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Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

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Properties of LTI System 

In the preceding chapters,we have already derived expressions for discrete as well as continuous time convolution operations.

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)       Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

We shall now discuss the important properties of convolution for LTI systems.

1) Commutative property : By the commutative property,the following equations hold true :

a) Discrete time:
Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Proof : We know that

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)   Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Hence we make the following substitution (n - k = l )

∴The above expression can be written as

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

So it is clear from the derived expression that

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Note :

1. 'n' remains constant during the convolution operation so 'n' remains constant in the substitution “n-k = l” even as 'k' and 'l' change.
2. “l” goes from  - ∞ to + ∞  , this would not have been so had 'k' been bounded.( e.g :- 0 < k < 11 would make n < l < n – 11)

b) Continuous Time:

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Proof:

We Know That 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) 

Making The Substitution t - λ = Ø               Limits  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

                                       dt  = - d λ 

 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Thus we proved that convolution is commutative in both discrete and continuous variables.

Thus the following two systems : One with input signal x(t)and impulse response h(t) and the other with input signal h(t) and impulse response x(t) both give the same output y(t) 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

2) Distributive Property :

By this property we mean that convolution isdistributive over addition.

a) Discrete :   Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

b) Continuous : Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

A parallel combination of LTI systems can be replaced by an equivalent LTI system which is described by the sum of the individual impulse responses in the parallel combination.

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)                         Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

 

3) Associative property 

a) Discrete time :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Proof : We know that

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Making the substitutions: p = k ; q = (l - k) and comparing the two equations makes our proof complete.
Note: As k and l go -∞ from to +∞  independently of each other, so do p and q, however p depends on k, and q depends on l and k.

b) Continuous time :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 Lets substitute

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

The Jacobian for the above transformation is

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Doing some further algebra helps us see equation (2) transforming into equation (1) ,i.e. essentially they are the same. The limits are also the same. Thus the proof is complete.

Implications

This property (Associativity) makes the representation y[n] = x[n]*h[n] *g[n] unambiguous. From this property, we can conclude that the effective impulse response of acascaded LTI system is given by the convolution of their individual impulse responses.

 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Consequently the unit impulse response of a cascaded LTI system is independent of the order in which the individual LTI systems are connected. Note :All the above three properties are certainly obeyed by LTI systemsbuthold for non-LTI systems in, as seen from the following example:

 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)
 

4) LTI systems and Memory 
Recall that a system is memoryless if its output depends on the current input only. From the expression :

It is easily seen that y[n] depends only on x[n] if and only if  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)
Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

5) Invertibility : 

A system is said to be invertible if there exists an inverse system which when connected in series with the original system produces an output identical to the input.

We know that

  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

6) Causality :

a) Discrete time :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

In order for a discrete LTI system to be causal, y[n] must not depend on x[k] for k > n. For this to be true h[n-k]'s corresponding to the x[k]'s for k > n must be zero. This then requires the impulse response of a causal discrete time LTI system satisfy the following conditions :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Essentially the system output depends only on the past and the present values of the input.

Proof : ( By contradiction )

Let in particular h[k] is not equal to 0, for some k<0

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

So we need to prove that for all x[n] = 0, n < 0, y[0] = 0

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Now we take a signal defined as

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

This signal is zero elsewhere. Therefore we get the following result :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

We have come to the result that y[0] ≠ 0, for the above assumption. ∴ our assumption stands void. So we conclude that y[n] cannot be independent of x[k] unless h[k] = 0 for k < 0

Note : Here we ensured a non-zero summation by choosing x[n-k]'s as conjugate of h[k]'s.

b) Continuous time :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

In order for a continuous LTI system to be causal, y(t) must not depend on x(v) for v > t . For this to be true h(t-v)’s corresponding to the x(v)’s for v > t must be zero. This then requires the impulse response of a causal continuous time LTI system satisfy the following conditions :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

As stated before in the discrete time analysis,the system output depends only on the past and the present values of the input.
Proof : ( By contradiction )

Suppose, there exists a > 0 such that h(-a)≠0

Now consider  x(t) = δ(t-α)

Since,

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

  • System is not causal, a contradiction. Hence,

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

7) Stability : 

A system is said to be stable if its impulse response satisfies the following criterion :

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Theorem: 

 

Stability Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

Stability Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

Proof of sufficiency:

Suppose   Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

We have  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

 

If x[n] is bounded i.e. 

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

But as Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

Proof of Necessity: 

Take any n

Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

If | h[k] | = 0, then x[n-k] is bounded with bound   Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

Then,  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

 

Hence  Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) But since the system is stableProperties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) which in turn implies thatProperties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE). Hence if y[n] is bounded then the condition Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) must hold.

Hence Proved A similar proof follows in continuous time when you replace Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) by integral .Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE)

The document Properties of LTI System - Notes | Study Signals and Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Signals and Systems.
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