---
title: "Single tone DSB-FC AM in time and frequency domain"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:single-tone-dsbfc-time-freq"
source: https://www.bctengineer.com/q/semester_6/communication_systems/single-tone-dsbfc-time-freq
site: BCT Engineering
---

# Single tone DSB-FC AM in time and frequency domain

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2075 Baisakh (CS I), 3+3+3 marks
- Video: [LECT-4  : SINGLE TONE AM  (DSB-FC) , TOTAL POWER  & TRANSMISSION EFFICIENCY.](https://www.youtube.com/watch?v=qLQcTXmKzEc) — EPOV CHANNEL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/single-tone-dsbfc-time-freq
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Find the time domain and frequency domain expressions for single tone DSB-FC AM modulated wave. Also, show the spectrum of the modulated signal.

## Short answer

For single-tone DSB-FC AM, time domain is s(t)=Ac[1+μ cos(2πfmt)]cos(2πfct) expanding to carrier plus upper and lower sidebands. Frequency domain has impulses at ±fc (weight Ac/2) and ±(fc±fm) (weight μAc/4). Bandwidth is 2fm.

## Answer

#### Time-domain expression

Let the message and carrier be

$$
m(t) = A_m\cos2\pi f_mt, \qquad c(t) = A_c\cos2\pi f_ct, \quad f_c \gg f_m
$$

In DSB-FC AM the carrier amplitude varies linearly with $m(t)$:

$$
\begin{aligned}
s(t) &= [A_c + A_m\cos2\pi f_mt]\cos2\pi f_ct \\
&= A_c[1 + \mu\cos2\pi f_mt]\cos2\pi f_ct, \quad \mu = \frac{A_m}{A_c}
\end{aligned}
$$

Using $\cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)]$:

$$
s(t) = A_c\cos2\pi f_ct + \frac{\mu A_c}{2}\cos2\pi(f_c+f_m)t + \frac{\mu A_c}{2}\cos2\pi(f_c-f_m)t
$$

Three components: carrier, upper side frequency (USF), lower side frequency (LSF).

```text
 s(t):   envelope Ac(1 + u cos wm t)
       Ac(1+u) _      ____
              / \/\/\/    \/\
   ..........|..............|....  0
              \ /\/\/\    /\/
       -Ac(1+u) -      ----
```

#### Frequency-domain expression

Taking the Fourier transform, using $\cos2\pi f_0t \leftrightarrow \tfrac12[\delta(f-f_0) + \delta(f+f_0)]$:

$$
\begin{aligned}
S(f) = {} & \frac{A_c}{2}[\delta(f-f_c) + \delta(f+f_c)] \\
& + \frac{\mu A_c}{4}[\delta(f-f_c-f_m) + \delta(f+f_c+f_m)] \\
& + \frac{\mu A_c}{4}[\delta(f-f_c+f_m) + \delta(f+f_c-f_m)]
\end{aligned}
$$

#### Spectrum

```text
                 S(f)
         Ac/2              Ac/2
          |                 |
   uAc/4  |  uAc/4   uAc/4  |  uAc/4
     |    |    |       |    |    |
 ----+----+----+---0---+----+----+---> f
 -fc-fm -fc -fc+fm   fc-fm  fc  fc+fm
```

- Carrier impulses of weight $A_c/2$ at $\pm f_c$.
- Side-frequency impulses of weight $\mu A_c/4$ at $\pm(f_c \pm f_m)$.
- Bandwidth: $B = (f_c + f_m) - (f_c - f_m) = 2f_m$.

**Power** (load $R$): $P_t = \dfrac{A_c^2}{2R}\left(1 + \dfrac{\mu^2}{2}\right)$, efficiency $\eta = \dfrac{\mu^2}{2 + \mu^2}$ (max 33.3% at $\mu = 1$).

## Check yourself

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

1. For m(t) = Am cos(2 pi fm t) and carrier Ac cos(2 pi fc t), the DSB-FC wave is:
   A. Am cos(2 pi fm t) cos(2 pi fc t)
   B. Ac[1 + mu cos(2 pi fm t)] cos(2 pi fc t) with mu = Am/Ac
   C. Ac cos(2 pi (fc + fm) t) only
   D. Ac[1 + mu cos(2 pi fc t)] cos(2 pi fm t)
   Answer: B. The carrier amplitude varies linearly with the message, giving Ac[1 + mu cos(2 pi fm t)] cos(2 pi fc t). (Key point: Time-domain expression)

2. Expanding the single-tone AM wave gives which components?
   A. Carrier and two side frequencies at fc + fm and fc - fm
   B. Only two sidebands, no carrier
   C. Carrier and message at fm
   D. Carrier at fc and a single sideband at fc + fm
   Answer: A. Using the product-to-sum identity, the wave has carrier amplitude Ac and two sidebands of amplitude mu Ac/2. (Key point: Three spectral components)

3. In the Fourier transform S(f), what are the impulse weights at +/-fc and at +/-(fc +/- fm)?
   A. Ac and mu Ac
   B. Ac/2 and mu Ac/2
   C. Ac/2 and mu Ac/4
   D. Ac/4 and mu Ac/2
   Answer: C. Each cosine splits into two impulses of half weight, so the carrier gives Ac/2 and each side frequency gives mu Ac/4. (Key point: Impulse weights)

4. For single-tone AM, the bandwidth and the maximum power efficiency are:
   A. fm and 100%
   B. 2 fm and 50%
   C. fm and 33.3%
   D. 2 fm and 33.3% at mu = 1
   Answer: D. The spectrum spans fc - fm to fc + fm, so B = 2 fm; efficiency mu^2/(2 + mu^2) is 33.3% at mu = 1. (Key point: Bandwidth and efficiency)

## Related in Communication Systems

- [Angle modulation; single tone FM time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-single-tone-fm-time-freq) — Communication Systems, chapter 3
- [Need for modulation; single tone AM expressions](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-need-am-single-tone) — Communication Systems, chapter 2, asked 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
- [Switching modulator for DSB-FC AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aswitching-modulator-dsbfc) — Communication Systems, chapter 2
- [Double-tone AM; DSB vs SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adouble-tone-am-dsb-vs-ssb) — Communication Systems, chapter 2
- [Demodulation of DSB-FC AM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-demodulation-method) — Communication Systems, chapter 2

## Related in other subjects

- [Modulation; AM and FM modulated waves](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Amodulation-am-fm-wave) — Basic Electronics Engineering, chapter 5