A system with input $$x(t)$$ and output $$y(t)$$ is defined by the input $$-$$ output relation:
$$y\left( t \right) = \int\limits_{ - \infty }^{ - 2t} {x\left( \tau \right)} d\tau .$$ The system will be

The transfer function of a linear time invariant system is given as $$G\left( s \right) = {1 \over {{s^2} + 3s + 2}}.$$ The steady state value of the output of this system for a unit impulse input applied at time instant $$t=1$$ will be

$$X\left( z \right) = 1 - 3\,\,{z^{ - 1}},\,\,Y\left( z \right) = 1 + 2\,\,{z^{ - 2}}$$ are $$Z$$-transforms of two signals $$x\left[ n \right],\,\,y\left[ n \right]$$ respectively. A linear time invariant system has the impulse response $$h\left[ n \right]$$ defined by these two signals as $$h\left[ n \right] = x\left[ {n - 1} \right] * y\left[ n \right]$$ where $$ * $$ denotes discrete time convolution. Then the output of the system for the input $$\delta \left[ {n - 1} \right]$$

A signal is processed by a causal filter with transfer function $$G(s).$$ For a distortion free output signal waveform, $$G(s)$$ must.

$$G\left( z \right) = a{z^{ - 1}} + \beta \,\,{z^{ - 3}}$$ is a low-pass digital filter with a phase characteristic same as that of the above question if