1.A "mixer" is an RF electric circuit that is commonly used in communication systems to shift the EM wave's frequency to a desired frequency band by multiplying an input sinusoidal wave with another sinusoidal wave. The configuration of a mixer is simply a combination of an electric multiplier and a band-pass filter (BPF) as shown in the following diagram. Assume the input vector space is generated by \(n\) input sinusoidal waves with the same non-zero amplitude \(A\), but different frequencies, \(\omega_1, \omega_2, \cdots, \omega_n\), and the output vector space is generated by these \(n\) "down-converted" sinusoidal waves. Note: "down-convert" means the frequency is reduced.