Is the System y(t) = x(t) + ∫(t - τ)x(τ)dτ Linear?

[magnifik]

y(t) = x(t) + $$\int$$ (t - $$\tau$$)x($$\tau$$)d$$\tau$$

for it to be linear, T[kx(t)] = kT[x(t)]

i'm wondering if i also multiply x(tau) by k.
at first i thought I'm not supposed to so i have

T[kx(t)] = kx(t) + $$\int$$ (t - $$\tau$$)x($$\tau$$)d$$\tau$$
and
kT[x(t)] = k[x(t) + $$\int$$ (t - $$\tau$$)x($$\tau$$)d$$\tau$$] = kx(t) + k$$\int$$ (t - $$\tau$$)x($$\tau$$)d$$\tau$$
so they aren't equal and aren't linear. however, I'm not sure about this answer because I'm not sure if x($$\tau$$) should also be multiplied by k, which would make it linear (under scaling...i tried superposition and using multiplying k by the tau-dependent x's is also linear)

any help would be appreciated. thx.

 

[KataKoniK]

Hello,

Thank you for your post. I can help clarify the concept of linearity in this context.

To determine if a system is linear, we need to consider two properties: scaling and superposition.

Scaling property: This means that if we multiply the input signal by a constant, the output should also be multiplied by the same constant. In this case, if we multiply x(t) by k, the output should also be multiplied by k. This is what the equation T[kx(t)] = kT[x(t)] represents.

Superposition property: This means that the output of the system for a sum of two input signals should be equal to the sum of the outputs for each individual input signal. In this case, if we have two input signals x1(t) and x2(t), the output for the sum of these signals should be equal to the sum of the outputs for x1(t) and x2(t) separately. Mathematically, this is represented as T[x1(t) + x2(t)] = T[x1(t)] + T[x2(t)].

Now, let's apply these properties to the equation y(t) = x(t) + \int (t - \tau)x(\tau)d\tau.

Scaling property: If we multiply x(t) by k, the output becomes y(t) = kx(t) + \int (t - \tau)kx(\tau)d\tau. This satisfies the scaling property since the output has been multiplied by k, just like the input.

Superposition property: If we have two input signals x1(t) and x2(t), the output for the sum of these signals becomes y(t) = x1(t) + x2(t) + \int (t - \tau)(x1(\tau) + x2(\tau))d\tau. Expanding the integral, we get y(t) = x1(t) + x2(t) + \int (t - \tau)x1(\tau)d\tau + \int (t - \tau)x2(\tau)d\tau. This satisfies the superposition property since the output is equal to the sum of the outputs for x1(t) and x2(t) separately.

Therefore, we can conclude that the equation y(t) = x(t) + \int (t - \tau)x(\tau)d\tau is linear under both scaling and superposition,