Module 7: AM, FM, and the spetru analyzer.
7.0 Introdution Eletroagneti signals ay be used to transit inforation very quikly, over great distanes. Two oon ethods by whih inforation is enoded on radio signals, aplitude and frequeny odulation, will be reviewed in this odule. Also, the proess of retrieving inforation fro enoded signals will be disussed. Finally, the basi oponents of the spetru analyzer will be exained. 7.1 Modulation With the proper equipent, radio signals an be transitted and reeived over large distanes. Inforation ay therefore be exhanged over large distanes by enoding inforation on radio waves. This is aoplished through odulation of radio signals. Modulation is the proess of enoding inforation onto a arrier signal whih has frequeny f. This arrier signal is alled the odulated signal, while the inforation arrying, or baseband signal is referred to as the odulating signal. Two types of odulation will be reviewed in this odule. Aplitude odulation onsists enoding inforation onto a arrier signal by varying the aplitude of the arrier. Frequeny odulation onsists of enoding inforation onto a arrier signal by varying the frequeny of the arrier. One a signal has been odulated, inforation is retrieved through a deodulation proess. 7. Suppressed arrier aplitude odulation (double sideband) A general sinusoidal signal an be expressed as f( t) A( t) os (t). where the aplitude A and phase angle ay, in general, be funtions of tie. It is onvenient to write tie varying angle (t) as t ( t), therefore the sinusoidal signal ay be expressed as f( t) A (t)os t (t). The ter A(t) is alled the envelope of the signal f(t), and is alled the arrier frequeny. The proess of aplitude odulation onsists of the aplitude of the arrier wave being varied in sypathy with a odulating signal. A atheatial representation of an aplitude odulated signal is obtained by setting (t) =0 in the expression for the general sinusoidal 7-1
1. Carrier Wave 0.6 0.0-0.6-1. 1. 0 4 6 8 10 1 14 Modulation Wave 0.6 0.0-0.6-1. 1 3 4 5 6 7 8 9 10 1. AM Signal 0.6 0.0-0.6-1. 0 4 6 8 10 1 14 Figure 1. Aplitude odulation. signal, and letting the envelope A(t) be proportional to a odulating signal f(t). What results is a new (odulated)signal, given by y(t) f (t )os( t). The spetru of the odulated signal y( t) an be found by using the odulation property of the Fourier transfor. In Chapter 3, the Fourier transfor pair was defined as The Fourier transfor of a signal f (t) 1 F( ) f ( t) e j 0 t is then f( t)e j 0 t 7- F( ) e j t d f (t)e j t dt. f (t)e j 0 t e j t dt f (t)e j( 0 )t dt.
f ( t) y( t) = f ( t)os( ω t) os( ω t) Figure : Aplitude odulation (suppressed arrier) Thus the Fourier transfor of f( t)e j 0 t ay be expressed f(t) e j 0 t F 0. The aplitude odulated signal y(t) ay be written in ters of oplex exponentials y (t ) f ( t)os t 1 j t f (t) e e! j t. When y(t) is expressed in this for, and fro the exaple above, it an be seen that the Fourier transfor of y(t) is given by f (t) os( t) 1 F( ) F("# ). ω ω ω ω + ω ω ω Lower side frequeny Upper side frequeny Figure 3. Single odulating frequeny AM signal spetru. ω 7-3
±$ Thus, the spetru of f (t ) is translated by. It is seen that the odulation proess auses the frequenies assoiated with the odulating signal to disappear. Instead, a new frequeny spetru appears, onsisting of two sidebands, known as the upper sideband (USB), and the lower sideband (LSB). The spetru of the odulated signal y( t) does not ontain the spetru of the original arrier, but is still entered about the arrier frequeny. Thus this type of odulation is referred to as double-sideband, suppressed-arrier aplitude odulation. A blok diagra of the suppressed arrier aplitude odulation operation is presented in Figure. If the odulating signal ontains a single frequeny $, then $ USB=$ +$, and $ LSB=$ - $ (Figure 3). If odulating signal f(t) has a bandwidth of $ bw, then F($ ), the spetru of f(t), will extend fro -$ bw to +$ bw. The upper sideband of the spetru of the odulated signal Y($ ) will extend fro $ to $ +$ bw. Likewise, the lower sideband will extend fro $ -$ bw to $. Both the negative and positive frequeny oponents of the odulating signal f(t) appear as positive frequenies in the spetru of the odulated signal y(t). It is also seen that the bandwidth of f(t) is doubled in the spetru of the odulated signal when this type of odulation is eployed. $ F( ω ) ω bw 0 ω bw ω 1 [ F( ω + ω ) + F( ω ω )] 1 F( 0) 1 F( 0) ω 0 ω ω ω bw ω bw Figure 4. Spetra of odulating wave and resulting AM wavefor 7-4
