In most communication systems, the modulated signal has the form

Transcription

1 Chaper : Basic Modulaion Techniques In mos communicaion sysems, he modulaed signal has he form x () A()cos[ ()], (-1) c c where c is known as he carrier frequency, A() is he envelope and () is he phase. Ampliude A() and phase () may depend on message m(). When A() depends linearly on he message, and is a consan independen of m, we have linear modulaion. When () depends on m(), we have nonlinear modulaion. Linear Modulaion Double Sideband (DSB) is he firs form of linear modulaion we will consider. The general form of a DSB signal is x () A m()cos[ ], (-) DSB c c 0 where A c and are consans. For convenience, we will assume ha = 0. Figures -1a hrough -1c depic a block diagram of a DSB modulaor, a sinusoidal message m and he DSB ime domain wave form x DSB (), respecively. Noe ha every sign change in m() resuls in a 180 phase shif in he ransmied signal x DSB (). DSB is very popular when used o ransmi digial daa. In his applicaion, m() is a digial waveform ha swiches beween +1 and -1 vols. Hence, m() swiches he phase of he ransmied carrier by radians. For his reason, for a ±1 binary message, he modulaion is called phase-shif keying. The Fourier ransform of x DSB is X DSB(j ) x A DSB() M(j j c ) M(j j c ) c F, (-3) Laes Updaes a hp:// -1

2 a) m() x DSB () = A c m()cos c A c cos c m() b) x DSB () c) Figure -1: a) Block diagram of a DSB modulaor. b) Sinusoidal message. c) The resuling x DSB (). where M(j) = F [m()] is he Fourier ransform of he message. As shown by Figure -, X DSB (j) is a scaled version of he message ranform ha has been ranslaed o ± c. As is usual, we will assume ha he message bandwidh is small compared o c, so x DSB () is a narrow-band signal. In general, X DSB (j) conains a discree carrier componen (a specral line) a c, an upper sideband (he USB is he porion of X c (j) ha lies above he carrier c ) and a lower sideband (he LSB is he porion of X c (j) which lies in he frequency range 0 < < c ). If m() has no DC componen, hen he carrier in X c (j) will be suppressed (any nonzero DC componen Laes Updaes a hp:// -

3 M(j) -W W X DSB (j) ½A c M(jj c ) ½A c M(jj c ) c c Figure -: Specrum M(j) of message and specrum X DSB (j) of DSB signal. of m() will lead o a nonzero carrier componen). In many applicaions, in order o improve efficiency, we seek o allocae no ransmier power o he carrier (he carrier conveys no informaion abou m() so i is desirable o allocaed no power o he carrier). Finally, noe ha he ransmission bandwidh is wice he message bandwidh. DSB Demodulaion We assume ha he received signal is a replica of he ransmied signal; ha is, he signal x DSB() Acm() cos c (-4) is received. As shown by Figure -3, demodulaion involves muliplying x DSB by a phase coheren replica of he carrier and hen low-pass filering he produc. The oupu of he demodulaor s muliplier is d() [Acm()cosc]cos c Acm() Acm()cos c. (-5) The low-pass filer (LPF) following he muliplier filers ou all componens cenered a c. The oupu of he LPF is Laes Updaes a hp:// -3

4 x DSB cos c a) x DSB () = A c m()cos c LPF A c m() cos c x DSB () b) x DSB (){cos c } c) Figure -3: a) DSB demodulaor, b)x DSB and c) produc of x DSB and coheren carrier. y () A m(). (-6) d c A fundamenal problem wih DSB is he need for a phase coheren reference (i.e., Acos c on Fig. -1a) a he receiver. Complicaing his problem is he fac ha a carrier may Laes Updaes a hp:// -4

5 no be ransmied, in many applicaions. Le us analyze he effecs of a phase error in he carrier used o demodulae x r. Assume ha our local reference is cos( c + ()), where () is a phase error erm. The muliplier s oupu is d() [A m()cos ]cos( ()) A m()cos () A m()cos( ()), (-7) c c c c c c and he oupu of he LPF is, a bes, y () A m() cos () d c (-8) (we assume his signal is wihin he pass band of he LPF). In y d, he ime varying erm cos() could inroduce serious disorion. On he oher hand, depending on he applicaion, i many no maer much, if kep small. When m() is human voice, we usually can olerae a small nonzero frequency error d/d and sill make ou wha is being said. On he oher exreme, when m() is digial daa, and a compuer inerpres he demodulaed y d, small phase errors can be devasaing. There are ways o regenerae a phase coheren carrier a he receiver, even if one is no ransmied. One commonly used mehod squares he received DSB signal o produce 1 1 r c c c c c x () A m ()cos A m () A m ()cos. (-9) If m() is a power signal, hen m () has a nonzero DC average. In his case, x r has a discree specral componen a c which can be exraced by a narrow band filer cenered a c. The exraced c componen is divided by wo in frequency (by a D flip-flop, for example) o generae a coheren reference a he receiver. For he case m() = ±1 is a binary daa sequence (i.e., PSK), Figure -4 depics a block Laes Updaes a hp:// -5

6 Inpu m() sin c -½cos c Phase Locked Loop BPF x ( x BPF c Loop Filer m() LPF sin c sin c VCO Figure -4: Block diagram of a squaring loop DSB demodulaor. The VCO oupu is divided by wo in frequency o obain a phase-coheren reference for coheren demodulaion of he inpu DSB signal. diagram of a DSB demodulaor ha uilizes a squaring operaion. In his applicaion, a phase lock loop (PLL) serves o recover he c componen in x (he PLL locks ono he c componen in x ). Tha is, he PLL acs like a narrow band-pass filer ha exracs he c componen from is inpu. Under proper operaion (i.e., when he closed loop phase error is small), he phase of he VCO oupu leads by / radians he phase of he PLL inpu. Hence, he VCO oupu is sin c, a resul ha is divided by wo in frequency o produce a coheren reference for demodulaing he DSB inpu. Since he demodulaor relies on he nonlinear operaion x, he demodulaor is ofen called a squaring loop. Ampliude Modulaion Ampliude modulaion was invened by Reginald A. Fessenden, a Canadian, who successfully ransmied, for he firs ime, he sound of human voice. He firs ransmied voice beween wo 50-foo owers on Cobb Island locaed in he Poomac River, Washingon D.C., December 3rd, Prior o AM, radio operaors used crude spark gap ransmiers o send only Morse code. A he ime, few people shared Fessenden's belief ha broadcasing he human voice was possible, much less pracical. When Fessenden asked he opinion of he grea Thomas Edison, Edison replied, "Fezzie, wha do you say are man's chances of jumping over he moon? I hink Laes Updaes a hp:// -6

