Define AM; switching modulator for DSB-FC

Amplitude modulation

Amplitude modulation is the process in which the amplitude of a high-frequency carrier is varied in proportion to the instantaneous value of the message, keeping frequency and phase constant:

$$s(t) = A_c[1 + k_am(t)]\cos 2\pi f_ct$$

Switching modulator

A switching modulator uses a diode as an ideal switch driven by a large carrier. The carrier and the message are added and applied to the diode with a resistive load, followed by a band-pass filter.

          D
 m(t)-->(+)--|>|----+-------> [ BPF  ] ---> s(t)
         ^          |         [ at fc]
 Accos   |          > RL      [BW=2W ]
 (wct)---+          >
                    |
                   GND

Assumptions: $A_c \gg |m(t)|$, so the carrier alone decides when the diode is on; the diode is ideal (zero resistance when on, open when off).

Input

$$v_1(t) = A_c\cos\omega_ct + m(t)$$

Diode as a switch: the diode conducts in the positive half cycles of the carrier and is off in the negative ones:

$$v_2(t) = \begin{cases} v_1(t), & c(t) > 0 \\ 0, & c(t) < 0 \end{cases}$$

So $v_2(t) = v_1(t)\,g_p(t)$, where $g_p(t)$ is a periodic pulse train of 50% duty cycle at $f_c$. Its Fourier series is

$$g_p(t) = \frac{1}{2} + \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{2n-1}\cos[(2n-1)\omega_ct] = \frac{1}{2} + \frac{2}{\pi}\cos\omega_ct - \frac{2}{3\pi}\cos3\omega_ct + \dots$$

Diode output

$$\begin{aligned} v_2(t) &= [A_c\cos\omega_ct + m(t)]\left[\frac{1}{2} + \frac{2}{\pi}\cos\omega_ct - \dots\right] \\ &= \frac{A_c}{2}\cos\omega_ct + \frac{2}{\pi}m(t)\cos\omega_ct \\ &\quad + \frac{1}{2}m(t) + \frac{2A_c}{\pi}\cos^2\omega_ct + \text{terms near } 3f_c, 5f_c, \dots \end{aligned}$$

The term $\frac{2A_c}{\pi}\cos^2\omega_ct = \frac{A_c}{\pi}(1 + \cos2\omega_ct)$ gives DC and $2f_c$ components.

Spectrum of $v_2(t)$: baseband $m(t)$ and DC (0 to $W$), the wanted AM around $f_c$, components at $2f_c$, $3f_c$ and so on.

Band-pass filter: centred at $f_c$ with bandwidth $2W$, it removes everything except the first two terms:

$$s(t) = \frac{A_c}{2}\cos\omega_ct + \frac{2}{\pi}m(t)\cos\omega_ct = \frac{A_c}{2}\left[1 + \frac{4}{\pi A_c}m(t)\right]\cos\omega_ct$$

This is a DSB-FC AM wave with amplitude sensitivity

$$k_a = \frac{4}{\pi A_c}$$

Conditions: $f_c > 2W$ so that the AM band ($f_c \pm W$) does not overlap the baseband or the $2f_c$ term, and $k_a|m(t)| \le 1$ to avoid over-modulation.