' ' 7.3 AM deodulation An AM signal is deodulated by first ixing the odulated signal of the sae arrier frequeny y(t) with another sinusoid The Fourier transfor of this signal is y( t) os($ t) % f (t) os ($ t) % 1 f (t) 1 & os( $ t). or { y ( t) os($ t) } % ' 1 f (t) 1 & os( $ t) y( t) os($ t) % 1 F( $ ) & 1 F( $(& $ ) & F( $() $ ). By using a low-pass filter, the frequeny oponents entered at ±$ an be reoved to leave only the 1/ F($ ) ter. It is obvious that in order to properly reover the original signal it is neessary that $ >$ bw. A blok diagra of the deodulation proess is shown in Figure 5. y( t) = f ( t)os( ω t) 1 ( 1 t ) f ( t) + os( ω ) Low Pass Filter 1 f ( t) os( ω t) Figure 5: AM deodulation. 7.4 Large arrier aplitude odulation (double sideband) In pratie, the deodulation of suppressed arrier aplitude odulated signals requires fairly opliated iruitry in order to aquire and aintain phase synhronization. A uh less opliated (and thus less expensive) reeiver an be used if a slightly different odulation shee is eployed. In large arrier aplitude odulation, the arrier wave inforation is inorporated as a part of the wavefor being transitted. It is onvenient to let the aplitude of the arrier be larger than any other part of the signal spetral density. While this akes the deodulation proess uh easier, low-frequeny response of the syste is lost. For soe signals however, frequeny response down to zero is not needed (suh as in audio signals). Consider a arrier wave with aplitude A, and frequeny $, represented by 7-5
( t) % A os($ t). The odulated wavefor of a large arrier AM signal an be then be expressed atheatially as y( t) % f (t)os($ t) & A os( $ t). The spetru assoiated with this odulated signal is given by Y($ ) % 1 F($& $ ) & 1 F($) $ ) &+* A, ($& $ ) &+* A, ($) $ ). It is seen that the spetru of the large arrier AM signal is the sae as that of the suppressed arrier AM signal with the addition of ipulses at ± $. The large arrier AM signal ay be rewritten y(t) % A & f (t) os($ t). In this for, y(t) ay be thought of as onsisting of a arrier signal os($ t) having aplitude A & f (t). If the aplitude of the arrier A is suffiiently large, then the envelope of the odulated wavefor will be proportional to f(t) (hene the nae large arrier AM). Deodulation in this ase is siply involves the detetion of the envelope of a sinusoid. Figure 6. Coparison of large arrier AM and suppressed arrier AM. 7-6
7.6 The envelope detetor An envelope detetor is any iruit whose output follows the envelope of an input signal. The siplest for of suh a detetor is a non-linear harging iruit whih has a fast harge tie and a slow disharge tie. This is easily ipleented by plaing a diode in series with a parallel obination of a apaitor and a resistor. The envelope of an input signal is deteted by the following proess: - The input wavefor (in this ase a large arrier AM signal) harges the apaitor to the axiu value of the wavefor during positive half-yles of the input signal. - As the input signal falls below axiu, the diode beoes reverse biased, and swithes off. - The apaitor then begins a relatively slow disharge through the resistor until the next positive half-yle. - When the input signal beoes greater than the apaitor voltage, the diode beoes forward biased, and the apaitor harges to a new peak value. For optiu operation, the disharge tie onstant RC is adjusted so that the axiu negative rate of the envelope never exeeds the exponential disharge rate. - If the tie onstant is too large, the envelope detetor ay iss soe positive half-yles of the arrier, and will not orretly reprodue the envelope of the input signal. - If the tie onstant is too sall, the detetor generates a ragged signal. Figure 7. Envelope detetor. 7-7
4 4 7.5 Frequeny odulation Frequeny odulation is a type of angle odulation. Angle odulation hanges the phase of a signal as well as its aplitude, where aplitude odulation leaves the phase unhanged. Phase odulation is another type of angle odulation very siilar to frequeny odulation. The general for of a sinusoidal signal an be written as f ( t) % A (t ) os[. ( t)]. The instantaneous angular frequeny of this signal, $ ( t) i, is the derivative of the phase $ (t) % d. ( t) i. dt This definition of instantaneous frequeny suggests two obvious ethods of angle odulation: - If the phase angle of a arrier with fundaental frequeny $ is varied linearly with a odulating wavefor f(t). ( t) %/$ t & k p f ( t) &0. 0 The odulated signal is said to be phase odulated. Here $, k p, and. 0 are onstants. The instantaneous frequeny of a phase odulated signal is given by $ i % d. dt %/$ & k p df dt. - If the instantaneous frequeny of a arrier with fundaental frequeny $ is varied in proportion to an input odulating signal f(t), suh that $ i %/$ & k f f (t ) the resulting odulated signal is said to be frequeny odulated. Here $ and k f, are onstants. The phase angle assoiated with the FM signal is given by t. ( t) %01 0 i (3 ) d3 whih ay also be expressed ( t) 5 t 671 t k f f (3 ) d36 0 0. 7-8