7 one is as likely as he oher." Forunaely, Edison was wrong. I ook six years for Fessenden o refine his invenion. Then, on Chrismas Eve 1906, Fessenden made he firs radio broadcas (of speech and music) in hisory from Bran Rock Saion, Massachuses. Radio operaors on ships in he Alanic were shocked o hear a human voice emiing from he equipmen hey used o receive Morse code. Many operaors called heir Capains o he radio room, where hey heard Fessenden make a shor speech, play a record, and give a rendiion of "O Holy Nigh" on his violin. Since he 190 s, AM has been used in commercial broadcasing. Also, i is sill used in civil aviaion and amaeur radio. Mos signal generaors can be AM modulaed by a buil-in modulaor. Also, oher ypes of es equipmen can modulae/demodulae AM. AM resuls when a DC bias A is added o message m() prior o he DSB modulaion process (in wha follows, we assume ha m() has a zero DC componen). This resuls in he ransmission of a carrier componen if bias A 0. The AM signal is defined as A m() 1 am () x () A cos A cos AM c c c n c Accosc Acam n()cosc carrier componen sideband componen, (-10) where Ac AA c, m() m n (), a min{m()}. (-11) min{m()} A m n () is message m() normalized so ha he minimum value of m n () is -1. Parameer a, a 0, is known as he modulaion index. The quaniy A 1 am () c n is known as he envelope of AM Laes Updaes a hp:// -7

8 m() A+m() x AM () A A c cos c Figure -5: AM modulaor signal x c (). For a 1, he envelope is never negaive, and he message appears o ride on op of he ransmied signal. For a > 1, he signal experiences a -radian phase shif a each zero crossing of he envelope (a fac of imporan significance as discussed below). See Figure -5 for a block diagram of an AM modulaor and Figure -6 for an example of a message and AM modulaed signal. In he frequency domain, he specrum of AM is m n () 1-1 T/4 T/ 3T/4 T A c (1+a) x AM A c (1-a) -A c (1-a) -A c (1+a) T/4 T/ 3T/4 T Figure -6: Message m n and AM waveform x AM. Laes Updaes a hp:// -8

9 F X ( ) F x () A cos A a m ()cos AM AM c c c n c Aa A c c( c) ( c) M n( c) M n( c). carrier specrum sideband specrum (-1) Noe he exisence of discree carrier specral lines a ± c. Also, he ranslaed message erms M n (± c ) conain upper and lower sidebands (M n (- c ), for > c, is an upper side band while M n (- c ), for 0 < < c, is a lower side band). Finally, noe ha he ransmission bandwidh of AM is wice he message bandwidh, jus like DSB. The ransmied signal power is divided beween he carrier and informaion conveying sidebands. Power allocaed o he carrier is (in he sense ha i does no convey informaion) wased. This leads o he noion of efficiency. Efficiency of AM The average ransmied power of he AM signal is x 1 c() [A m()] Ac cos c [A m()] A c (1 cos c). (-13) If m() is slowly varying wih respec o cos c, his las equaion leads o he approximaion 1 1 x c() A c A A m() m() A c A m(), (-14) since m() = 0 by assumpion. Define efficiency as he percenage of oal power ha conveys informaion. More precisely, efficiency is he percenage of oal ransmied power ha is in he sidebands. From he las equaion, we can wrie Laes Updaes a hp:// -9

10 m Efficiency (100%) A m. (-15) Since m() = aam n () we have mn a Efficiency (100%). (-16) 1 a m n For a 1 he maximum efficiency is 50% (for a square wave message wih a = 1). If m() is a sine wave, and a = 1, hen efficiency = 33%. For mos complex messages, such as voice, efficiency is under 10%. Example -1: Deermine he efficiency and oupu x AM () for an AM modulaor operaing wih a modulaion index of.5. The carrier power is 50 was, and he message signal is m 9 m() 4 cos[ ] sin[ ], (-17) m a graph of which is depiced by Figure -7. Soluion: Observe he message signal shown in Fig..7. The minimum value of m() is , m() m Min a m (.435) Laes Updaes a hp:// -10

11 and he minimum falls a m = (.435). The normalized message signal is given by 1 m n() 4cos[ m ] sin[ 9 m].9166cos[ m ].4583sin[ 9 m] (-18) The mean-square value of m n () is 1 1 n (-19) m () (.9166) (.4583).551 Finally, he efficiency is.5(.551) Efficiency (100%) 11.60% 1.5(.551) (-0) Since he carrier power is 50 was, we have 1 (A c) 50 (-1) which implies ha A 10. Since sin(x) = cos(x - /), we can wrie c x c() cos( m ).4583cos( 9 m ) cos c. (-) Transmied Power in AM Signal The ransmied AM signal is given by Laes Updaes a hp:// -11

12 AM c n c x () A 1 am () cos (-3) The insananeous ransmied power is x AM (). The average power in x AM is given by 1 (A c ) PAVG x AM () (A c ) 1 am n () (1cosc ) 1a m n (), (-4) was. To obain his resul, we used he fac ha message m() has an average value of zero. Ofen, power is specified in erms of peak envelope power. The envelope A c [1 + am n ()] is slowly varying wih respec o he RF carrier cos c. Over every cycle of he RF carrier, he envelope is approximaely consan. The peak envelope power (PEP) is he insananeous power [x AM ()] averaged over he RF cycle having he greaes ampliude. Hence, we can wrie (A c ) PPEP max1 am n (). (-5) c For m n = cos m and a = 1, we ge P AVG = 3 A /4 and P PEP = A, so P PEP is abou.7 imes P AVG. For a message consising of a human voice, he PEP power migh be wo or hree imes (or more) he average power. AM Coheren Demodulaion Ampliude modulaion can be demodulaed coherenly, see Figure -8. The demodulae oupu conains a consan DC erm ha is usually eliminaed by a lack of DC response in he audio sages ha follow he demodulaor. The coheren reference needed by he demodulaor c x () A AM c 1 am n () cosc LPF A c [1 a m ()] n cos c Figure -8: Coheren demodulaion of AM. Laes Updaes a hp:// -1

