---
title: "Define AM; switching modulator for DSB-FC"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:define-am-switching-modulator"
source: https://www.bctengineer.com/q/semester_6/communication_systems/define-am-switching-modulator
site: BCT Engineering
---

# Define AM; switching modulator for DSB-FC

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2067 Mangsir (CS I), 2+8 marks
- Video: [Generation of AM signal: Switching Modulator Explained](https://www.youtube.com/watch?v=M4P5QCEo9WM) — ALL ABOUT ELECTRONICS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/define-am-switching-modulator
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Define Amplitude Modulation. With block diagram and necessary derivations, show that switching modulator can be used to generate Double Sideband Full Carrier AM signal.

## Short answer

Amplitude modulation varies carrier amplitude in proportion to message. In a switching modulator, a diode switched by a large carrier multiplies the sum of message and carrier with a square wave; after bandpass filter, output is s(t) = (Ac/2)[1 + (4/(pi Ac)) m(t)] cos(omega_c t), a DSB-FC wave.

## Answer

#### 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.

```text
          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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does amplitude modulation vary?
   A. The carrier frequency in proportion to the message
   B. The carrier phase in proportion to the message
   C. The carrier amplitude in proportion to the message, with frequency and phase constant
   D. The message amplitude in proportion to the carrier
   Answer: C. In AM the high-frequency carrier amplitude follows the message while frequency and phase stay constant. (Key point: Definition of AM)

2. Why is the carrier amplitude chosen much larger than |m(t)| in the switching modulator?
   A. So the carrier alone decides when the diode is on or off
   B. So the message is amplified
   C. So the BPF can be omitted
   D. So the diode becomes linear
   Answer: A. With Ac >> |m(t)| the diode conducts during positive carrier half cycles regardless of m(t), so it acts as a switch driven by the carrier. (Key point: Diode as a switch)

3. After the BPF at fc, which output remains?
   A. (1/2)m(t) + Ac/π
   B. (Ac/2)cos ωc t + (2/π) m(t) cos ωc t
   C. (2/π) m(t) cos 3ωc t
   D. (2Ac/π) cos^2 ωc t
   Answer: B. The pulse-train term 1/2 times Ac cos ωc t and the 2/π cos ωc t term times m(t) are the only components near fc. (Key point: Wanted term)

4. Writing the output as (Ac/2)[1 + ka m(t)] cos ωc t, what is ka?
   A. 2/(π Ac)
   B. 4/(π Ac)
   C. π/(4Ac)
   D. 1/(2Ac)
   Answer: B. Factoring Ac/2 out of (2/π) m(t) gives ka = (2/π)/(Ac/2) = 4/(π Ac). (Key point: Amplitude sensitivity)

## Related in Communication Systems

- [Switching modulator for DSB-FC AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aswitching-modulator-dsbfc) — Communication Systems, chapter 2
- [SSB vs full carrier AM; ring modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-vs-am-ring-modulator) — Communication Systems, chapter 2
- [Square-law modulator for DSB-AM; characteristics of AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asquare-law-dsbam-characteristics) — Communication Systems, chapter 2, asked 2×
- [DSB-FC generation using square law modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-square-law-modulator) — Communication Systems, chapter 2
- [DSB-AM vs SSB; ring modulator for DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbam-ssb-compare-ring-modulator) — Communication Systems, chapter 2
- [DSB-FC single tone expression and modulation index](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-single-tone-index) — Communication Systems, chapter 2

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