Given the following systemT{X[n]} = X[n] + 3u[n+1]Where u[x] represents unit step functions-Which of the following is a correct representation of the system?a)Non Linear Time Invariant Systemb)Non Linear Time Variant Systemc)Linear Time Invariant Systemd)Linear Time Variant SystemCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Introduction:
In this question, we are given a system represented by the equation T{X[n]} = X[n] + 3u[n - 1], where T{} represents the system and X[n] represents the input signal. We need to determine whether the given system is a linear time-invariant (LTI) system, a linear time-variant (LTV) system, a non-linear time-invariant (NLTI) system, or a non-linear time-variant (NLTV) system.

Solution:

1. Linearity:
To determine linearity, we need to check if the system satisfies the superposition principle. Let's consider two input signals X1[n] and X2[n] with corresponding outputs Y1[n] and Y2[n].

Superposition principle:
If T{aX1[n] + bX2[n]} = aT{X1[n]} + bT{X2[n]}

Let's substitute the given values into the equation:
T{aX1[n] + bX2[n]} = aX1[n] + bX2[n] + 3u[n - 1]
aT{X1[n]} + bT{X2[n]} = a(X1[n] + 3u[n - 1]) + b(X2[n] + 3u[n - 1])

Comparing the two equations, we can see that they are not equal. Therefore, the system does not satisfy the superposition principle, indicating that it is a non-linear system.

2. Time-Invariance:
To determine time-invariance, we need to check if a time shift in the input signal results in a corresponding time shift in the output signal. Let's consider an input signal X[n] and its corresponding output Y[n].

Let's shift the input signal by k units to the right and denote it as X[n - k]. The output of the shifted input signal will be T{X[n - k]} = X[n - k] + 3u[n - k - 1].

If the system is time-invariant, the output should be Y[n - k]. Let's compare the two equations:
Y[n - k] = X[n - k] + 3u[n - k - 1]

Comparing the two equations, we can see that they are not equal. Therefore, the system does not exhibit time-invariance, indicating that it is a time-variant system.

Conclusion:
Based on the analysis above, we can conclude that the given system is a non-linear time-variant (NLTV) system.

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