Which condition must hold for a system to be linear?

a1x1+a2x2 gives a1y1+a2y2 for all inputs and constants

Shifting the input shifts the output by the same amount

Bounded input always gives bounded output

Output depends only on present and past inputs

What does x[n]=0 give for y[n]=2x[n]+5, and what does that show?

y=5, which is not 0, so the system is not linear

y=5, which shows the system is time-varying

y=2, which shows the system has memory

In the superposition test the weighted sum of outputs contains 5(a1+a2) while the response to the combined input contains only 5. When do these agree?

Only when a1+a2=1, not for all constants

For y[n]=2x[n]+5, x=1 gives y=7. What does x=2 give, and what does this show?

y=9, not 14, so homogeneity fails

y=9, so the system is time-varying

y=12, so the system is incrementally linear

2076 Bhadra · 4 marks

Is y[n] = 2x[n] + 5 linear

Short answer

The system y[n] = 2x[n] + 5 is non‑linear because zero input yields output 5, and superposition does not hold; it is incrementally linear.

Check whether the discrete-time system described by the following input-output relation is linear or non-linear. y[n] = 2 × x[n] + 5

A system is linear if it satisfies superposition (additivity and homogeneity):

a_1x_1[n]+a_2x_2[n]\ \rightarrow\ a_1y_1[n]+a_2y_2[n]

Test

Let $x_1[n]\rightarrow y_1[n]=2x_1[n]+5$ and $x_2[n]\rightarrow y_2[n]=2x_2[n]+5$ .

Response to the combined input $x_3=a_1x_1+a_2x_2$ :

y_3[n]=2\big(a_1x_1[n]+a_2x_2[n]\big)+5=2a_1x_1[n]+2a_2x_2[n]+5

Weighted sum of the individual outputs:

a_1y_1[n]+a_2y_2[n]=2a_1x_1[n]+2a_2x_2[n]+5(a_1+a_2)

These are equal only if $a_1+a_2=1$ , not for all constants. So $y_3[n]\ne a_1y_1[n]+a_2y_2[n]$ in general.

Quick check (zero-input test): for a linear system, zero input gives zero output. Here $x[n]=0$ gives $y[n]=5\ne0$ .

Numerical example: $x[n]=1\rightarrow y=7$ ; doubling the input, $x[n]=2\rightarrow y=9\ne 14$ .

Answer: the system $y[n]=2x[n]+5$ is non-linear. It is called incrementally linear: it is a linear system ( $2x[n]$ ) plus a constant offset (5), so differences of outputs do respond linearly to differences of inputs. (It is memoryless, causal, time-invariant and stable.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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Is y[n] = 2x[n] + 5 linear

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