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Simple Modulation

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.

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Simple Modulation

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  1. 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. 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. 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. Simple Modulation

  2. 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”.

  3. The Math The Modulated sinusoid might take a form like: e(t) = vI(t)cos(wct) where wc 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 . . . . .

  4. More Math e(t) ={VMAX + vI(t)}cos(wct) > 0 + x e(t) vI(t) + cos(wct) VMAX 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 wm, we can extend what we learn to arbitrary waveforms using Fourier analysis. Now . . .

  5. . . . and More Math e(t) ={Ec + Emcos(wmt)}cos(wct) Emis the amplitude of the “modulating” sinusoid. Ecis the DC offset, larger than Em . If Ec >> Em , or Em is zero, then we have e(t) =Ec cos(wct) Therefore, we refer to Ec as the amplitude of the “unmodulated” carrier.

  6. e(t) ={Ec + Emcos(wmt)}cos(wct) Define Modulation Index: ma = Em/Ec “Modulation Envelope” e(t) =Ec {1+ macos(wmt)}cos(wct)

  7. vp(peaks of modulation) = EC + EM vv(valleys of modulation) = EC - EM

  8. Spectral Content Lower Sideband Upper Sideband Carrier – wm +wm wc – wm wc wc+wm

  9. Spectral Power If e(t) is applied to a radiation resistance R, then the transmitted power is: Since all the “information” is in the sidebands, PC is wasted power. PLSB Pc PUSB – wm +wm wc – wm wc wc+wm

  10. Complex Spectra Let vI(t) = v1cos(w1t) + v2cos(w2t) + v3cos(w3t) + . . . m1 = v1/VMAXm2 = v2/VMAXm3 = v3/VMAX . . . m < 1 or “overmodulation” occurs (carrier phase inversion). Let vI(t) be a complex waveform with Fourier transform VI(w): After vI(t) modulates a carrier with frequency wc . . . VI(w) -wc wc

  11. The Barstool Perspective Here’s Don, gettin’ a buzz on, down at the old “Complex Plane” bar and grill . . . Spin ‘til you puke, Don . . . wc Carrier Phasor wc

  12. What we see . . . What Don sees . . . Carrier Carrier (stationary) LSB USB + LSB wc USB LSB wc - wm - wm + wm USB wc + wm - wc

  13. T G I F T G I W

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