13 + + x AM () C R v ou () m() - - Figure -9: A simple envelope deecor. can be supplied by phase locking a PLL ono he carrier componen of x AM. The PLL acs like a narrow-band filer ha exracs he carrier componen of he signal. Noe ha coheren demodulaion can be used regardless of he modulaion index a. AM Demodulaion - Envelope Deecion If modulaion index a is equal o, or less han, uniy (a 1), AM can be demodulaed by a very simple echnique called envelope deecion. On he oher hand, if a > 1, envelope deecion will no work; he deecor oupu audio will be highly disored. The reason for his is simple. For a > 1, he signal experiences a 180 phase change a each envelope sign change, and envelope deecors are insensiive o signal phase. So, an envelope deecor will no respond o sign changes in he AM signal envelope, and disorion of he recovered audio resuls. A simple envelope deecor will only work if 0 a 1. Figure -9 depics a schemaic diagram of a simple envelope deecor. As long as envelope Ac 1 am n () is non-negaive, message m() appears o ride on op of half-wave recified x c (). In his case, a close approximaion of can be obained by smoohing he oupu of he diode wih an RC circui. The ime consan of he RC smoohing circui is no exremely criical. However, as a general rule of humb, bes resuls can be obained if Ac 1 am n () 1 RC 1, (-6) f f c B where f c is he carrier frequency in Hz, and f B is he message bandwidh, in Hz. The diode is assumed o have a small forward on resisance; he charging ime consan is exremely small (charging occurs when x AM > v ou ). Excep for he drop across he diode, Laes Updaes a hp:// -13

14 oupu v ou follows inpu x AM when he diode is conducing. When x AM < v ou, he diode is no conducing, and capacior volage v ou discharges hrough he resisor. If he discharging ime consan RC is oo small, a severe saw-ooh-like buzz, a frequency f c, will be imposed on he demodulaed message. If RC is oo large, he oupu will floa on envelop peaks, and severe disorion will occur. I is imporan o realize ha, due o he nonlinear swiching acion of he diode, he role of he RC circui is o smooh he oupu and form a signal ha follows closely he modulaion envelope. In his nonlinear circui, do no hink of he RC circui as jus a convenional, single-pole low-pass filer. A relaively simple upper bound can be obained on ime consan RC for he case of a sinusoidal message. As shown on Figure -10, assume ha he capacior discharges from he carrier peak value E 0 = A 1 acos c m0 a ime 0. Noe ha 0 is associaed wih a peak in a cycle of he carrier, no he message or envelope (cos m 0 can be any value beween 1 and +1). For a range of beween 0 and 0 + 1/f c, he capacior is discharging, so he capacior volage is ( )/RC () 0e V 0 c E. (-7) v OUT x AM x AM E /f c Figure -10: Posiive half of x AM shown as doed-line graph. Oupu v OUT () depiced as solid line graph. E 0 = v OUT ( 0 ) = A c [1+acos m 0 ], a local peak in he carrier (he message and envelope may no peak a 0 ). Laes Updaes a hp:// -14

15 The ime inerval beween wo successive carrier peaks is 1/f c = / c. Since RC >> 1/ c, he quaniy /RC is small for ime beween carrier peaks and () E 0 c 0 1 RC V. (-8) If V c () is o follow he envelope, hen i is required ha acos ( 1/f ). (-9) RCfc 1acosm0 m 0 c Since m << c, we have (use he ideniy cos(+) = cos cos - sin sin and he fac ha cos and sin for small 1acos ( 1/f ) 1acos( /f ) m 0 c m 0 m c m 1 a cos( m m0 )cos a sin( m0 )sin fc fc (-30) m 1acos( m0 ) a sin( m0 ). fc Now, he las wo equaions combine o yield 1 m0 a sin( m0 ). (-31) RCf f 1 acos m c c This resul can be wrien as 1 a cos m0 a m sin m0 RC RC (-3) Laes Updaes a hp:// -15

16 or m m 0 m 0 m m 0 m a sin cos a sin an (1/ RC) RC RC 1 RC. (-33) Since m o is arbirary, we mus have a 1 1 m RC RC (-34) and RC 1 a a m, (-35) he desired upper bound on ime consan RC. Malab Envelope Deecor Simulaion The Malab program lised in Figure -11 envelope deecs he AM signal v in () 1 asin() sin(w) (-36) over he ime period 0. The resuls are depiced by Figs. -1 hrough -15; hese figures show he inpu v in as he hin line plo, and hey show he diode-based envelope deecor oupu as a hick line plo (riding on op of v in ). As given by (-36), AM signal v in () uses m = 1 rad/sec, and (-35) yields an upper bound of RC < 1 a /a for floaing Laes Updaes a hp:// -16

17 disorion no o occur. %Envelope.m %Envelope.m deecs an AM waveform global RC, alpha, W, D; = 0 : *pi/1000 : *pi; %Allocae memory for inpu and oupu arrays Vin = zeros(1,1001); Vou = zeros(1,1001); %Define inpu array Vin = ( 1 + alpha*sin() ).*sin(w*); %Firs poin of oupu is he iniial value of he envelope Vou(1) = 1; %Compue oupu over all poins for i = :1001 if Vin(i) > Vou(i-1); Vou(i) = Vin(i); else Vou(i) = Vou(i-1)*exp(-D/RC); end end %Plo inpu hen pause plo(, Vin) axis([0 *pi -1-alpha 1+alpha]) pause %Hi any key o plo oupu plo(, Vou) axis( [0 *pi 0 1+alpha] ) Figure -11: Malab program for simulaion of an envelope deecor. To aid visualizaion, he values of m = 1, W = 50 and D = /1000 (oupu ime sep) were used in all plos. The carrier frequency is 50 imes he message frequency (a raio of 50 is smaller han wha you would normally encouner in pracice). Also, a =.5 (50% modulaion deph) was used for Figures -1 hrough -14. For Fig. -1, he RC ime consan is /10, a lile bi oo small. The value RC = /10 =.68 is significanly below upper bound 1.5 /.5 Laes Updaes a hp:// -17

18 Figure -1: Inpu AM signal (hin line plo) and oupu of envelope deecor (hick line plo). The modulaion index is ½. The RC ime consan is /10, a value ha is a bi oo small. = 1.73 (so no floaing disorion occurs). For Fig. -13, he RC ime consan is /5, a value ha is jus abou righ. The value RC = /5 = 1.6 is less han upper bound 1.5 /.5= 1.73 (so no floaing disorion occurs). For Fig. -14, he RC ime consan is /3, a value ha is a bi oo large. The value RC = /3 =.09 is above upper bound disorion occurs. 1 a /a = 1.73, and floaing As should be eviden by now, a good value for RC depends on modulaion index a. As Figure -13: Inpu AM signal (hin line plo) and oupu of envelope deecor (hick line plo). The modulaion index is ½. The RC ime consan is /5, a value ha is jus abou righ. Laes Updaes a hp:// -18

