Angle Modulation Frequency Modulation

Transcription

1 Angle Modulation Frequeny Modulation Consider again the general arrier v t =V osω t + φ ωt + φ represents the angle o the arrier. There are two ways o varying the angle o the arrier. a) By varying the requeny, Frequeny Modulation. b) By varying the phase, Phase Modulation 1

2 Frequeny Modulation In FM, the essage signal (t) ontrols the requeny o the arrier. Consider the arrier v t =V osω t then or FM we ay write: FM signal v t =V os π + requeny deviation t will depend on (t). s,where the requeny deviation Given that the arrier requeny will hange we ay write or an instantaneous arrier signal V os where i is the instantaneous angle = ω t=v osπ t=v osφ i ω t = i π t i i and i is the instantaneous requeny. i

3 Frequeny Modulation Sine φ = π t i i then dφ dt i = π i or i = 1 π dφ dt i i.e. requeny is proportional to the rate o hange o angle. I is the unodulated arrier and is the odulating requeny, then we ay dedue that i = + Δ os is the peak deviation o the arrier. ω t 1 dφ = π dt i 1 dφ π dt i Hene, we have = + Δ osω t dφ dt,i.e. i = π + πδ osω t 3

4 Frequeny Modulation Ater integration i.e. ω + πδ osω t φ i πδ = ω t + φ i = ω t + Δ sin ω sin Hene or the FM signal, v t =V osφ s ω dt ω i t t v s Δ t =V os ω t + sinω t 4

5 Frequeny Modulation The ratio Δ is alled the Modulation Index denoted by i.e. β = Peak requeny deviation odulating requeny Note FM, as ipliit in the above equation or v s (t), is a non-linear proess i.e. the priniple o superposition does not apply. The FM signal or a essage (t) as a band o signals is very oplex. Hene, (t) is usually onsidered as a 'single tone odulating signal' o the or t =V ω t os 5

6 Frequeny Modulation The equation v t =V os ω t + sinω t series (Bessel untions) s v s Δ t =V J βos ω +nω n= n ay be expressed as Bessel where J n () are Bessel untions o the irst kind. Expanding the equation or a ew ters we have: t v s ( t) V J 0 ( )os( t ) Ap V J ( )os( ) t Ap V J1( )os( ) t Ap V J ( )os( ) t Ap V J 1 ( )os( ) t Ap 6

7 FM Signal Spetru. The aplitudes drawn are opletely arbitrary, sine we have not ound any value or J n () this sketh is only to illustrate the spetru. 7

8 Generation o FM signals Frequeny Modulation. An FM deodulator is: a voltage-to-requeny onverter V/F a voltage ontrolled osillator VCO In these devies (V/F or VCO), the output requeny is dependent on the input voltage aplitude. 8

9 V/F Charateristis. Apply V IN, e.g. 0 Volts, +1 Volts, + Volts, -1 Volts, - Volts,... and easure the requeny output or eah V IN. The ideal V/F harateristi is a straight line as shown below., the requeny output when the input is zero is alled the undeviated or noinal arrier requeny. Δ The gradient o the harateristi ΔV denoted by per Volt. is alled the Frequeny Conversion Fator, 9

10 V/F Charateristis. Consider now, an analogue essage input, t =V ω t os As the input (t) varies ro + V 0 V the output requeny will vary ro a axiu, through, to a iniu requeny. 10

11 V/F Charateristis. For a straight line, y = + x, where = value o y when x = 0, = gradient, hene we ay say and when V IN = (t) OUT = = + OUT αv IN +αt,i.e. the deviation depends on (t). Considering that axiu and iniu input aplitudes are +V and -V respetively, then ax in = = +αv αv on the diagra on the previous slide. The peak-to-peak deviation is ax in, but ore iportantly or FM the peak deviation is Peak Deviation, Δ = αv Hene, Modulation Index, β = Δ = αv 11

12 Suary o the iportant points o FM In FM, the essage signal (t) is assued to be a single tone requeny, t =V ω t os The FM signal v s (t) ro whih the spetru ay be obtained as v s t =V J βos ω +nω n= where J n () are Bessel oeiients and Modulation Index, n t β = Δ = αv Hz per Volt is the V/F odulator, gradient or Frequeny Conversion Fator, per Volt is a easure o the hange in output requeny or a hange in input aplitude. Peak Deviation (o the arrier requeny ro ) Δ = αv 1

