Simple Modulation

A sinusoidal electrical signal (voltage/current) can be generated at a high frequency. The fields associated with such a signal can practically be made to propagate through free space and be recovered at a distance through coupling structures which we call antennas.

A pure, invariant sinusoid carries no information except for the fact that it can be detected, and it’s presence or absence represents one bit of information.

By varying certain aspects of a transmitted sinusoid, the variations can be detected and measured at the receiver, and interpreted as useful information. The most commonly varied aspects of transmitted sinusoids are its amplitude, its frequency, or its phase.

We refer to the pure, invariant sinusoid as a “carrier”.

We refer to the systematic variation of one or more of its properties as “modulation.

Amplitude Modulation

The simplest (and oldest) method of modulation is accomplished by varying the amplitude of the carrier by a second signal, call it vI(t), which contains the information we wish to recover at a remote location. As an example, let vI(t) be the voltage generated by a microphone.

If vI(t) can be recovered at the remote location, amplified, and used to drive a speaker, then the sounds picked up by the microphone will be reproduced at the remote location.

We call this “communication”.

The MathThe Modulated sinusoid might take a form like:

e(t) = vI(t)cos(ct) where c is the carrier frequency.

This is called “Double Sideband Suppressed Carrier” (DSBSC) modulation.

The problem with this type of modulation is that since vI(t) is bipolar, the phase of the modulated carrier will flip by 180 degrees whenever vI(t) goes negative, and we don’t want to do any phase modulation ( . . . yet).

This problem is solved by adding a DC offset to vI(t) which is greater than the maximum peak voltage that vI(t) ever exhibits . . . . .

More Mathe(t) ={VMAX + vI(t)}cos(ct) > 0

We have considered vI(t) to be an arbitrary waveform, but Fourier teaches us that any arbitrary waveform can be represented by a set of sinusoids.

Good to know. That means that if we first study the case where vI(t)

is a sinusoid having frequency m, we can extend

what we learn to arbitrary waveforms using Fourier analysis. Now . . .

x e(t)

VMAX

vI(t)

cos(ct)

+

+

. . . and More Mathe(t) ={Ec + Emcos(mt)}cos(ct)

Em is the amplitude of the “modulating” sinusoid.

Ec is the DC offset, larger than Em .

If Ec >> Em , or Em is zero, then we have

e(t) =Ec cos(ct)

Therefore, we refer to Ec as the amplitude of the “unmodulated” carrier.

e(t) ={Ec + Emcos(mt)}cos(ct)

Define Modulation Index: ma = Em/Ec

e(t) =Ec {1+ macos(mt)}cos(ct)

“Modulation Envelope”

vp (peaks of modulation)

= EC + EM

vv (valleys of modulation)

= EC - EM

2

2p v c m c m m

ap v c m c m c

v v E E E E Em

v v E E E E E

Spectral Content

( ) 1 cos cos

cos cos cos

cos cos cos2

cos cos cos2 2

c a m c

c c c a m c

ac c c c m c m

a ac c c c m c c m

e t E m t t

E t E m t t

mE t E t t

m mE t E t E t

Carrier Lower Sideband

Upper Sideband

c – m c c+m

– m +m

Spectral Power

c – m c c+m

( ) cos cos cos2 2

a ac c c c m c c m

m me t E t E t E t

If e(t) is applied to a radiation resistance R, then the transmitted power is:

2 2

2

2 2 2 2 2

2

2

2 22 2 2

2 2 4 2 4

12

4

a ac c

c

c c a c a

ac LSB USB c

aLSB USB c

m mE E

EP

R R R

E E m E m

R R R

mP P P P

mP P P

PLSB Pc PUSB

– m +m

Since all the “information” is in the sidebands, PC is wasted power.

Complex Spectra

c

VI()

Let vI(t) = v1cos(1t) + v2cos(2t) + v3cos(3t) + . . .

m1 = v1/VMAX m2 = v2/VMAX m3 = v3/VMAX . . .

2 2 21 2 3m m m m

c

m < 1 or “overmodulation” occurs (carrier phase inversion).

Let vI(t) be a complex waveform with Fourier transform VI():

After vI(t) modulates a carrier with frequency c . . .

The Barstool Perspective

Carrier Phasorc

c

Here’s Blake, gettin’ a buzz on, down at the old “Complex Plane” bar and grill . . .

Spin ‘til you puke, Blake . . .

What we see . . . What Blake sees . . .

Carrier

USB

LSB

c c - m

c + m

Carrier(stationary)

- m

- c

+ m

LSBUSB

USB + LSB

T G I FT G I M