A discrete-time system is defined as y(n)= x(n^3). The system is 1 pointCausal, non-linear and time-invariantnon-causal, linear and time-invariantCausal, linear and time-invariantCausal, linear, and time-variantnon-causal, non-linear and time-variantnon-causal, linear and time-variant
Question
A discrete-time system is defined as y(n)= x(n^3). The system is 1 pointCausal, non-linear and time-invariantnon-causal, linear and time-invariantCausal, linear and time-invariantCausal, linear, and time-variantnon-causal, non-linear and time-variantnon-causal, linear and time-variant
Solution
The system y(n) = x(n^3) is non-causal, non-linear and time-variant.
Here's why:
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Causality: A system is causal if the output at any time depends only on the present and past inputs, not on future inputs. In this case, y(n) depends on x(n^3), which means the output at time n depends on the input at time n^3. Since n^3 can be greater than n, the system is non-causal.
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Linearity: A system is linear if it satisfies the principles of superposition and homogeneity. The given system does not satisfy these principles because if we input a sum of two signals or a scaled signal into the system, the output will not be the sum or the scaled version of the individual outputs. Hence, the system is non-linear.
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Time-invariance: A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. In this case, if we shift the input signal, the output will not be an identical shift because of the cubic power. Therefore, the system is time-variant.
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