AMPLITUDE MODULATION AND DEMODULATION

Transcription

1 Modulation is a tehnique to transit inforation via radio arrier wavefor. It is a non-linear proess that generates additional frequenies, as we will see. Aplitude Modulation (AM) works by varying the aplitude ( strength ) of a arrier signal, in proportion to aplitude of a odulation signal that is to be transitted to one or ore reeivers. The arrier signal ( is just a sinusoid with a frequeny f and an aplitude A. This is atheatially expressed as ( = A os( π f = A os( ω where the angular frequeny ω (in radians/se) is π ties the ordinary frequeny f (in hertz). Let s assue that the odulation signal (a.k.a. baseband signal) is a onstant audio tone. That is, a sinusoid with frequeny f and aplitude A : ( A os( π f = A os( ω =, To siplify the following analysis, without hanging the essene, we hoose A = and ipose the ondition that A <, hene ( <. The AM odulation proess onsists of adding a positive DC offset (bias) to the odulation signal, and ultiplying the result with the arrier signal. For a DC offset equal to (again, for onveniene), the resulting AM odulated signal a( is: a t ( ) = [ + ( ] ( = [ + A os( ω ] os( ω = os( ω + A os( ω os( ω Figure illustrates the proess: Figure : the AM odulation proess The DC offset is required: without it, the transitted AM signal annot be deodulated with a siple envelope-detetor reeiver. Instead, a ore opliated synhronous (oheren detetor is required, with a loal osillator frequeny that exatly athes the arrier frequeny. Let s use the following standard trigonoetry produt-to-su identity (that we all learned in shool, right?), to expand the produt ter in the above equation: Hene, os a t ( x) os( y) = [ os( x + y) + os( x y) ] ( ) = os( ω + A os( ω os( ω = os A ( ω + os( ω t + ω + os( ω t ω Clearly, the frequeny spetru of this signal onsists of three disrete spetral lines (on the positive half of the frequeny axis). The frequenies are f, f + f, and f f. That is, the original arrier A 07 Frank Dörenberg N4SPP V.

2 frequeny, as well as the su and differene frequeny. The latter two are known as the lower sideband (LSB) and upper sideband (USB). They are loated syetrially about the arrier frequeny. The AM signal is also known as DSB-FC or DSB-LC: Double Sideband Full (or Large) Carrier, as opposed to DSB-SC (Double Sideband - Suppressed Carrier). Norally, f >> f. Figure : tie doain signals and the equivalent frequeny spetru The equation above shows that the height of the sideband lines is different fro that of the residual arrier. The relative height is 0log(M) db, where M is the non-negative Modulation Index (a.k.a. odulation depth). This is the ratio of the odulation aplitude and the aplitude of the unodulated arrier. For M = (i.e., 00% odulation), the envelope of the odulated signal periodially reahes zero, and the sidebands are 6 db below the arrier. For saller M, the sidebands are lower (saller) than that (i.e., by ore than 6 db). For very sall M, ost of the transitter power is used for the arrier frequeny, rather than the transitted inforation (i.e., the odulation). For M >, the odulated signal is distorted whenever the odulating aplitude exeeds the arrier aplitude. At the reeiver, the reeived AM signal ust be deodulated. That is, the original odulating (baseband) signal ust be reovered (reonstruted). There are several basi ways to do this: envelope detetion Square-Law deodulation synhronous produt detetion asynhronous produt detetion A siple envelope detetor onsists of a full-wave or half-wave diode retifier, followed by a lowpass filter. The filter uts off the arrier frequeny, but not the odulating frequeny. The output of the filter is the envelope of the AM signal plus a DC offset. The latter is proportional to the aplitude of the reeived arrier signal, and is reoved with a bloking apaitor (or transforer) in series with the filter output. As diodes are inherently non-linear, the output signal of the detetor has soe distortion on it. Envelope detetion using the Hilbert Transfor is beyond the sope of this disussion. 07 Frank Dörenberg N4SPP V.

3 Figure 3: input & output of a diode-retifier envelope detetor As the nae suggests, with Square-Law deodulation, the reeived a( signal is squared, i.e., ultiplied by itself. The resulting signal sqa( is the square of the odulating signal + (, plus that sae squared signal ties the seond haroni ( = double frequeny) of the arrier signal: Again, using the produt-to-su identity ( = [ + ( ] os ( ω sqa( = a we obtain os + ( x) = os( x) os( x) = [ os( 0) + os( x) ] = [ os( x) ] sqa( = [ + ( ] os ( ω = [ + ( ] [ + os( ω ] [ + ( ] + [ + ( ] os( ω The latter high frequeny signal ω t is filtered out, and the reaining squared odulating signal is passed through a square root iruit. After reoving the DC ontent, ( is obtained. A synhronous detetor is a produt detetor (ixer) with a loal osillator (LO) signal that is synhronized to the arrier of the reeived signal (both frequeny and phase, i.e., oheren. The ultipliation perfored by the ixer is referred to as heterodyning, or beating. The LO signal is derived fro the reeived arrier signal itself, e.g. with a Phase-Loked Loop iruit (PLL). The ixer 07 Frank Dörenberg N4SPP V.

