CM · Modulation basics

Why We Modulate

A low-frequency signal needs a huge antenna and collides with its neighbors. Riding it on a carrier slides its spectrum up the frequency axis — antennas shrink and many signals line up side by side.

Slot your station into the empty gap

Slide the carrier frequency to place your signal (gold) without overlapping the neighbors. One multiplication moves the whole signal to that frequency.

Carrier frequency fcfc = 0.30

Sweep the carrier with the slider.

The frequency shift the multiply makes

S(f) = ½ M(f − fc) + ½ M(f + fc)

Overlapping

Why baseband is awkward

An antenna works well only when it is a sizable fraction of a wavelength. Voice sits at a few kilohertz, whose wavelength stretches tens of kilometers — no one builds an antenna that big. Worse, if everyone sits in the same low band, the signals scramble together.

Multiplying shifts the spectrum

Multiply the signal m(t) by a carrier cos(2π fc t), and on the frequency axis the whole spectrum of m slides over by fc. The shape is untouched; only its location moves. That single line is the heart of modulation.

Observes(t) = m(t) × cos(2π fc t)

Multiplying by the carrier is modulation.

Choosem(t) × cos → M(f − ?)

A product in time becomes a shift in frequency.

So many signals stack side by side

Give each station a different carrier frequency and each spectrum lands in its own spot. As long as the spacing is wider than the bandwidth, they never overlap. Sharing the axis this way is frequency-division multiplexing.

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

A cosine carries both +fc and −fc, so two copies appear.

On your ownBtot = ?

Place N channels of bandwidth B side by side and the total band is N × B.

Back to the first screen

You could slot your signal into the gap because multiplying by the carrier moved its whole spectrum to that spot. Push the frequency too low and it overlapped a neighbor and turned red; lift it far enough and it separated into gold. Modulation is, in the end, carrying a signal up to an empty seat on the frequency axis.

Modulation multiplies a signal m(t) by a carrier cos(2π fc t), moving its spectrum up by the frequency fc. The shape is preserved and only the position changes, so using different fc lets many signals share the axis without overlap — multiplexing.

In the next unit

The multiply that shifts the spectrum is settled. Next we watch that multiply as a real waveform. In AM the signal rides on the carrier’s amplitude, so the envelope traces the original signal itself.