13 FM Signal Waveors. The diagras below illustrate FM signal waveors or various inputs At this stage, an input digital data sequene, d(t), is introdued the output in this ase will be FSK, (Frequeny Shit Keying). 13

14 FM Signal Waveors. Assuing d( t) V V or 1's or 0's OUT OUT 1 0 V V or 1's or 0's the output swithes between 1 and 0. 14

15 FM Signal Waveors. The output requeny varies gradually ro to ( + V ), through to ( - V ) et. 15

16 FM Signal Waveors. I we plot OUT as a untion o V IN : In general, (t) will be a band o signals, i.e. it will ontain aplitude and requeny variations. Both aplitude and requeny hange in (t) at the input are translated to (just) requeny hanges in the FM output signal, i.e. the aplitude o the output FM signal is onstant. Aplitude hanges at the input are translated to deviation ro the arrier at the output. The larger the aplitude, the greater the deviation. 16

17 FM Signal Waveors. Frequeny hanges at the input are translated to rate o hange o requeny at the output. An attept to illustrate this is shown below: 17

18 FM Spetru Bessel Coeiients. The FM signal spetru ay be deterined ro n v ( t) V J ( )os( n ) t s n The values or the Bessel oeiients, J n () ay be ound ro graphs or, preerably, tables o Bessel untions o the irst kind. 18

19 FM Spetru Bessel Coeiients. J n () =.4 = 5 In the series or v s (t), n = 0 is the arrier oponent, i.e. V J 0 ( )os( t), hene the n = 0 urve shows how the oponent at the arrier requeny,, varies in aplitude, with odulation index. 19

20 FM Spetru Bessel Coeiients. Hene or a given value o odulation index, the values o J n () ay be read o the graph and hene the oponent aplitudes (V J n ()) ay be deterined. A urther way to interpret these urves is to iagine the in 3 diensions 0

21 Exaples ro the graph = 0: When = 0 the arrier is unodulated and J 0 (0) = 1, all other J n (0) = 0, i.e. =.4: Fro the graph (approxiately) J 0 (.4) = 0, J 1 (.4) = 0.5, J (.4) = 0.45 and J 3 (.4) = 0. 1

22 Signiiant Sidebands Spetru. As ay be seen ro the table o Bessel untions, or values o n above a ertain value, the values o J n () beoe progressively saller. In FM the sidebands are onsidered to be signiiant i J n () 0.01 (1%). Although the bandwidth o an FM signal is ininite, oponents with aplitudes V J n (), or whih J n () < 0.01 are deeed to be insigniiant and ay be ignored. Exaple: A essage signal with a requeny Hz odulates a arrier to produe FM with a odulation index = 1. Sketh the spetru. n J n (1) Aplitude Frequeny V V V V Insigniiant Insigniiant

23 Signiiant Sidebands Spetru. As shown, the bandwidth o the spetru ontaining signiiant oponents is 6, or = 1. 3

24 Signiiant Sidebands Spetru. The table below shows the nuber o signiiant sidebands or various odulation indies () and the assoiated spetral bandwidth. No o sidebands 1% o Bandwidth unodulated arrier e.g. or = 5, 16 sidebands (8 pairs). 4

25 Carson s Rule or FM Bandwidth. An approxiation or the bandwidth o an FM signal is given by BW = (Maxiu requeny deviation + highest odulated requeny) Bandwidth ( ) Carson s Rule 5

26 Narrowband and Wideband FM Narrowband FM NBFM Fro the graph/table o Bessel untions it ay be seen that or sall, ( 0.3) there is only the arrier and signiiant sidebands, i.e. BW =. FM with 0.3 is reerred to as narrowband FM (NBFM) (Note, the bandwidth is the sae as DSBAM). Wideband FM WBFM For > 0.3 there are ore than signiiant sidebands. As inreases the nuber o sidebands inreases. This is reerred to as wideband FM (WBFM). 6