19 Figure -14: Inpu AM signal (hin line plo) and oupu of envelope deecor (hick line plo). The modulaion index is ½. The RC ime consan is /3, a value ha is a bi oo large (he deecor oupu floas above he rue envelope over par of he modulaion period). index a approaches uniy, you mus use smaller values of RC o preven demodulaor oupu floaing wih is associaed harmonic disorion (some oupu disorion is unavoidable for near-uniy index values). For example, Figure -13 shows good resuls wih RC = /5 and a = ½. However, significan disorion occurs if he same value of RC is used wih a =.95, as can be seen from examining Figure Figure -15: Inpu AM signal (hin line plo) and oupu of envelope deecor (hick line plo). A value of modulaion index a =.95 was used o obain his plo. The RC ime consan is /5, a value ha is oo large (for a =.95) as is eviden by he significan amoun of deecor oupu floaing. Laes Updaes a hp:// -19

20 ( ) LPF A c 1 am a m vin A[1 c am]cos n c v0 n n Figure -16: A square-law deecor. Square-Law Deecor An ampliude modulaed signal can be demodulaed by a square law deecor, if he modulaion index is sufficienly small. As depiced by Figure -16, a square law deecor forms is oupu v o by low-pass filering he square of he inpu v in. If v in = ge 1 am () 1 c n c c n n c A cos A 1 a m () a m () (1 cos ). (-37) Ac 1 am n () cosc we The low-pass filer removes he c componen o produce he oupu Ac v o() 1am n() a m n(). (-38) The second-order erm m n () inroduces second-order harmonic disorion ha can be severe if modulaion index a is no small. On he oher hand, if a << 1 and a blocking capacior is used o remove he DC componen, his las resul can be approximaed by o() Ac a m n() v. (-39) AM Inpu oupu Figure -17: Approximaion o Square Law Deecor Laes Updaes a hp:// -0

21 The nonlinear squaring operaion can be implemened by a diode ha is forward biased ino he knee of is i-v characerisic, see Fig The DC volage source (he baery on he figure) serves o forward bias he diode ino is square law region. I is imporan o remember ha he inpu signal is small compared o he DC bias so ha he diode always is forward biased (unlike he envelope deecor). Unlike is use in he envelope deecor, he RC nework serves as a band-pass filer o exrac he demodulaed message. To some degree of efficiency, a wide range of nonlinear operaions will demodulae AM. In fac, in he presence of a srong ransmied AM signal, i is hard o preven demodulaion of he signal by recifying connecions/juncions in elephone ses, loudspeaker coils, ec. Ofen, i is necessary o place by-pass capaciors across devices in order o shor-circui picked-up radio frequency (RF) currens and preven unwaned demodulaion of a srong AM modulaed signal. Single Sideband Modulaion In DSB, eiher sideband conains sufficien informaion o reconsruc he message m(). Eliminaion of one of he sidebands resuls in single sideband modulaion (SSB). The signal is known as lower sideband (LSB) if he upper sideband is eliminaed, and i is known as upper sideband (USB) if he lower sideband is eliminaed. Figure -18 depics single-sided specral plos of he message M(j), double sideband X DSB (j), lower sideband X LSB (j), and upper sideband X LSB (j). In 1915, SSB was paened by John Carson. Originally, in he elephone sysem, i was used o frequency-division muliplex (FDM) muliple voice channels ono one cable. In radio communicaion oday, SSB is very popular for he ransmission of voice informaion, especially in he HF porion (3 30Mhz) of he radio specrum. SSB has some obvious advanages. Firs, i requires only half of he ransmission bandwidh, as compared o DSB and AM. In his era of governmen sponsored specrum aucions, specrally efficien forms of modulaion can improve boh sysem performance and one s boom line. When sysem design and bandwidh are opimized for a given modulaion forma, DSB and SSB offer similar performance in erms of receiver oupu signal-o-noise Laes Updaes a hp:// -1

22 M(j) a) b) X DSB (j) LSB USB 0 W c -W c c +W Posiive Frequency Axis c) X USB (j) d) X LSB (j) c c +W Posiive Frequency Axis c -W c Posiive Frequency Axis Figure -18: Single-sided specral plos of a) message M(j), b) double sideband X DSB (j), c) upper sideband X USB (j) and lower sideband X LSB (j). (SNR) raio (for a given received signal power and noise specral densiy). However, when compared o AM wih is high percenage of power allocaed o he carrier, SSB offers much improvemen over AM in erms of receiver oupu SNR (for a given received signal power and noise specral densiy). There are wo commonly used mehods o generae SSB. The firs is called he phasing mehod, and i gained populariy early in he pracical developmen and use of SSB (primarily in he 1950 s). The second mehod is called he filer mehod. In he early days of SSB developmen, good sideband filers were expensive and hard o obain (so he phasing mehod was dominan). However, in he 1960 s and 1970 s, significan echnical advances were made in he design and manufacure of crysal band-pass filers, and good sideband filers became inexpensive. For his reason, he filer mehod of SSB generaion is dominan oday (however, wih he adven of powerful DSP echnology, he phasing mehod is making a comeback). Laes Updaes a hp:// -

23 x DSB () = A c m()cos c m() Sideband Filer x LSB () or x USB () A c cos c Figure -19: Filer mehod of single sideband generaion. Filer Mehod of Single Sideband Generaion Figure -19 depics a simplified block diagram of a filer-ype SSB generaor. Firs, double sideband is generaed. Then, wih he aid of a seep-skir, band-pass filer (known as a sideband filer in he lieraure), he desired sideband is seleced and he undesired sideband is filered ou. Of course, in pracice, he filering process is imperfec, and a small amoun of he unwaned sideband is ransmied (along wih a small amoun of unbalanced carrier). However, 40dB (or more) of unwaned sideband suppression is obained easily wih commercially available sideband filers. Obviously, he sideband signal is generaed a a fixed frequency c = f c. Commercially available, quarz crysal-based-echnology sideband filers are available a f c = 9Mhz and oher sandard frequencies. Also, mechanical filers are available a f c = 455Khz and oher sandard frequencies (ypically, mechanical filers are under 1Mhz). Afer generaion a a fixed frequency, he single sideband signal is heerodyned (using one-or-more mixer sages) o he desired ransmi frequency. Then, by using a linear power amplifier, he signal is increased in power and sen up he ransmission line o he ransmiing anenna. Phasing Mehod of Single Sideband Generaion We develop he phasing mehod for generaing LSB firs. LSB will be generaed if a DSB signal is passed hrough an ideal low-pass filer ha exends from - c o + c, as depiced by Figure -0. Filer H L (j) can be represened as Laes Updaes a hp:// -3