Carrier 0 1 3 4 5 6 7 8 9 f(t) 0 1 3 4 5 6 7 8 9 FM Signal 0 1 3 4 5 6 7 8 9 PM Signal 0 1 3 4 5 6 7 8 9 Figure 8. FM and PM signals It an be seen in Figure 8 that phase and frequeny odulation are losely related. For both phase and frequeny odulation, the odulating signal auses the arrier to inrease and derease fro the fundaental frequeny, while the aplitude reains onstant. For the FM signal, the frequeny rises with positive odulating aplitudes and falls with negative odulating aplitudes. For the PM signal, the frequeny rises with inreasing odulating aplitudes, and dereases with dereasing odulating aplitudes. Frequeny odulation has several advantages over aplitude odulation: 8 The radiated signal level reains onstant, therefore transitters an be run at a onstant power output. 8 Aplitude variations due to external interferene soures are not interpreted as signals. 8 Seletive fading does not our beause the aplitude of the arrier is onstant. 8 It is possible to design systes having better dynai range and signal-to-noise ratio. The obvious disadvantage of frequeny odulation is that a greater bandwidth is required than for aplitude odulation. Another signifiant differene between aplitude and angle odulation has to do with the 7-9
4 4 relationship between the odulated and odulating signals. When signals are aplitude odulated, a one-to-one orrespondene exists between the odulated and odulating signals. In this ase the odulation is said to be linear. This linear relationship does not always exist for phase and frequeny odulation. As a result, the sidebands assoiated with angle odulation do not obey the priniple of superposition. 7.6 Spetral ontent of FM signals Unlike aplitude odulation, frequeny odulation produes (theoretially) an infinite nuber of sidebands. It is not possible to evaluate the Fourier transfor of a general FM signal, therefore, for the sake of sipliity, the ase of a sinusoidal odulating signal is onsidered. In this ase and the instantaneous frequeny is f( t) 5 A os( t) whih ay be expressed i ( t) 5 6 Ak f os ( t) i ( t) 5 where 9 is alled the peak frequeny deviation. The phase angle of the FM signal ay be expressed whih ay be expressed where : =9 or (t) 5 6 9 t ( t) 501 0 os( i (3 ) d3 t 6 : sin( / is the odulation index of the FM signal. The resulting FM signal ay be expressed in phasor notation as y( t) 5 Re Ae j; (t) t) t) y( t) < Re Ae j= t e j> sin= t. The seond exponential ter in the expression above an be expanded in a Fourier series 7-10
N where e j> sin= t <@?BA n CED A n e jn= t Making a hange of variable GIHIJ n < 1 T F T/ D T/ t H (K /T)tgives n H 1 e j> sin= t e D jn= t dt. e KML j(p sinq O nq ) d G/H J n (R ) O N where J n (R ) is the Bessel funtion of the first kind of order n. Using this result gives e jp sins t H@TBU n VEW U J n (X )e jny t therefore y(t) Z Re Ae jy t [B\ n ]E^ \ J n (_ )e jn` t a A bd n eef J n (g )os(h i nh )t. Fro this it is evident that an FM wavefor with sinusoidal odulation has an infinite nuber of sidebands. However, the agnitudes of the spetral oponents of the higher-order sidebands are negligible. The nuber of sidebands whih are signifiant depend on the order of the Bessel funtion n, and the value of g. ω ω 4ω ω ω ω + ω ω + 4ω ω ω ω 3ω ω ω ω + ω Figure 9. Spetru of an FM wavefor. ω + 3ω ω 7-11