27 VHF/FM VHF/FM (Very High Frequeny band = 30MHz 300MHz) radio transissions, in the band 88MHz to 108MHz have the ollowing paraeters: Max requeny input (e.g. usi) 15kHz Deviation Modulation Index 5 75kHz V For = 5 there are 16 sidebands and the FM signal bandwidth is 16 = 16 x 15kHz = 40kHz. Applying Carson s Rule BW = (75+15) = 180kHz. 7

28 Coents FM The FM spetru ontains a arrier oponent and an ininite nuber o sidebands at requenies n (n = 0, 1,, ) FM signal, n v ( t) V J ( )os( n ) t s n In FM we reer to sideband pairs not upper and lower sidebands. Carrier or other oponents ay not be suppressed in FM. The relative aplitudes o oponents in FM depend on the values J n (), where V thus the oponent at the arrier requeny depends on (t), as do all the other oponents and none ay be suppressed. 8

29 Coents FM Coponents are signiiant i J n () For <<1 ( 0.3 or less) only J 0 () and J 1 () are signiiant, i.e. only a arrier and sidebands. Bandwidth is, siilar to DSBAM in ters o bandwidth - alled NBFM. Large odulation index WBFM. eans that a large bandwidth is required alled The FM proess is non-linear. The priniple o superposition does not apply. When (t) is a band o signals, e.g. speeh or usi the analysis is very diiult (ipossible?). Calulations usually assue a single tone requeny equal to the axiu input requeny. E.g. (t) band 0Hz 15kHz, = 15kHz is used. 9

30 Power in FM Signals. Fro the equation or FM n n s t n J V t v ) )os( ( ) ( we see that the peak value o the oponents is V J n () or the n th oponent. Single noralised average power = ) ( RMS pk V V then the n th oponent is ) ( ) ( n n J V J V Hene, the total power in the ininite spetru is Total power n n T J V P )) ( ( 30

31 Power in FM Signals. By this ethod we would need to arry out an ininite nuber o alulations to ind P T. But, onsidering the waveor, the peak value is V, whih is onstant. Sine we know that the RMS value o a sine wave is V pk V and power = (V RMS ) then we ay dedue that P T V V n V J n ( ) Hene, i we know V or the FM signal, we an ind the total power P T or the ininite spetru with a siple alulation. 31

32 Power in FM Signals. Now onsider i we generate an FM signal, it will ontain an ininite nuber o sidebands. However, i we wish to transer this signal, e.g. over a radio or able, this iplies that we require an ininite bandwidth hannel. Even i there was an ininite hannel bandwidth it would not all be alloated to one user. Only a liited bandwidth is available or any partiular signal. Thus we have to ake the signal spetru it into the available hannel bandwidth. We an think o the signal spetru as a train and the hannel bandwidth as a tunnel obviously we ake the train slightly less wider than the tunnel i we an. 3

33 Power in FM Signals. However, any signals (e.g. FM, square waves, digital signals) ontain an ininite nuber o oponents. I we transer suh a signal via a liited hannel bandwidth, we will lose soe o the oponents and the output signal will be distorted. I we put an ininitely wide train through a tunnel, the train would oe out distorted, the question is how uh distortion an be tolerated? Generally speaking, spetral oponents derease in aplitude as we ove away ro the spetru entre. 33

34 Power in FM Signals. In general distortion ay be deined as D Power in total spetru- Power in Bandliited spetru Power in total spetru D P T P T P BL With reerene to FM the iniu hannel bandwidth required would be just wide enough to pass the spetru o signiiant oponents. For a bandliited FM spetru, let a = the nuber o sideband pairs, e.g. or = 5, a = 8 pairs (16 oponents). Hene, power in the bandliited spetru P BL is P BL a na ( V J n ( )) = arrier power + sideband powers. 34

35 Power in FM Signals. Sine P T V Distortion D V V V a na ( J n ( )) 1 a na ( J n ( )) Also, it is easily seen that the ratio D Power in Bandliited spetru Power in total spetru P P BL T a na ( J n ( )) = 1 Distortion i.e. proportion p power in bandliited spetru to total power = a na ( ( )) J n 35

36 Exaple Consider NBFM, with = 0.. Let V = 10 volts. The total power in the ininite spetru V a = 50 Watts, i.e. ( J n ( )) = 50 Watts. na Fro the table the signiiant oponents are n J n (0.) Ap = V J n (0.) ( Ap) Power = P BL = 49.5 Watts 49.5 i.e. the arrier + sidebands ontain or 99% o the total power 36

