For each of the following discrete time systems, where y[n] and x[n] are respectively theoutput and input sequences, determine whether the system is (i) linear, (ii) causal, (iii)stable and (iv) shift-invariant.(a) y[n] = n2x[n],(b) y[n] = x4[n],(c) y[n] = αx[-n], α is a non-zero constant and(d) y[n] = x[n-5]e) y[n] = x[2n]f) y[n] = x[n]x[n-1]
Question
For each of the following discrete time systems, where y[n] and x[n] are respectively theoutput and input sequences, determine whether the system is (i) linear, (ii) causal, (iii)stable and (iv) shift-invariant.(a) y[n] = n2x[n],(b) y[n] = x4[n],(c) y[n] = αx[-n], α is a non-zero constant and(d) y[n] = x[n-5]e) y[n] = x[2n]f) y[n] = x[n]x[n-1]
Solution
(a) To determine whether the system is linear, we need to check if it satisfies the properties of additivity and homogeneity.
Additivity: Let's assume we have two input sequences x1[n] and x2[n], and their corresponding output sequences y1[n] and y2[n]. If we apply the system to the sum of the input sequences, x1[n] + x2[n], the output sequence should be the sum of the individual outputs, y1[n] + y2[n].
In this case, the system is y[n] = n^2x[n]. Let's consider two input sequences x1[n] and x2[n], and their corresponding output sequences y1[n] and y2[n].
For x1[n] + x2[n], the output sequence would be (n^2)(x1[n] + x2[n]).
On the other hand, if we apply the system to x1[n] and x2[n] separately, the output sequences would be (n^2)x1[n] and (n^2)x2[n] respectively.
Therefore, the system is additive.
Homogeneity: Let's assume we have an input sequence x[n] and its corresponding output sequence y[n]. If we apply the system to a scaled version of the input sequence, ax[n], the output sequence should be scaled by the same factor, ay[n].
In this case, if we apply the system to ax[n], the output sequence would be (n^2)(ax[n]) = a(n^2)x[n].
Therefore, the system is homogeneous.
Since the system satisfies both additivity and homogeneity, it is linear.
To determine if the system is causal, we need to check if the output at any given time depends only on the current and past values of the input.
In this case, the system is y[n] = n^2x[n]. The output at time n depends on the current and past values of the input, x[n], x[n-1], x[n-2], and so on. Therefore, the system is causal.
To determine if the system is stable, we need to check if bounded inputs result in bounded outputs.
In this case, the system is y[n] = n^2x[n]. Since the output is proportional to the square of the input, it is not guaranteed that bounded inputs will result in bounded outputs. Therefore, the system is not stable.
To determine if the system is shift-invariant, we need to check if shifting the input sequence results in a corresponding shift in the output sequence.
In this case, the system is y[n] = n^2x[n]. If we shift the input sequence by k, the output sequence would be (n-k)^2x[n-k]. Since the output sequence is not a simple shift of the input sequence, the system is not shift-invariant.
(b) Following the same steps as above, we can determine that the system y[n] = x^4[n] is nonlinear, causal, unstable, and shift-invariant.
(c) The system y[n] = αx[-n] is linear, causal, stable, and shift-invariant.
(d) The system y[n] = x[n-5] is linear, causal, stable, and shift-invariant.
(e) The system y[n] = x[2n] is linear, causal, stable, and shift-invariant.
(f) The system y[n] = x[n]x[n-1] is nonlinear, causal, unstable, and shift-invariant.
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