24 H L (j) X DSB (j) c 0 c X LSB (j) = H L (j)x DSB (j) c 0 c Figure -0: Developmen of lower sideband. 1 L c c H (j ) [sgn( ) sgn( )], (-40) a resul ha is depiced by Figure -1. Apply he DSB signal o H L ; in he frequency domain, we wrie sgn( + c ) c 0 c -sgn( - c ) c 0 c H L (j) = ½[sgn( + c )-sgn( - c )] 1 c 0 Figure -1: Consrucion of H L for he generaion of lower sideband. Laes Updaes a hp:// -4 c

25 X (j ) X (j )H (j ) LSB DSB L A c [M(j j c ) M(j j c )][sgn( c ) sgn( c )] 4 A c [M(j j c )sgn( c ) M(j j c )sgn( c )] 4. (-41) A c [M(j j c )sgn( c ) M(j j c )sgn( c )] 4 Noe ha here are four erms on he righ-hand-side of his las equaion. The second and hird erms combine o form ¼A c [M( + c ) + M( - c )]. The firs and fourh erms combine o produce ¼A c [M( + c )sgn( + c ) - M( - c )sgn( - c )]. Hence, we can wrie A X c LSB(j ) [M(j j c ) M(j j c )] 4 A c [M(j j c )sgn( c ) M(j j c )sgn( c )] 4. (-4) The ime-domain LSB signal is jus he inverse ransform of his las resul. Firs, noe ha -1 A c A M(j j c) M(j j c) c m()cosc 4 F. (-43) Furher, noe ha F[m()] ˆ = -jsgn( )M(j ) F ˆ j c [m()e ] = -jsgn( c)m(j j c) (-44) so ha Laes Updaes a hp:// -5

26 Audio Channel #1 Audio Message m() Hilber Transformer ˆm() Audio Channel # ½A c cos c Carrier Oscillaor ½A c sin c + + for LSB - for USB Modulaor Oupu F A [M(j j c )sgn( c ) M(j j c )sgn( c ) 4-1 c Figure -: Simplified block diagram of a phasing mehod SSB generaor. j A c m()e ˆ m()e ˆ 4 j c j c (-45) Ac ˆm()sin c Finally, Equaions (-43) and (-45) can be used o deermine he inverse ransform of (-4); his leads o he desired resul x LSB() 1 X LSB(j ) 1 Acm()cosc 1 A cm()sin ˆ c F, (-46) a useful formula for LSB. A similar developmen leads o x USB() 1 X USB(j ) 1 Acm()cosc 1 A cm()sin ˆ c F (-47) for upper sideband. The block diagram depiced by Figure - illusraes how o implemen Equaions (-46) and (-47). Laes Updaes a hp:// -6

27 In a pracical phasing SSB modulaor, he Hilber ransformer would be replaced by a wide-band 90 phase shif nework, a nework/sysem ha acceps m() as inpu and produces wo nearly equal ampliude and nearly orhogonal messages o feed he balanced modulaors. Figure -3 depics a block diagram of such a sysem. The nework s magniude response from inpu o eiher oupu would be nearly consan over he message bandwidh of ineres. Also, over he message bandwidh, here would be (nearly) a 90 differenial phase shif beween he wo oupus. The phase relaionship beween he inpu and eiher oupu is no imporan. In a pracical phasing modulaor, he funcionaliy described by Figure -3 would be used o produce he wo base band audio signals ha are fed o he balanced modulaors depiced on Figure -. Alernae Developmen of SSB As shown by Figure -4, le M p (j) and M n (j) denoe he posiive and negaive, respecively, pars of he ransform M(j) of message m(). From inspecion of Figure -4a, we can wrie m 1 () m() Wideband 90 Phase Shif Nework m () -90 M(j 1 ) M (j ) M(j ) M(j ) consan (over message bandwidh) M ( j ) jm ( j ) (over message bandwidh) 1 Figure -3: Wide-band 90 phase shif nework. Laes Updaes a hp:// -7

28 M(j) M p (j) M n (j) -W W W -W (a) X USB (j) ½A c M n (j + j c ) ½A c M p (j j c ) ½AcM p (j + j c ) c (b) X LSB (j) c ½A c M n (j j c ) c (c) Figure -4: Alernae Developmen of SSB c 1 p M (j ) F[m() jm()] ˆ 1 n M (j ) F[m() jm()] ˆ. (-48) By definiion, an USB signal has a frequency domain represenaion (see Fig. -4b) 1 1 USB c p c c n c X (j ) A M (j j ) A M (j j ) (-49) Take he inverse ransform of (-49) o obain Laes Updaes a hp:// -8

29 1 j c 1 j c USB c ˆ 4 4 c ˆ x () A [m() jm()]e A [m() jm()]e, (-50) j jc jc jc jc 1 e e 1 e e Am() c Am() c ˆ 1 1 c c c ˆ c A m() cos( ) A m() sin( ) he desired formula for an upper sideband modulaed signal. The formula for a lower sideband signal can be developed in a similar manner. Inspecion of Figure -4c reveals 1 1 LSB c p c c n c X (j ) A M (j j ) A M (j j ). (-51) The inverse ransform of his signal is ˆ c ˆ 1 j 1 jc LSB 4 c 4 c x () A m() jm() e A m() jm() e, (-5) j jc jc jc jc 1 e e 1 e e Am() c Am() c ˆ 1 1 c c c ˆ c Am()cos Am()sin he desired formula for a lower sideband modulaed signal. Demodulaion of SSB - Produc Deecors As depiced by Figure -5, SSB can be demodulaed by muliplying i by a phase coheren carrier and filering he produc by a low-pass filer. The produc of he SSB signal and he coheren reference yields Laes Updaes a hp:// -9

30 x LSB () or x USB () X d() Lowpass Filer y D () 4cos( c + ()) Correc Demodulaion Requires () = 0 Figure -5: SSB demodulaor. d() 1A 1 cm()cosc A cm()sin ˆ c 4cos( c ()) Ac m()cos () m()cos(c ()) m()sin ˆ () m()sin( ˆ c ()) (-53) Low-pass filering produc d() produces y () A m()cos () A m()sin (). (-54) d c c ˆ For () = 0 we obain he desired resul. Depending on he applicaion, if () 0, he erm Am()sin () c ˆ may inroduce serious disorion. In he case of human speech, i is possible o undersand he message even if small frequency errors are presen (d/d 0). For speech, d/d is adjused manually by a lisener who adjuss a radio uning dial unil he/she can copy he ransmission. (The ear/brain is no sensiive o saic phase errors; i is sufficien o make d/d small.) Demodulaion of SSB - Carrier Reinserion/Envelope Deecion SSB can be demodulaed by he mehod illusraed by Figure -6. The oupu of he summer operaion is 1 1 c c c ˆ c c e() A m()cos A m()sin Kcos 1Am() 1 c Kcosc Am()sin c ˆ c. (-55) Laes Updaes a hp:// -30