37 Exaple P P T BL Distortion = P T or 1%. Atually, we don t need to know V, i.e. alternatively Distortion = 1 1 n1 ( J n (0.)) (a = 1) D = 1 (0.99) (0.0995) P 1 BL Ratio ( J ( )) 1 D P T n1 n 37

38 FM Deodulation General Priniples. An FM deodulator or requeny disriinator is essentially a requeny-to-voltage onverter (F/V). An F/V onverter ay be realised in several ways, inluding or exaple, tuned iruits and envelope detetors, phase loked loops et. Deodulators are also alled FM disriinators. Beore onsidering soe speii types, the general onepts or FM deodulation will be presented. An F/V onverter produes an output voltage, V OUT whih is proportional to the requeny input, IN. 38

39 FM Deodulation General Priniples. I the input is FM, the output is (t), the analogue essage signal. I the input is FSK, the output is d(t), the digital data sequene. In this ase IN is the independent variable and V OUT is the dependent variable (x and y axes respetively). The ideal harateristi is shown below. We deine V o as the output when IN =, the noinal input requeny. 39

40 FM Deodulation General Priniples. The gradient V is alled the voltage onversion ator i.e. Gradient = Voltage Conversion Fator, K volts per Hz. Considering y = x + et. then we ay say V OUT = V 0 + K IN ro the requeny odulator, and sine V 0 = V OUT when IN = then we ay write V OUT V 0 KV IN where V 0 represents a DC oset in V OUT. This DC oset ay be reoved by level shiting or AC oupling, or the F/V ay be designed with the harateristi shown next 40

41 FM Deodulation General Priniples. The iportant point is that V OUT = KV IN. I V IN = (t) then the output ontains the essage signal (t), and the FM signal has been deodulated. 41

42 FM Deodulation General Priniples. Oten, but not always, a syste designed so that V OUT = (t). A oplete syste is illustrated. K 1, so that K = 1 and 4

43 FM Deodulation General Priniples. 43

44 Methods Tuned Ciruit One ethod (used in the early days o FM) is to use the slope o a tuned iruit in onjuntion with an envelope detetor. 44

45 Methods The tuned iruit is tuned so the, the noinal input requeny, is on the slope, not at the entre o the tuned iruits. As the FM signal deviates about on the tuned iruit slope, the aplitude o the output varies in proportion to the deviation ro. Thus the FM signal is eetively onverted to AM. This is then envelope deteted by the diode et to reover the essage signal. Note: In the early days, ost radio links were AM (DSBAM). When FM ae along, with its advantages, the links ould not be hanged to FM quikly. Hene, NBFM was used (with a spetral bandwidth =, i.e. the sae as DSBAM). The arrier requeny was hosen and the IF ilters were tuned so that ell on the slope o the ilter response. Most FM links now are wideband with uh better deodulators. A better ethod is to use siilar iruits, known as a Foster-Seeley Disriinator 45

46 Foster-Seeley Disriinator This gives the oposite harateristis shown. Diode D eetively inverts the tuned iruit response. This gives the harateristi S type detetor. 46

47 Phase Loked Loops PLL A PLL is a losed loop syste whih ay be used or FM deodulation. A ull analytial desription is outside the sope o these notes. A brie desription is presented. A blok diagra or a PLL is shown below. Note the siilarity with a synhronous deodulator. The loop oprises a ultiplier, a low pass ilter and VCO (V/F onverter as used in a requeny odulator). 47

48 Phase Loked Loops PLL The input IN is applied to the ultiplier and ultiplied with the VCO requeny output O, to produe = ( IN + O ) and = ( IN O ). The low pass ilter passes only ( IN O ) to give VOUT whih is proportional to ( IN O ). I IN O but not equal, V OUT = V IN, IN O is a low requeny (beat requeny) signal to the VCO. This signal, V IN, auses the VCO output requeny O to vary and ove towards IN. When IN = O, V IN ( IN O ) is approxiately onstant (DC) and O is held onstant, i.e loked to IN. As IN hanges, due to deviation in FM, O traks or ollows IN. V OUT = V IN hanges to drive O to trak IN. V OUT is thereore proportional to the deviation and ontains the essage signal (t). 48