Need for modulation; single tone AM expressions

Modulation

Modulation is the process in which some characteristic (amplitude, frequency or phase) of a high-frequency carrier wave is varied in accordance with the instantaneous value of the message (modulating) signal. In amplitude modulation (AM) the carrier amplitude is varied.

Need for modulation

1. Practical antenna height: antenna length $\approx\lambda/4$; for 10 kHz audio it would be 7.5 km, but for a 1 MHz carrier only 75 m.
2. Multiplexing: different stations use different carriers, so many signals share the same channel without interference.
3. Efficient radiation and long range: high-frequency signals radiate and propagate much better.
4. Narrowbanding: reduces the ratio of highest to lowest frequency, so one antenna and amplifier design works.
5. Noise reduction: wideband schemes (FM) improve SNR.
6. Channel matching: shifts the signal into the frequency band suited to the medium.

Time-domain expression of single-tone AM

Let the message be $m(t)=A_m\cos\omega_mt$ and carrier $c(t)=A_c\cos\omega_ct$, with $f_c\gg f_m$. In standard AM the envelope is $A_c+m(t)$:

$$\begin{aligned} s(t)&=[A_c+A_m\cos\omega_mt]\cos\omega_ct\\ &=A_c[1+\mu\cos\omega_mt]\cos\omega_ct,\qquad \mu=\frac{A_m}{A_c} \end{aligned}$$

where $\mu$ is the modulation index ($0\le\mu\le1$). Using $\cos A\cos B=\frac{1}{2}[\cos(A+B)+\cos(A-B)]$:

$$s(t)=\underbrace{A_c\cos\omega_ct}_{\text{carrier}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t}_{\text{USB}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t}_{\text{LSB}}$$

       Ac(1+mu)
   .-.   envelope    .-.
  /   \             /   \
 /     \    .-.    /     \   <- carrier inside envelope
        \  /   \  /
         '-     '-  Ac(1-mu)

Frequency-domain expression

Using $\cos2\pi f_0t\leftrightarrow\frac{1}{2}[\delta(f-f_0)+\delta(f+f_0)]$:

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

                 S(f)
       Ac/2               Ac/2
 muAc/4 |  muAc/4  muAc/4 |  muAc/4
   |    |    |       |    |    |
 --+----+----+---+---+----+----+--> f
  -fc-fm -fc -fc+fm 0 fc-fm fc fc+fm

Key results

• Bandwidth: $B=(f_c+f_m)-(f_c-f_m)=2f_m$.
• Power: $P_c=\frac{A_c^2}{2}$, each sideband $\frac{\mu^2A_c^2}{8}$, so total $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$.
• Efficiency: $\eta=\frac{\mu^2}{2+\mu^2}$; maximum 33.3% at $\mu=1$.

Example: $A_c=10$ V, $\mu=0.5$, $f_c=1$ MHz, $f_m=1$ kHz gives lines of 10 V at 1 MHz and 2.5 V at 0.999 MHz and 1.001 MHz; bandwidth 2 kHz.