To determine whether it's LTI system

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In summary, to prove that the given system is linear and time-invariant, we need to show that it satisfies the superposition principle for linearity and that the output at a specific time is the same regardless of when the input is applied for time-invariance.
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glorimda
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Homework Statement



this system satisfies the condition of initial rest.

Y[n] =(1/2)y[n-1] + x[n]

I'm trying to prove that this is linear and time invariant.

The Attempt at a Solution



I'm trying it this way.
To be LTI system,
It has to satisfy below.
aX1[n] -> aY1[n]
bX2[n] -> bY2[n]
aX1 + bX2 -> aY1 + bY2

1. firstly, I don't see
aX[n] -> aY[n]

Since if I put aX[n] then (1/2)Y[n-1] + aX[n] =! {aY[n]= a(1/2)Y[n-1]+aX[n]}
i.e., aX[n] -> aY[n] is not true in my view.

2. Secondly,
Just let me try this without multiplying constant

X1[n] -> Y1[n]
X2[n] -> Y2[n]
X1 + X2 -> Y1 + Y2

Y1[n] = (1/2)Y1[n-1] + X1[n]
Y2[n] = (1/2)Y2[n-1] + X2[n]

X3[n] = X1[n] + X2[n]

Y3[n] = (1/2)Y3[n-1] + X3[n] = (1/2)Y3[n-1] + {X1[n] + X2[n]}

After this I don't seem to find what to do with Y3[n-1] and since this, I can't prove
Y3[n] = Y1[n] + Y2[n]

Could somebody help me and tell me what I'm doing wrong?
Appreciate for it.
 
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  • #2


Hi there,

First of all, let me clarify that I am a scientist and not a mathematician, so I will try my best to explain this in a simple and understandable way.

To prove that this system is linear and time-invariant, we need to show that it satisfies both properties, linearity and time-invariance.

Linearity:
To prove linearity, we need to show that the system satisfies the superposition principle, which states that the output of the system when two inputs are applied simultaneously is equal to the sum of the outputs when each input is applied separately.

In this case, let's consider two inputs, X1[n] and X2[n], and their corresponding outputs, Y1[n] and Y2[n].

Now, if we apply both inputs simultaneously, the output will be:

Y[n] = (1/2)Y[n-1] + X1[n] + X2[n]

But, if we apply each input separately, the outputs will be:

Y1[n] = (1/2)Y1[n-1] + X1[n]
Y2[n] = (1/2)Y2[n-1] + X2[n]

Therefore, according to the superposition principle, the output when both inputs are applied simultaneously is equal to the sum of the outputs when each input is applied separately. Hence, the system is linear.

Time-invariance:
To prove time-invariance, we need to show that the output of the system when an input is applied at a specific time is the same as the output when the same input is applied at a different time.

In this case, let's consider an input, X[n], and its corresponding output, Y[n]. Now, if we apply the input X[n] at time n, the output will be:

Y[n] = (1/2)Y[n-1] + X[n]

But, if we apply the same input X[n] at a different time n+1, the output will be:

Y[n+1] = (1/2)Y[n] + X[n+1]

We can see that the output at time n+1 is dependent on the output at time n, which means the system is time-invariant.

Therefore, we can conclude that the system described by the given equation is linear and time-invariant.

I hope this explanation helps. Let me know if you have any further questions. Good
 

1. What is an LTI system?

An LTI (Linear Time-Invariant) system is a mathematical model used to describe a system that produces an output based on a linear combination of its input signals, and whose behavior does not change over time. This means that the output of the system is only dependent on its input, and is not affected by any external factors.

2. How do you determine if a system is LTI?

To determine if a system is LTI, we can use the mathematical properties of linearity and time-invariance. Linearity means that the output of the system is a linear combination of its input signals, while time-invariance means that the behavior of the system does not change over time. If a system satisfies both of these properties, it can be considered an LTI system.

3. What is the importance of determining if a system is LTI?

Determining if a system is LTI is important because it allows us to use various mathematical tools and techniques to analyze and predict its behavior. LTI systems have well-defined characteristics and responses, making them easier to study and control compared to non-LTI systems.

4. How do external factors affect the linearity and time-invariance of a system?

External factors such as noise, disturbances, and nonlinear inputs can affect the linearity and time-invariance of a system. If the output of a system is not a linear combination of its input signals or if its behavior changes over time due to these external factors, then it cannot be considered an LTI system.

5. Can a system be partially LTI?

No, a system cannot be partially LTI. It either satisfies the properties of linearity and time-invariance or it does not. If a system does not satisfy both of these properties, it cannot be considered an LTI system.

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