ ( n - 2 ) x ( 2 ) h ( n - 2 ) x ( n ) δ ( n - m ) = x ( m ) δ ( n - m ) x ( m ) h ( n - m ) = y ( n )


y ( n ) = m = - x ( m ) h ( n - m )

and changing variables gives

y ( n ) = m = - h ( n - m ) x ( m )

If the system is linear but time varying, we denote the response to an impulse at n = m by δ ( n - m ) h ( n , m ) . 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 ( n ) = h ( n , m ) * x ( n ) = m = - h ( n , m ) x ( m ) .

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 ( n ) = h ( n ) * x ( n ) = m = - h ( m ) x ( n - m ) .

If the system is causal, h ( n ) = 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 ( n ) = h ( n ) * x ( n ) = m = 0 n h ( n - m ) x ( m )

or, showing the operations commute

y ( n ) = h ( n ) * x ( n ) = m = 0 n h ( m ) x ( n - m ) .

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.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Damian Reply
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sciencedirect big data base
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
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of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Source:  OpenStax, Brief notes on signals and systems. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10565/1.7
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