31 In erms of magniude and phase, Equaion (-55) can be wrien as e() R()cos( c ()), (-56) where 1 1 c c ˆ R() [ A m() K] [ A m()] () Tan 1 1 c 1 c Am() ˆ Am() K (-57) are he envelope and phase, respecively, of e(). Now, he oupu of he demodulaor depiced by Figure -6 is given by 1 1 y d() [ A cm() K] [ A cm()] ˆ. (-58) If consan K is chosen large enough so ha 1 1 [ A cm() K] [ A cm()] ˆ, (-59) we can approximaed y d () as x LSB () or x USB () + + e() Envelope Deecor y D () Kcos( c ) Figure -6: Demodulaion of SSB using carrier reinserion Laes Updaes a hp:// -31

32 d 1 y () A m() K, (-60) c a resul ha conains he desired message. Transmied Power in SSB Waveform The SSB signal is represened as 1 1 SSB c c c ˆ c x () A m()cos A m()sin. (-61) The insananeous ransmied power is x SSB(). The average ransmied power is 1 PAVG x SSB() A 4 c m ()cos c m ˆ ()sin c A c m () m ˆ () A c m (). (-6) The peak-envelope-power (PEP) is of ineres. The SSB signal can be represened as Ac SSB ˆ x () m () m () cos( c an 1 m/m ˆ ). (-63) The envelope and phase are slowly varying relaive o he carrier cos c. Over every RF cycle he envelope and phase are approximaely consan. The peak envelope power (PEP) is he insananeous power averaged over he RF cycle having he greaes ampliude. Hence, he PEP power is 1 Ppep Ac max m () m ˆ () 8 (-64) Laes Updaes a hp:// -3

33 For m() = cos m, we have P avg = P pep = A c /8. For m() a human voice, a general rule of humb is ha P pep is beween wo and hree imes P avg. Angle Modulaion The general angle modulaed signal is described by x c() Accos[ c ()], (-65) where A c and c are consans, and angle depends on he message m(). Unlike he modulaion mehods discussed so far, an angle-modulaed signal is a nonlinear funcion of he message. Phase modulaion and frequency modulaion are wo forms of commonly-used angle modulaion. The insananeous phase of signal x c is given by c (). (-66) Ofen, angle is called he insananeous phase deviaion. The insananeous frequency of x c is. (-67) d d c d d The quaniy d/d is called he insananeous frequency deviaion. The peak frequency deviaion d peak max d (-68) is an imporan parameer in pracical FM sysem design. The wo basic ypes of angle modulaion are 1) phase modulaion (PM) and ) frequency Laes Updaes a hp:// -33

34 modulaion (FM). We will consider boh; however, we will place mos of our emphasis on FM. In PM, he phase () = Kpm() (-69) is proporional o message m(). Consan K p > 0 is he modulaion index for PM, and i has unis of radians/vol. The PM signal is x c() Accos[ c Kpm()]. (-70) In FM, he frequency deviaion is proporional o he message so ha d () Km() f d, (-71) or () = Kf m(x)dx. (-7) Posiive K f is he frequency deviaion consan, expressed in radians/second-vol. Someimes, frequency deviaion is specified in Hz. In his case, he relevan consan is f d where K f f, (-73) d and f d is expressed in Hz/vol. The FM signal is x c() Accos[ c Kf m(x)dx]. (-74) Laes Updaes a hp:// -34

35 1 m() cos( c ) cos( c + K p m()) cosc Kf m Figure -7: Message m(), unmodulaed carrier cos( c ), phase modulaed carrier cos( c + K p m()) (wih K p = /) and frequency modulaed carrier cos( c + K f m) (K f c in magniude ). Figure -7 depics a uni sep message, an unmodulaed carrier, a phase modulaed carrier and a frequency modulaed carrier. Noe ha he ransmied power is consan and independen of message m(). The ransmied power is x c() A 1 1 c cos [ c ()] A c (1 cos[c ()]) Ac, (-75) Laes Updaes a hp:// -35

36 he approximaion (a very good one in pracice!) due o he fac ha varies slowly relaive o cos c. Unlike AM, DSB and SSB, he angle modulaed ransmier duy cycle is 100%. Hence, an angle-modulaed ransmier mus have a huskier power supply, and more conservaively raed componens, han an equivalen-power AM/DSB/SSB ransmier. Since is envelope is consan, an angle modulaed signal can be amplified by a simple non-linear power amplifier (unlike an AM/DSB/SSB signal which mus employ a linear power amplifier). Narrow-Band Angle Modulaion The angle modulaion is said o be narrow band if () << 1 for all ime. For his case, we can wrie he ransmied signal as x () A cos[ ()] A cos () cos A sin () sin c c c c c c c A cos A ()sin c c c c (-76) () = Kpm(), for PM. (-77) () = K m(x)dx, for FM f In his las resul, we have use he fac ha cos 1 and sin. Narrow-band angle modulaion is used by all municipal services (i.e., police, fire, ciy, sae personnel, ec.), he miliary, amaeur radio operaors and many oher groups. Consider aking he Fourier ransform of he narrow-band angle modulae signal. The ransform is F x () FA cos A ()sin c c c c c A A c ( c) ( c) j ( c) ( c) c, (-78) where Laes Updaes a hp:// -36

37 p ( ) F () K M(j ), for PM. (-79) K j f M( j ), for FM From his we conclude ha if m() has message bandwidh W << c (he usual case in applicaion), hen x c () will be a band-pass signal wih bandwidh W. (Ofen, narrow-band angle modulaion is defined o be he case where he ransmission bandwidh is wice he message bandwidh.) In he ampliude specrum of FM, noe ha low frequencies in M(j) = F [m()] are emphasized more han high frequencies, because of he facor K f / in (j). Figure -8 depics a message ampliude specrum, he ampliude specrum of he resuling narrow band M(j) A m -W W Ampliude Specrum of Message A c A c A c A m K p / c -W c c +W Ampliude Specrum of Narrowband PM c -W c c +W A c A c AA c mkf c AA c mkf c c -W c c +W Ampliude Specrum of Narrowband FM c -W c c +W Figure -8: a) Message ampliude specrum, b) ampliude specrum of narrow band PM, c) ampliude specrum of narrow band FM. Laes Updaes a hp:// -37

