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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

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 )

or

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.

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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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