Properties of LTI System | Signals and Systems - Electrical Engineering (EE) PDF Download

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

Proof : We know that

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

So it is clear from the derived expression that

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

Proof:

We Know That 

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

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

                                       dt  = - d λ 

 

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

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

 

 

3) Associative property 

a) Discrete time :

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

Proof : We know that

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

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

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

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

 Lets substitute

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

The Jacobian for the above transformation is

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

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

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

 

6) Causality :

a) Discrete time :

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

Now we take a signal defined as

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

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

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

  • System is not causal, a contradiction. Hence,

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

Theorem: 

 

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

 

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

 

Proof of sufficiency:

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

 

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

 

 

If x[n] is bounded i.e. 

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

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

 

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

 

Proof of Necessity: 

Take any n

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

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

 

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

 

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

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

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

1. What are the properties of an LTI system?
Ans. The properties of a Linear Time-Invariant (LTI) system include linearity, time-invariance, causality, stability, and memorylessness. Linearity means that the system follows the principle of superposition, where the output is a linear combination of the inputs. Time-invariance implies that the system's behavior remains constant over time. Causality states that the output of the system depends only on present and past inputs, not future inputs. Stability indicates that the system's output remains bounded for any bounded input. Memorylessness means that the output at any given time depends only on the input at that specific time.
2. How can we determine if a system is linear or not?
Ans. To determine if a system is linear or not, we can apply the principle of superposition. If the system satisfies the condition of superposition, where the output is a linear combination of the inputs, then it is linear. Mathematically, if y1(t) represents the response of the system to input x1(t) and y2(t) represents the response to input x2(t), then for any constants a and b, the system is linear if y(a*x1(t) + b*x2(t)) = a*y1(t) + b*y2(t) holds true.
3. What is the significance of time-invariance in an LTI system?
Ans. Time-invariance is significant in an LTI system because it ensures that the system's behavior remains constant over time. It means that the system's response to an input signal does not change with time. This property simplifies the analysis and design of LTI systems as we can focus on understanding the system's behavior at a specific time and then apply that knowledge to any other time. Time-invariance allows us to use time-domain techniques, such as convolution, to analyze and manipulate signals in an LTI system.
4. How does causality affect the behavior of an LTI system?
Ans. Causality is an important property of an LTI system as it determines the cause-effect relationship between the input and output signals. A causal LTI system means that the output at any given time depends only on the present and past inputs, not future inputs. This property aligns with our intuitive understanding of cause and effect, where the output is a consequence of the input. Causality allows us to predict the system's response based on the input history, making it easier to analyze and design LTI systems.
5. Why is stability a desirable property in an LTI system?
Ans. Stability is a desirable property in an LTI system because it ensures that the output remains bounded for any bounded input. In other words, a stable system does not exhibit uncontrolled or excessive oscillations or growth. Stability is crucial to prevent instability in feedback control systems, where the output of the system is used to control the input. It ensures that the system's response does not become unpredictable or chaotic. Stable LTI systems provide reliable and predictable behavior, making them suitable for various applications in engineering and signal processing.
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