 # Simple Modulation A sinusoidal electrical signal (voltage/current) can be generated at a high frequency. The fields associated with such a signal can practically.

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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 v I (t), which contains the information we wish to recover at a remote location. As an example, let v I (t) be the voltage generated by a microphone. If v I (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 Math The Modulated sinusoid might take a form like: e(t) = v I (t)cos(  c t) 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 v I (t) is bipolar, the phase of the modulated carrier will flip by 180 degrees whenever v I (t) goes negative, and we don’t want to do any phase modulation (... yet). This problem is solved by adding a DC offset to v I (t) which is greater than the maximum peak voltage that v I (t) ever exhibits.....

More Math e(t) ={V MAX + v I (t)}cos(  c t ) > 0 We have considered v I (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 v I (t) is a sinusoid having frequency  m, we can extend what we learn to arbitrary waveforms using Fourier analysis. Now... x e(t) V MAX v I (t) cos(  c t) + +

... and More Math e(t) ={E c + E m cos(  m t ) }cos(  c t ) E m is the amplitude of the “modulating” sinusoid. E c is the DC offset, larger than E m. If E c >> E m, or E m is zero, then we have e(t) =E c cos(  c t ) Therefore, we refer to E c as the amplitude of the “unmodulated” carrier.

e(t) ={E c + E m cos(  m t ) }cos(  c t ) Define Modulation Index: m a = E m /E c e(t) =E c {1+ m a cos(  m t ) }cos(  c t ) “Modulation Envelope”

v p (peaks of modulation) = E C + E M v v (valleys of modulation ) = E C - E M

Spectral Content Carrier Lower Sideband Upper Sideband  c –  m  c  c +  m –  m +  m

Spectral Power  c –  m  c  c +  m If e(t) is applied to a radiation resistance R, then the transmitted power is: P LSB P c P USB –  m +  m Since all the “information” is in the sidebands, P C is wasted power.

Complex Spectra cc VI()VI() Let v I (t) = v 1 cos(  1 t) + v 2 cos(  2 t) + v 3 cos(  3 t) +... m 1 = v 1 /V MAX m 2 = v 2 /V MAX m 3 = v 3 /V MAX...  c m < 1 or “overmodulation” occurs (carrier phase inversion). Let v I (t) be a complex waveform with Fourier transform V I (  ): After v I (t) modulates a carrier with frequency  c...

The Barstool Perspective Carrier Phasor cc c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  c -  m  c +  m Carrier (stationary) -  m -  c +  m LSBUSB USB + LSB

T G I FT G I M

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