Modulation

Throughout this problem two signals are used. Be careful which signal is referred to in each of the parts. The first signal, x₁(t), is a strictly band-limited signal, i.e., the spectrum X₁(f) of this signal equals zero for frequencies |f| > f_m. The second signal, x₂(t), is a square wave with period T and amplitude 1, i.e., the signal alternates between amplitudes +1 and -1.

1.

The square wave signal x₂(t) is to be amplitude modulated and non-coherent envelope detection is to be used at the receiver. The carrier frequency equals $f_0=\frac{2}{T}$. Give an expression for the amplitude modulated signal. For any parameters in your expression indicate what value they should assume.

2.

Sketch and accurately label the amplitude modulated signal from part (a).

3.

Assume now that the band-limited signal x₁(t) is to be transmitted. Again, amplitude modulation with carrier frequency $f_0 \gg f_m$ is to be employed. Compute an expression for the spectrum of the amplitude modulated signal in terms of X₁(f). Sketch and accurately label the magnitude spectrum assuming a ``typical'' X₁(f).

4.

Now, the square wave signal x₂(t) is to be frequency modulated. Give an expression for the frequency modulated signal. For any parameters in your expression indicate what value they should assume.

5.

Assume now that the carrier frequency equals $f_0=\frac{2}{T}$ and that the frequency deviation constant $f_d=\frac{1}{T}$ (i.e., $k_f=\frac{2\pi}{T}$). Sketch and accurately label the frequency modulated signal from part (d).