38 PM signal, and he ampliude specrum of he resuling narrow band FM signal. Tone Modulaion Suppose m() is a simple one. To keep hings simple, assume ha m() Amsin m for PM. (-80) =A cos for FM m m For his sinusoidal message we have () K m() K A sin for PM p p m m A. (-81) K m(x)dx K sin for FM m f f m m To summarize his we wrie () sin m, (-8) where K A K p f m A m m for PM for FM. (-83) Consan is known as he modulaion index for one modulaion (symbol is used only when one modulaion is under consideraion). Narrow band modulaion requires ha << 1. Finally, noe ha K f A m is he peak frequency deviaion for narrow band FM. For police, municipal services and for amaeur radio, narrow-band FM uilizes a peak frequency deviaion of around 5 khz; he peak frequency deviaion occurs on voice peaks. Laes Updaes a hp:// -38

39 Phasor Diagram for Narrow-Band Tone FM/PM For narrow-band one modulaion wih = sin m we have x () A cos A ( sin )sin c c c c m c. (-84) Ac Ac Accos c cos[( c m)] cos[( c m)] A phasor diagram for x c is given by Figure -9. The carrier is he reference, so i remains saionary. The componen a c + m (he USB) is increasing in angle relaive o he carrier and is drawn as roaing m radians/second in a couner clockwise direcion. The componen a c - m (he LSB) is decreasing in angle relaive o he carrier and is drawn as roaing m radians/second in a clockwise direcion. Noe ha for narrow-band angle modulaion, x c conains a componen ha is orhogonal o he carrier. This conrass wih he case for AM. For AM wih one modulaion we have x c() Ac 1a cos m cos c aa A c ccos c cos[( c m) cos[( c m)]. (-85) The phasor diagram for AM is given by Figure -30. Noe ha x c conains no componen orhogonal o he carrier. For AM, he ransmied signal is always in phase wih he carrier; he resulan x c () A c (LSB) m m A c / (USB) carrier (phase reference) Figure -9: Phase diagram for narrow-band angle modulaion. Noe ha x c has a componen ha is orhogonal o he carrier. Laes Updaes a hp:// -39

40 A c (LSB) carrier (phase reference) m m A c / (USB) resulan x c () Figure -30: Phasor diagram for ampliude modulaion. Noe ha x c has no componen orhogonal o he carrier. informaion is conained in he ampliude and no in he phase. Wide-Band Tone Modulaion (Large ) In pracice, wide-band angle modulaion is considered o be he case where he ransmission bandwidh is large compared o wice he message bandwidh. A wide-band onemodulaed signal (eiher FM or PM) can be wrien as jc jsinm x c() Accos( csinm) AcRee e. (-86) The j sin m e erm is periodic in. I can be expanded ino an exponenial Fourier series. The series expansion has he form e jsinm jnm cne n (-87) where m / m jsinm jnm 1 j(nxsin x) cn e e d e dx J / n ( ) m (-88) However, his las inegral is he well-known Bessel funcion of he firs kind of order n wih real Laes Updaes a hp:// -40

41 argumen, denoed as J n (). Hence, he Fourier series can be wrien as jsinm n jnm n e J ( )e 1 j(nxsin x) J n( ) e dx, (-89) a version of he well-known Jacobi-Anger formula. Use his expansion in he formula for x c o obain j c jnm x c() AcRee J n( )e Ac J n( )cos[( c n m)] n n. (-90) Noe ha x c has a carrier wih ampliude J 0 (), and i has an infinie number of sidebands, in A c J -1 A c J 1 Single-Sided Ampliude Specrum A c J - A c J - A c J A c J 3 A c J 0 A c J - A c J 4 Single-Sided Phase Specrum Figure -31: Single-sided ampliude and phase specrum. Laes Updaes a hp:// -41

42 heory. The n h sideband pair has ampliude J n (). Figure -31 depics an example of an angle-modulaed signal conaining a carrier and four pairs of modulaion sidebands. Tables and compuer programs can provide values of J n () for n 0. For negaive inegers n, we can use J n( ) J n( ), n even. (-91) J ( ) J ( ), n odd n n The ineger-order Bessel funcions saisfy a recursion relaionship. This relaionship is n J n1( ) J n1( ) J n( ). (-9) This recursion can be used in a forward direcion; given J n and J n-1, i can be used o compue J n+1. However, he forward recursion is numerically unsable. Any iniial error will grow rapidly and, afer a few ieraions of he recursion, make he resuls unusable. On he oher hand, he backward recursion is numerically sable. Given J n+1 and J n, he recursion can be used o calculae a very accurae value for J n-1, and he backward recursion can be repeaed as ofen as desired wihou fear of numerical insabiliy. Figure -3 depics graphs of a few low-order Bessel funcions. A few properies of Bessel funcions are of ineres. 1. For << 1, J 0 () and J 1 () srongly dominae J k (), k, and (-90) can be wrien as x c() Ac J n( )cos[( c n m)] n. (-93) AJ()cos c 0 caj()cos( c 1 c m)aj c 1()cos( c m) Laes Updaes a hp:// -4

43 1.0 J 0 () 0.8 J n () as a Funcion of J 1 () J () J 4 () J 6 () Figure -3: Graphs of some Bessel funcions. This approximaion represens he narrow band modulaion case.. For fixed n, J n () oscillaes wih increasing. However, he ampliude of oscillaion decays quickly wih large. 3. For fixed, he maximum value of J n () decreases wih increasing n. In fac, for sufficienly large n, we have he asympoic relaionship J n ( ) n n n!. (-94) 4. J 0 () is zero for =.4048, 5.501, ,. The signal x c () will no conain a carrier componen for hese values of. Observaion #4 provides a pracical mehod of measuring K f for an FM ransmier. Feed he oupu of an audio generaor o he FM ransmier under es. Observe he oupu of he ransmier on a specrum analyzer. Use any convenien audio frequency (say 1KHz). Increase Laes Updaes a hp:// -43

44 he audio level unil he carrier vanishes and hen calculae K f. Bandwidh of Angle Modulaed Signal - Tone Message Case Sricly speaking, a signal ha is angle modulaed wih a one message conains power over an infinie bandwidh of frequencies. However, only a finie number of sideband pairs have significan power, so he bandwidh of he signal is finie, for all pracical purposes. The power raio P r (k) is he raio of he power conained in he carrier and firs k sideband pairs o he oal power. For he case of a one message, we wrie he power raio as k ½ Ac J n( ) k nk r 0 n ½Ac n1 P(k) J ( ) J ( ). (-95) For a paricular applicaion, bandwidh can be deermined by 1) defining an accepable P r, ) solving for he required value of k (using a able of Bessel funcions), and 3) compuing bandwidh B = k m. The resuls of his procedure are given in Table 1 below. The value of k for P r.7 is indicaed by a single underscore in he able; he value of k for P r.98 is indicaed by a double underscore. For example, for =, a double underscore occurs in he fourh row (in he = column) of he able. For his case, a carrier and hree sideband pairs conain 98% of he oal ransmied power; he ransmied bandwidh is B = 6 m. For P r =.98 and ineger values of, i is noed ha k is equal o 1 +. Hence, for P r =.98 and ineger we approximae Bandwidh ( 1) m, (-96) where m is he frequency of he one message. For P r =.98 and non-ineger, his formula gives a useful approximaion o bandwidh. Laes Updaes a hp:// -44