4 output o( of this detetor onsists of a DC signal, the odulating signal, and the seond-order haroni ( = double) of the arrier frequeny: o ( a( A os( ω = [ + A os( ω ] A A os ( ω LO LO Using we obtain os + o ( ω = [ os( ωt ω + os( ωt + ω ] = [ os( ωt )] ( [ + A os( ω ] A ALO os ( ω [ + A os( ω ] [ + os( ω ] = [ + A os( ω ] + [ + A os( ω ] os( ω That is, a DC ter plus the odulating signal f, plus the oplete AM signal, shifted fro f to f : Figure 4: AM odulation and synhronous de-odulation The higher-frequeny produt signals are filtered out with a low-pass filter that has a ut-off frequeny between the odulating frequeny f and f f. Note that with synhronous detetion, the DC-ter in o( serves no purpose, and injetion of the DC bias during the odulation proess is also not neessary. An asynhronous detetor that uses a ixer to ultiply ( = heterodyne) the AM signal with a sinusoidal signal fro a loal osillator (LO). Unlike the uh ore opliated synhronous detetor, the LO of this detetor is now not synhronized to the arrier signal. I.e., it has a different frequeny and, hene, different phase. This auses the arrier of the reeived odulated signal not to be shifted to zero frequeny (f f LO = 0), but to the non-zero differene between that arrier frequeny and the LO frequeny (f f LO = Δf): 07 Frank Dörenberg N4SPP V.

5 o ( a( ALO os( ωlo = [ + A os( ω ] A os( ω ALO os( ωlo [ + A os( ω ] os( ω os( ωlo = [ + A os( ω ] { os[ ( ω ωlo ) t] + os[ ( ω + ωlo ) t] } [ + A os( ω ] os[ ( ω ωlo ) t] + [ + A os( ω ] os[ ( ω + ωlo ) t] = a ( + a ( Hene, the assoiated frequeny spetru shows two oplete AM signals (arrier plus sidebands): one entered on the differene between the arrier and LO frequeny (i.e., at f = f f LO ), the other entered on the su of the arrier and LO frequeny (i.e., at f = f + f LO ): Figure 5: AM odulation and asynhronous de-odulation If f LO is large opared to f, then the AM signal at f = f + f LO an be filtered out. The filter output an then be passed through envelope detetor, to reover the original odulating signal. An AM-reeiver oprising an asynhronous ixer and an envelope detetor is alled a superheterodyne reeiver, or superhet for short. For the speial ase where f LO = f, we basially obtain the synhronous detetion ase (besides the effets of phase differene). As os(x) = os(-x), it basially does not atter on whih side of f the f LO is loated (through there ay be pratial reasons to selet a speifi side). If the LO frequeny f LO is signifiantly higher than the odulation frequeny f, (e.g., relatively lose to the arrier frequeny f ), then the seond ter (with the su beat frequeny f = f + f LO ) an easily be filtered out. If f LO is hosen properly, the reaining filtered AM signal an then be deodulated with an envelope detetor as desribed above. Why heterodyne the odulated signal down to an interediate frequeny (IF)? This allows sharp filters with a fixed enter frequeny - plaed after the detetor, instead of sharp filters - with an adjustable enter frequeny that ust trak the LO frequeny - plaed in front of the ultiplier (ixer). 07 Frank Dörenberg N4SPP V.

6 What happens if we have not one, but two AM radio transitters: eah with a -tone odulating signal (f and f ), and with arrier frequenies (f and f, respetively) that are spaed by less than the HF, IF, and audio bandwidth of the radio reeiver. I.e., what if the radio an reeive both AM signals siultaneously (in broadast radio, this would be alled interferene )? The resulting transitted RF spetru siply oprises two independent AM signals: Figure 6: spetru of two separate arriers, eah AM-odulated with a onstant tone Note that this is not the sae spetru as for a single arrier that is AM odulated with two separate onstant tones: 07 Frank Dörenberg N4SPP V.

7 Figure 7: spetru of a single arrier, AM-odulated with two onstant tones If the reeiver has an asynhronous produt detetor, then both of the reeived AM signals in Figure 6 will be shifted downward (i.e., oved to the left on the frequeny sale) by an aount equal to f LO, where f LO f. At the sae tie, two additional AM signals appear: the sae reeived AM signals, but now both shifted upward by an aount equal to f LO : Figure 8: output of the ixer of an asynhronous produt detetor If f LO is large opared to f and f, then the AM signals at f + f LO and f + f LO an be filtered out. The base signals at f and f an then be reovered with an envelope detetor. Note that this approah does not work with a synhronous deodulator: f LO annot be equal to f and to f at the sae tie. E.g., if the LO is synhronized to f (i.e., f LO = f ), then the ixer will shift the AM signal at f to f - f LO 0. Hene, the latter AM signal will not be deodulated orretly: the baseband odulating tone at f will sound like a tone at f = f - f LO - f f. 07 Frank Dörenberg N4SPP V.

8 For baseband signals other than a single tone, the story is exatly the sae. But instead of a disrete spetral line at a distane f on either side of the arrier, there will be sidebands with a ertain bandwidth: Figure 9: output AM spetru for odulating signal that is not a single tone 07 Frank Dörenberg N4SPP V.