Consider the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

**Input-Output Relationship**

The given system has the following input-output relationship:

y [n] = (1 (-1)n) x [n]

where x[n] is the input and y[n] is the output.

**Invertibility**

A system is said to be invertible if different inputs produce different outputs. In other words, if two different inputs x1[n] and x2[n] result in the same output y[n], then the system is not invertible.

Let's consider two different inputs: x1[n] = 1 and x2[n] = (-1)n.

For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1

For x2[n] = (-1)n,
y2[n] = (1 (-1)n) * (-1)n
= -(-1)n
= (-1)n+1

For n even, y2[n] = (-1)n+1 = -1
For n odd, y2[n] = (-1)n+1 = 1

Therefore, for n even, y1[n] = y2[n], and for n odd, y1[n] = -y2[n]. This implies that the system is not invertible, as two different inputs x1[n] = 1 and x2[n] = (-1)n result in the same output for n even.

**Time Invariance**

A system is said to be time-invariant if a time shift in the input results in a corresponding time shift in the output. In other words, if x[n] produces y[n], then x[n - k] should produce y[n - k], where k is a constant.

Let's consider the input x1[n] = 1.

For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1

Now, let's consider the input x2[n] = 1, delayed by k samples.

For x2[n - k] = 1,
y2[n - k] = (1 (-1)n) * 1
= 1

Therefore, for both x1[n] = 1 and x2[n] = 1 delayed by k samples, the output y[n] remains the same. Thus, the system is time-invariant.

**Conclusion**

Based on the analysis, the given system is non-invertible and time-invariant. Therefore, the correct answer is option D: non-invertible and time varying.

Similar GATE Doubts

A system with an input x(t) and output y(t) is described by the relation: y(t) = tx(t). This system is

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?

Consider a linear time-invariant system whose input r (t ) and output y (t) are related by the following differential equation.The poles of this system are at

Consider the code segmentint i, j, x, y, m, n;n = 20;for (i = 0, i {for (j = 0; j {if (i % 2){x + = ((4j) + 5i);}}}m = x + y;Which one of the following is false?

Top Courses for GATE

Suggested Free Tests

Related Content

Mock Test Series

Subject-wise Courses

Other Engg Branches

Related Exams

Company