CM · Modulation basics

Why We Modulate

Broadcast a low-frequency signal such as a voice as it is, and the antenna would have to be tens of kilometers long, while every station would crowd into the same low band and garble the others. Multiplying the signal by a carrier lifts it up the frequency axis, and that single step solves both problems.

Fitting your station into an open slot

The plot is a spectrum laid out along the frequency axis. The dashed bump at the far left is the baseband signal before it is moved, and the two gray bumps are stations A and B, already on the air. Move the carrier frequency and your signal's bump moves to that frequency as a whole: red when it overlaps a neighbor, orange when it is close, gold when it sits cleanly apart.

Carrier frequency fcfc = 0.30

It starts on top of station A, in red. It turns gold midway between the two stations or at the far right end.

The frequency shift the multiply makes

S(f) = ½ M(f − fc) + ½ M(f + fc)

Overlapping

Why baseband alone won't work

An antenna radiates well only when its length is a sizable fraction of the wavelength, typically about 1/4. A voice occupies a few kilohertz, so its wavelength runs to tens or hundreds of kilometers, and no one can build an antenna that size. Worse, if every station used the same low band, a receiver would hear all the voices mixed together. The dashed bump at the left of the plot marks exactly that baseband position.

Multiplication shifts the spectrum

Multiply a signal m(t) by the carrier cos(2π fc t), and the spectrum of m slides up the frequency axis by fc with its shape unchanged. The dashed gold arrow joining the baseband bump to your signal's bump in the plot stands for that shift. One multiplication in time becomes a relocation in frequency, and that one line holds the whole idea of modulation.

Observes(t) = m(t) × cos(2π fc t)

Multiplying by a carrier is modulation.

Choosem(t) × cos → M(f − ?)

A product in time becomes a shift in frequency.

So many signals can sit side by side

Give each station its own carrier frequency and each spectrum rises to its own place. As long as neighboring carriers are spaced wider than the bandwidth, the bumps never overlap. The gap between stations A and B in the plot, with room for your signal, is such a place. Sharing the spectrum this way is called frequency-division multiplexing.

Fill inS(f) = ½ M(f − fc) + ½ M(f ? fc)

A cosine holds both +fc and −fc, so two copies of the spectrum appear. The plot shows only the positive-frequency side.

On your ownBtot = ?

N channels of bandwidth B placed side by side take N×B in total.

Back to the first screen

At first your signal sat on top of station A and showed red. Raising the carrier frequency into the gap between the two stations turned the bump gold, and its width matched the baseband bump exactly. That happened because multiplying by the carrier carried the whole spectrum into the open slot. Modulation, in the end, is moving a signal to a free place on the frequency axis.

Modulation multiplies a signal m(t) by a carrier cos(2π fc t), shifting its spectrum up by fc. The shape is kept and only the position changes, so giving each station a different fc lets many signals be multiplexed without overlap.

In the next unit

The multiplication that moves a spectrum is settled. The next unit, AM, looks at the same multiplication as a waveform in time. Load the signal onto the carrier's amplitude, and the curve joining the waveform's peaks, the envelope, follows the shape of the original signal.