By a change of variable n−k=k,n−k=k, size 12{n - k=k rSup { size 8{,} } } {}, or k=n−k,k=n−k, size 12{k=n - k rSup { size 8{,} } } {} in the formula for convolution

y ( n ) = ∑ k = − ∞ ∞ x ( k ) h ( n − k ) = ∑ k , = − ∞ ∞ x ( n − k , ) h ( k , ) y ( n ) = ∑ k = − ∞ ∞ x ( k ) h ( n − k ) = ∑ k , = − ∞ ∞ x ( n − k , ) h ( k , ) size 12{y \( n \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( k \) h \( n - k \) } = Sum cSub { size 8{k rSup { size 6{,} } = - infinity } } cSup { infinity } {x \( n - k rSup { size 8{,} } \) h \( k rSup { size 8{,} } \) } } {}

and by replacing the temporary variable k’ by k, we get

y ( n ) = ∑ k = − ∞ ∞ x ( n − k ) h ( k ) = ∑ k = − ∞ ∞ h ( k ) x ( n − k ) = h ( k ) ∗ x ( k − n ) y ( n ) = ∑ k = − ∞ ∞ x ( n − k ) h ( k ) = ∑ k = − ∞ ∞ h ( k ) x ( n − k ) = h ( k ) ∗ x ( k − n ) size 12{y \( n \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {x \( n - k \) h \( k \) } = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {h \( k \) x \( n - k \) =h \( k \) *x \( k - n \) } } {}

That is the order of convolution is reversed. Thus we have two formulae of convolution:

and

In practive we usually let the longer sequence stay fixed, and shift the shorter one.

The commutative characteristic of convolution means that we can swap the input signal with the impulse response of a system without affecting the output. This idea is depicted in Figure 2.

It can be shown that

Figure 3 shows the system meaning of the associativity, where two systems in series (in cascade) can be replaced by only one whose impulse response is the convolution of the two individual impulse responses.

First ∣a∣∣a∣ size 12{ lline a rline } {} and ∣b∣∣b∣ size 12{ lline b rline } {} should be smaller than 1 to ensure the convergence of the sequences. Notice that both impulse responses are causal. The overall impulse response is

h ( n ) = h 1 ( n ) ∗ h 2 ( n ) = ∑ k = − ∞ ∞ h 1 ( k ) h 2 ( n − k ) h ( n ) = h 1 ( n ) ∗ h 2 ( n ) = ∑ k = − ∞ ∞ h 1 ( k ) h 2 ( n − k ) size 12{h \( n \) =h rSub { size 8{1} } \( n \) * h rSub { size 8{2} } \( n \) = Sum cSub { size 8{k= - infinity } } cSup { size 8{ infinity } } {h rSub { size 8{1} } \( k \) h rSub { size 8{2} } \( n - k \) } } {}

The actual limits of summation are k=0k=0 size 12{k=0} {} and k=nk=n size 12{k=n} {}(see Section 4 later), hence

h ( n ) = ∑ k = 0 n a k b k − n u ( k ) u ( k − n ) = b n ∑ k = 0 n ( a b ) k h ( n ) = ∑ k = 0 n a k b k − n u ( k ) u ( k − n ) = b n ∑ k = 0 n ( a b ) k alignl { stack { size 12{h \( n \) = Sum cSub { size 8{k=0} } cSup { size 8{n} } {a rSup { size 8{k} } b rSup { size 8{k - n} } u \( k \) u \( k - n \) } } {} # matrix { {} # {} # {} } =b rSub { size 8{n} } Sum cSub { size 8{k=0} } cSup { size 8{n} } { \( { { size 8{a} } over { size 8{b} } } \) rSup { size 8{k} } } {} } } {}

Using the formula of finite geometric series.

here x=a/bx=a/b size 12{x= {a} slash {b} } {}, we get

h ( n ) = b n 1 − ( a b ) n + 1 1 − a b = b n + 1 − a n + 1 b − a h ( n ) = b n 1 − ( a b ) n + 1 1 − a b = b n + 1 − a n + 1 b − a size 12{h \( n \) =b rSup { size 8{n} } { {1 - \( { { size 8{a} } over { size 8{b} } } \) rSup { size 8{n+1} } } over {1 - { { size 8{a} } over { size 8{b} } } } } = { {b rSup { size 8{n+1} } - a rSup { size 8{n+1} } } over {b - a} } } {}

It can be shown

The system meaning is illustrated in Figure 4 where two systems connected in parallel can be replaced by one whose impulse response is the sum of the two ones.

Since impulse response is a characterization (among other characterizations) of systems. As such, the causality of a system would be reflected on its impulse response. From the convolution Figure the output at instant n0n0 size 12{n rSub { size 8{0} } } {} is:

y ( n 0 ) = ∑ k = 0 ∞ h ( k ) x ( n 0 − k ) + ∑ k = − ∞ − 1 h ( k ) x ( n 0 − k ) y ( n 0 ) = ∑ k = 0 ∞ h ( k ) x ( n 0 − k ) + ∑ k = − ∞ − 1 h ( k ) x ( n 0 − k ) size 12{y \( n rSub { size 8{0} } \) = Sum cSub { size 8{k=0} } cSup { size 8{ infinity } } {h \( k \) x \( n rSub { size 8{0} } - k \) } + Sum cSub { size 8{k= - infinity } } cSup { size 8{ - 1} } {h \( k \) x \( n rSub { size 8{0} } - k \) } } {}

In order the output signal y(n0)y(n0) size 12{y \( n rSub { size 8{0} } \) } {} does not depend on future (n>n0)(n>n0) size 12{ \( n>n rSub { size 8{0} } \) } {}values of input signal x(n)x(n) size 12{x \( n \) } {}, the second term of above equation should be zero, i.e. h(k)=0h(k)=0 size 12{h \( k \) =0} {} for k<0k<0 size 12{k<0} {}. As k is a dummy variable, we conclude

h ( n ) = 0 at n < 0 h ( n ) = 0 at n < 0 size 12{h \( n \) =0 matrix { {} # {} } ital "at" matrix { {} # {} } n<0} {}

Thus, the causality of a system implies that its impulse response is zero and vice versa. The output at time n0n0 size 12{n rSub { size 8{0} } } {} is now the first term of the equation

y ( n 0 ) = ∑ k = 0 ∞ h ( k ) x ( n 0 − k ) y ( n 0 ) = ∑ k = 0 ∞ h ( k ) x ( n 0 − k ) size 12{y \( n rSub { size 8{0} } \) = Sum cSub { size 8{k=0} } cSup { size 8{ infinity } } {h \( k \) x \( n rSub { size 8{0} } - k \) } } {}

For any time n,

Had the convolution x(n)∗h(n)x(n)∗h(n) size 12{x \( n \) *h \( n \) } {} been used, the result would be

In above, only the causality of the system is considered. Now, the imput signal is also causal, the result is

And equivalently