7.7 FM deodulation Many ethods of reovering inforation fro an FM wavefor exist. One ethod involves the use of a syste that has a linear frequeny-to-voltage transfer harateristi. This type of syste is referred to as a disriinator. The siplest suh devie is an ideal differentiator. A general FM wavefor ay be expressed If A is a onstant then y( t) j A os k t l t k f f(n ) dn. 0 dy dt jpo A k l k f f(t) sin k t l t k f f (n )dn. 0 If, then the expression above has the fro of an AM signal with envelope k f f ( t ) «k A k 1 l k f k f(t) having frequeny k l k f f (t). Thus, the differentiator has onverted the FM signal into an AM signal with a slight frequeny variation (assuing that k f f ( t ) «k ). The original wavefor ay now be reovered by an envelope detetor. 7.8 The spetru analyzer Spetru analyzers are instruents that are used to easure the spetru of periodi wavefors. The ost oon type of spetru analyzer ontains a superheterodyne reeiver. A different type of spetru analyzer that has had inreasing popularity is alled a Fast Fourier Transfor (FFT) spetru analyzer. These two types of spetru analyzers will be disussed in this setion. qqq q Superheterodyne spetru analyzers A spetru analyzer is an exaple of a superheterodyne reeiver. Eah frequeny oponent of a signal is ixed down to an interediate frequeny where it an be easured 7-1
q q + Low Pass Filter Mixer IF Bandpass Filter Detetor Input Signal - Loal Osillator Rap Generator Display Figure 10. Superheterodyne spetru analyzer and anipulated. A siple blok diagra of a superheterodyne spetru analyzer is shown above. q An input signal passes through a low pass filter to a ixer, where it ixes with a signal fro a loal osillator. q The output of the ixer inludes not only the signals fro the loal osillator and the low pass filter, but also the haronis, and the sus and differenes of the original signal frequenies and their haronis. Any of the ixed signal that falls within the passband of the interediate frequeny filter is proessed further, retified by a detetor, digitized, and applied to the vertial plates of a athode-ray tube to produe a vertial defletion on the CRT sreen. A rap generator deflets the CRT bea horizontally aross the sreen fro left to right. The rap generator also tunes the loal osillator so that its frequeny hanges in proportion to the rap voltage. The spetru analyzer shown above bears a strong reseblane to a superheterodyne AM radio that reeives ordinary AM radio broadasts. The output of the spetru analyzer is the sreen of a CRT instead of a speaker, and the loal osillator is tuned eletronially instead of ehanially by a seletor knob. qqq q detetors To onvert the IF signal fro the IF bandpass filter to a onstant voltage that the display an orretly proess, soe sort of detetor ust be used. Coon types of detetors used in spetru analyzers are peak, quasi-peak, and envelope detetors. Many regulatory agenies speify separate liits for these detetors. This is done beause infrequently ourring events will result in a easured quasi-peak level that is saller than what would be easured with a 7-13
peak detetor. Sheati diagras of peak and quasi-peak detetors are shown below. These diagras are siple approxiations of atual detetors that ay be ore opliated. In the peak detetor, the IF signal is retified by the diode. The positive voltage that reahes the apaitor harges the apaitor to a axiu value. This voltage is then proessed and displayed. Therefore, even infrequent events will be easured by the peak detetor. The quasi-peak detetor has a resistor in parallel with the apaitor. This resistor provides a urrent path through whih the apaitor an disharge. Therefore, infrequent events ight give a lower easureent with a quasi-peak detetor beause the voltage stored in the apaitor deays faster in tie than in the peak detetor. Peak Detetor + + V in - - V out Quasi-Peak Detetor + + V in - - V out Figure 11. Peak and Quasi-Peak Detetors The reason for the distintion between detetors is that the intent of the regulatory liits is to prevent interferene in wire and radio ouniations reeivers. Infrequent spikes and other short duration noise events do not substantially prevent the reeption of desired inforation. qqq q Fast Fourier Transfor spetru analyzers Another type of spetru analyzer is a Fast Fourier Transfor (FFT) spetru analyzer. The FFT spetru analyzer works by sapling and digitizing the input signal and then perforing a disrete FFT on the digitized signal. FFT spetru analyzers are able to preserve the phase inforation of a signal, whih is diffiult for a siple superheterodyne spetru analyzer. Soe of the disadvantages of an FFT spetru analyzer are that the sensitivity, frequeny range, and overall dynai range are lower than urrent superheterodyne spetru analyzers Referenes 1. Streler, F., Introdution to Couniations Systes, Addison Wesley Publishing Copany, third edition, 199.. White, G., Mobile Radio Tehnology, Newnes, 1995. 3. HP Appliation Note 150, Spetru Analysis Basis 7-14