Amplitude Modulation: From Baseband to Sidebands

1. baseband
2. mixing
3. sidebands
4. DSB-SC
5. DSB-TC

Every step is a consequence of the one before it. The multiplication is the only operation we perform; sidebands, bandwidth, the envelope and the modulation index all follow from it.

Part 1

Multiplication and sidebands

Sections 01–04 — the fundamental operation, the single-tone result, and the bandwidth of a general message.

Let \(m(t)\) be a baseband information signal and let the carrier be

Multiplication—also called mixing or heterodyning—moves the information spectrum from around zero frequency to the carrier region.

The translated components below \(f_c\) form the lower sideband (LSB); those above \(f_c\) form the upper sideband (USB).

SSB is the subject of Week 4, where one sideband is removed using the Hilbert transform.

For suppressed-carrier AM:

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

One message tone and one carrier tone have become two transmitted tones — and neither of them is at \(f_c\).

Open the AM Visualizer — DSB-SC →

A practical message contains many frequency components. If its highest frequency is \(B\), multiplication with the carrier creates mirror-image translated sidebands:

This result holds for both DSB forms: adding a carrier line at \(f_c\) does not widen the occupied band.

The lower group is reversed in frequency order; the upper group keeps the order of the message.

Part 2

Why transmit a carrier?

Sections 05–07 — the design question, transmitted-carrier AM, and a side-by-side comparison.

This is motivation only; demodulator circuits are intentionally deferred.

The DC offset \(A_0\) produces a carrier-frequency component.

For \(m(t)=A_m\cos(2\pi f_m t)\), define \(\mu=A_m/A_0\). Absorbing the constant product into a carrier amplitude \(A_c'\) gives the familiar form:

The spectrum contains an LSB at \(f_c-f_m\), a carrier at \(f_c\), and a USB at \(f_c+f_m\).

The positive and negative envelopes are \(\pm A_c'[1+\mu\cos(2\pi f_m t)]\): a slow message-shaped boundary around fast RF oscillation.

Part 3

Modulation index

Section 08 — how far the envelope is allowed to swing, and what happens when it swings too far.

For the single-tone form, \(\mu\) measures the message amplitude relative to the DC term. The interesting threshold is \(\mu=1\).

Part 4

MATLAB, and checking yourself

Sections 09–11 — a computational comparison of the two forms, two reasoning checks, and the reference boundary.

Change \(f_m\), \(f_c\), \(A_m\), \(A_c\), and \(\mu\). Confirm that DSB-SC has no line at \(f_c\), DSB-TC does, and both occupy \(2B\) for the same baseband bandwidth.

Download the Week 3 MATLAB script ↓

%% Parameters
Fs=200e3; T=0.02; t=0:1/Fs:T-1/Fs;
fm=2e3; fc=15e3; Am=1; Ac=1; mu=0.5;
m=Am*cos(2*pi*fm*t); c=Ac*cos(2*pi*fc*t);
sSC=m.*c;
sTC=Ac*(1+mu*cos(2*pi*fm*t)).*cos(2*pi*fc*t);

%% Time-domain comparison
figure;
subplot(4,1,1); plot(t*1e3,m);  xlim([0 2]); grid on; title('Message');
subplot(4,1,2); plot(t*1e3,c);  xlim([0 .5]); grid on; title('Carrier');
subplot(4,1,3); plot(t*1e3,sSC); xlim([0 2]); grid on; title('DSB-SC');
subplot(4,1,4); plot(t*1e3,sTC); xlim([0 2]); grid on; title('DSB-TC'); xlabel('Time (ms)');

%% Centered double-sided spectra
N=length(t); f=(-N/2:N/2-1)*(Fs/N);
SSC=fftshift(fft(sSC)/N); STC=fftshift(fft(sTC)/N);
figure;
subplot(2,1,1); plot(f/1e3,abs(SSC)); xlim([-22 22]); grid on; title('DSB-SC: no carrier line');
subplot(2,1,2); plot(f/1e3,abs(STC)); xlim([-22 22]); grid on; title('DSB-TC: carrier plus sidebands'); xlabel('Frequency (kHz)');

%% Explore envelope conditions
for mu=[0.5 1 1.5]
    s=Ac*(1+mu*cos(2*pi*fm*t)).*cos(2*pi*fc*t);
    figure; plot(t*1e3,s); xlim([0 2]); grid on;
    title(sprintf('DSB-TC, modulation index mu = %.1f',mu)); xlabel('Time (ms)');
end

This lecture stops at DSB-TC. VSB and AM demodulation follow later and are not taught here; single-sideband modulation is the subject of Week 4.

EE 351 · Week 3

Both sidebands carry the same information

You can now derive the two sidebands from a multiplication, state the DSB bandwidth \(2B\), and explain what a transmitted carrier buys and costs. One question is left open: if the two sidebands are mirror images, must we transmit both?

Continue to Week 4 — SSB & Hilbert Transform →