A base band signal g(t) band limited to 100 Hz modulates a carrier of frequency $${{f_0}}$$ Hz. The modulated signal g(t) $$\cos 2\,\pi \,{f_0}t$$ is transmitted over a channel whose input x and output y are related by $$y = 2x + {x^2}$$. The spectrum of g(t) is shown in Fig. Sketch the spectrum of the transmitted signal and the spectrum of the received signal.

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.

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.

A signal, f(t) = $${e^{ - at}}$$ u(t), where u(t) is the unit step function, is applied to the input of a low-pass filter having $$\left| {H(\omega )} \right| = {b \over {\sqrt {{\omega ^2} + {b^2}} }}$$.

Calculate the value of the ratio, $${a \over b}$$, for which 50% of the input signal energy is transferred to the output.