45 k =.1 =. =.5 = 1 = = 5 = 8 = Table 1: Table of J k (), 0 k 11. Bandwidh as a Funcion of m For FM, noe ha = A m K f / m, he peak frequency deviaion A m K f divided by he one frequency m. As m decreases, increases and so does he number of significan sidebands. However, wih decreasing m, he sidebands come closer ogeher in frequency, and he required bandwidh approaches a consan (his is prediced by he formula: for small m, we have Bandwidh ( 1) m A m K f ). Bandwidh for Non-sinusoidal Modulaion For arbirary m() he ransmission bandwidh is difficul o define and compue (insead of signal bandwidh, i is beer o hink in erms of how much bandwidh is required o ransmi he signal wihou excessive disorion). Hence, we resor o a general rule of humb esimae. For arbirary m() we define he deviaion raio Laes Updaes a hp:// -45

46 peak frequency deviaion D. (-97) one-sided bandwidh of message m() For FM, his definiion is equivalen o Kf max m() D. (-98) one-sided bandwidh of message m() Deviaion raio D plays he same role for non-sinusoidal modulaion as he modulaion index plays for sinusoidal messages. Hence he ransmission bandwidh can be approximae by Bandwidh (D 1)W, (-99) where W is he one-sided message bandwidh. This las approximaion is known as Carson s Rule. (Also, in 1915, Carson paened SSB.) Experimenal daa shows ha Carson s rule provides good resuls for D << 1 (bandwidh = W, which is he narrow band modulaion case) and for D >> 1. (Carson s rule works well for D > 5.) Commercial FM Broadcasing In he Unied Saes, commercial FM broadcasing uses wideband FM in he channelized FM band ha exends from 88Mhz o 108Mhz. The Federal Communicaion Commission (FCC) assigns saions o carrier frequencies ha are spaced 00KHz apar in he FM band. The peak frequency deviaion is limied o 75KHz. The message bandwidh is limied o 15KHz. Hence, for commercial FM 75KHz D 5. (-100) 15KHz Laes Updaes a hp:// -46

47 Figure -33: Edwin H. Armsrong Carson s rule gives a bandwidh of (D+1)W = 180KHz. Experimenal daa shows his o be a lile low; a bandwidh of 00KHz is closer o realiy. Edwin H. Armsrong The Faher of Wideband FM In he early days of commercial broadcasing, ampliude modulaion was king. However, AM is plagued by saic caused by amospheric elecrical aciviy, especially during summerime hundersorms. Many researchers sough mehods for reducing he effec of saic on commercial broadcasing. Prior o he mid 1930 s, a generally acceped axiom was ha you could only reduce radio saic by decreasing ransmission/recepion bandwidh. Because of his, many leading researchers of he ime hough (and wroe in scholarly journals) ha wideband FM was worhless. This belief was no held by Edwin H. Armsrong, a leading radio/elecronics researcher of he ime. Abou 1934, Armsrong showed ha, under cerain specified condiions, you could Laes Updaes a hp:// -47

48 rade FM ransmission/recepion bandwidh for FM deecor oupu signal-o-noise raio. Tha is, wihin limis and under specified condiions, you could reduce he influence of radio noise by increasing ransmission/recepion bandwidh! A he ime, his resul by Armsrong was hard o diges, especially by hose in he AM broadcasing indusry!! The discovery of he benefis of wide-band FM was no Armsrong s firs major conribuion o he ar/science of radio. While in high school, Armsrong was an avid experimener/hobbyis in he nascen field of radio (he was a ham radio operaor). While a junior elecrical engineering suden a Columbia Universiy (abou 1913), Armsrong invened he regeneraive deecor, he use of which made radio receivers orders of magniude more sensiive (his was a big discovery a he ime). He wen on o inven he super regeneraive deecor. While serving in he US Army during WWI (in Paris France), Armsrong invened he superheerodyne receiver, he dominan receiver archiecure used oday. So, by he ime of his wide-band FM work, Armsrong was a well-known invenor in he field of radio. To help esablish his belief in he superioriy of FM for commercial broadcasing, Armsrong sared, in 1938, one of he firs commercial FM saions a Alpine, New Jersey, across he Hudson River from Yonkers, NY. (His massive and all ransmission ower sill sands oday!) In addiion, hroughou New England, he esablished he Yankee Nework of FM saions. Afer WWII, commercial FM broadcasing was pushed (by he FCC in response o a reques by David Sarnoff, presiden of RCA and a major backer of TV broadcasing) from is exising 4-o-50Mhz frequency allocaion o is curren allocaion, 88-o-108 MHz. This made exising FM equipmen obsolee, and i se FM broadcasing back by many years. During his ime, Armsrong was engaged in many paen infringemen law suies, rying o proec his several paens. This long liigaion process severely sressed Armsrong, according o his many friends. Unforunaely, healh, marial, legal and financial problems drove Armsrong, in 1954, o ake his life by jumping ou of his River House aparmen window in Manhaan. Laes Updaes a hp:// -48

49 A c ()cos( c + () ) Ideal FM Disc. y () 1 K d d d d FM Discriminaor Oupu y d () f c Slope = K D Insananeous Frequency 1 d fc d a) b) Figure -34: Ideal FM discriminaor. The consan f c is he discriminaor cener frequency, and K D is he discriminaor gain consan. Consans f c and K D compleely characerize he ideal FM discriminaor. I was no unil he 1970 s ha FM broadcasing surpassed AM broadcasing in he numbers of liseners and adverising revenue. However, oday, FM commercial broadcasing is king, rumping AM when i comes o broadcasing high-fideliy music and enerainmen (mos AM saions have survived by changing o an all spors/news/alk forma). Ideal FM Discriminaor The ideal FM discriminaor yields an oupu ha is proporional o he frequency deviaion of he discriminaor inpu, as illusraed by Figure -34. If he inpu o he ideal FM discriminaor is x () A ()cos[ ()], A () 0 for all, (-101) c c c c hen he oupu is y d() 1 K d D d, (-10) where K D is he discriminaor gain consan (unis are vols/hz). Noe ha he ideal discriminaor oupu is unaffeced by changes in he ampliude A c () of inpu x c () (we assume Laes Updaes a hp:// -49