Properties of Convolution and System Interconnections

5.6.3 Properties of Convolution and System Interconnections

Now let us consider properties of convolution. These properties are used in series and parallel inter connection of systems.

1. Commutative Property

The convolution is commutative operation. Consider the discrete time convolution.

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Let us define the new index of summation as, m = n - k and hence k = n - m.

The limits m = will be same as k, i.e (-∞, ∞).

Hence above equation becomes,

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Here observe that 'm' is dummy index and it can be replaced by any character, the meaning remains same. Hence replacing 'm' by 'k' in the above equation we get,

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From equation (1) & (2) it is clear that

.'. x(n) * h(n) = h(n) * x(n) = y(n).

The equation shows that discrete convolution is also commutative.

2. Series (cascade) connection of systems (Associative property).

The convolution is associative. Where the systems are connected in cascade, the impulse response of cascade connection is equal to convolution of impulse responses of individual systems.

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Proof:

Consider the series connection of two systems in figure 5.38 we can write.

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putting for y₁(k) in first equation,

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Let k - p = w, then limits of summations will remain same.

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Since convolution is commutative

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Putting for h(n) from above two equations in equation (1)

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3. Parallel connection of systems (Distributive property)

The overall impulse response in parallel connected systems is equal to sum of impulse responses of individual systems.

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Proof:

Consider the two systems connected in parallel as shown in figure 5.39. The overall output is,

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Thus when the two systems are connected in parallel, thier impulse responses get added.

Problem 11:

Find the overall impulse response of the system shown in figure 5.40.

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Solution:

Step I:

Convolution of aⁿ u(n) and δ(n-1) are cascaded. Hence their overall impulse response will be convolution. i.e.,

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Step II:

Convolution of δ(n-2) and aⁿ u(n)

The two blocks in lower link, having impulse response δ(n-2) and an u(n) are cascaded. Hence their overall impulse response will be convolution. i.e.,

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Step III:

Figure 5.41 Shows the simplified block diagram

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Here the blocks are in parallel. Hence their overall impulse response will be sum of individual impulse responses. i.e.,

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Here the two blocks are in parallel. Hence their overall impulse response will be sum of individual impulse responses. i.e.,

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Problem 12:

Find the overall impulse response of the causal system shown in figure 5.42.

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Solution:

Figure 5.42 shows the simplification of system. From figure 5.42 it is clear that the overall impulse response of the system is,

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Putting the values in above equation.

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This is the overall impulse response.