A system having a unit impulse response $$h\left( n \right)$$ = $$u\left( n \right)$$ is excited by a signal $$x\left( n \right)$$ $$ = \,{\alpha ^n}\,\,u\left( n \right).\,$$ Determine the output $$y\left( n \right)$$

A rectangular pulse of duration T is applied to a filter matched to this input. The output of the filter is a

A signal 3 sin $$\left( {\pi \,\,{f_0}t} \right) + \,5\,\,\cos \,\,\,(3\pi \,\,{f_0}t)$$ is applied to an RC low pass filter of 3 dB cutoff frequency $${f_0}$$. Determine and plot the output power spectrum and aslo calculate the total input and output normalized power.

An input signal A exp $$\left( { - \alpha \,t} \right)$$ u(t) with $$\alpha > 0$$ is applied to a causal filter, the impulse response of which is A exp $$\,( - \alpha \,\,t)$$. Determine the filter output, sketch it as a function of time and lable the important points.