Exponential Carrier Wave Modulation

Exponential Carrier Wave Modulation

S-72.1140 Transmission Methods in Telecommunication Systems (5 cr) Exponential Carrier Wave Modulation Exponential modulation: Frequency (FM) and phase (PM) modulation 2 FM and PM waveforms Instantaneous frequency and phase Spectral properties narrow band arbitrary modulating waveform tone modulation - phasor diagram wideband tone modulation Transmission BW Generating FM - signals de-tuned tank circuit

narrow band mixer modulator indirect modulators Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Contents (cont.) 3 Detecting FM/PM FM-AM conversion followed by envelope detector Phase-shift discriminator Zero-crossing detection (tutorials) PLL-detector (tutorials) Effect of additive interference on FM and PM analytical expressions and phasor diagrams implications for demodulator design FM preemphases and deemphases filters Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Phase modulation (PM) Carrier Wave (CW) signal: xC (t ) AC cos( C t (t )) C (t ) In exponential modulation the modulation is in the exponent or in the angle xC (t ) AC cos( C (t )) AC Re[exp( j C (t ))] Note that in exponential modulation superposition does not apply: xC (t ) A cos C t k f a1 (t ) a2 (t ) A cos C t A cos k f a1 (t ) a2 (t ) In phase modulation (PM) carrier phase is linearly proportional to

the modulation x(t): 3 x PM ( t ) AC cos( C t ( t ) ) x ( t ), Angular phasor has the C (t ) instantaneous frequency (phasor rate) C 2 f C (t ) t C 4 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Instantaneous frequency vq (t )

(t ) Constant frequency carrier: 2 2 rad/s, f 1Hz 0 0 Angle modulated carrier t 1s t/s vi (t ) Angular frequency (rate) is the derivative of the phase (the same way as the velocity v(t) is the derivative of distance s(t)) For continuously changing frequency instantaneous frequency is defined by differential changes:

t Compare to v (t ) ds (t ) s2 (t ) s1 (t ) d (t ) (t ) (t ) ( ) d linear motion: dt t t 2 1 dt 5 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen t

d (t ) (t ) (t ) ( ) d dt 6 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen rad/s rad/s2 7 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Frequency modulation (FM) In frequency modulation carriers instantaneous frequency is linearly proportional to modulation: (t ) C (t ) 2 [ fC f x(t )] dC (t ) / dt C (t ) 2 [ fC f x (t )]dt

Hence the FM waveform can be written as z t xC (t ) AC cos( C t 2f t x ( )d ), t t0 0 C (t ) Therefore for FM f (t ) f C f x(t ) and for PM (t ) x(t ) 8

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen t (t ) ( )d integrate AM, FM and PM waveforms Instantaneous frequency directly proportional to modulation waveform constant frequency follows derivative of the modulation waveform x PM ( t ) AC cos( C t x (t )) z xFM (t ) AC cos( C t 2f x ( )d ) t 9

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Assignment (i) Briefly summarize what is the main difference between FM and PM ? (ii) How would you generate FM by using a PM modulator? 10 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Assignment (i) Briefly summarize what is the main difference between FM and PM ? (ii) How would you generate FM by using a PM modulator? Solution (i) In PM instantaneous phase is directly proportional to modulating signal amplitude, in FM instantaneous frequency is directly proportional to modulating signal amplitude. (ii) One need to integrate the modulating signal before applying it to the PM modulator 11

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Narrowband FM and PM (small modulation index, arbitrary modulation waveform) The CW presentation: xC (t ) AC cos[ C t (t )] The quadrature CW presentation: xC (t ) xci (t )cos( C t ) xcq (t )sin( C t ) xci (t ) AC cos (t ) AC [1 (1 / 2 !) 2 (t )...] xcq (t ) AC sin (t ) AC [ (t ) (1 / 3!) 3 (t )...] Narrow band (small angle) condition: (t ) 1rad xci (t ) AC xcq (t ) AC (t ) Hence the Fourier transform of XC(t) is F xC (t ) F AC cos( C t ) AC (t )sin( C t )

1 j XC ( f ) AC ( f fC ) AC ( f fC ), f 0 2 2 F cos(2 f 0t ) F cos(2 f 0t ) x(t ) 1 1 ( f f0 ) ( f f0 ) X ( f f 0 )exp( j ) jX ( f f 0 )exp( j ) 212 2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen cos( ) cos( ) cos( ) sin( ) sin( ) Narrow band FM and PM spectra Remember the instantaneous phase in CW presentation:

xC (t ) AC cos[ C t (t )] (t ) x (t ) PM (t ) 2f FM zx( )d , t t t t0 0 The small angle assumption produces compact spectral presentation both for FM and AM: 1 j

XC ( f ) AC ( f fC ) AC ( f fC ), f 0 2 2 ( f ) F[ (t )] What does it mean to set this component to zero? X ( f ), PM jf X ( f ) / f , FM t g ( ) d t 0 13 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen G ( ) G (0) ( ) j

XFM ( f ) Example XPM ( f ) Assume: x(t ) sinc2Wt X ( f ) 1 f 2W 2W 1 j X C ( f ) AC ( f f C ) AC ( f f C ), f 0 2 2 PM ( f ) F [ PM (t )] X ( f ) FM ( f ) F [ FM (t )] jf X ( f ) / f

1 j f fC X PM ( f ) AC ( f f C ) AC 2 4W 2W , f 0 1 f f fC X FM ( f ) AC ( f f C ) AC 2 4 f fC W 2W 14 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

, f 0 Example 1 f sinc2Wt 2W 2W sin(2Wt ) 1 f expand sinc( x) 2Wt 2W 2W 2W sin(2Wf ) t swap f t , regroup 2W 2Wf

2W t 2W sinc(2Wf ) denote pulse width in time domain by 2W 2W t sinc( f ) f 1 2W 15 1 2W Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen t W W

Tone modulation with PM and FM: modulation index Remember the FM and PM waveforms: x PM (t ) AC cos[ C t x (t )] (t ) Assume tone modulation A sin( t ), PM R x ( t ) S TA cos( t ), FM m m z

xFM (t ) AC cos[ C t 2f x ( )d ] t (t ) m m Then x(t ) Am sin( mt ), PM (t ) x( )d ( Am f / f m )sin( mt ), FM 2 f t 16 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

FM and PM with tone modulation and arbitrary modulation index Time domain expression for FM and PM: xC (t ) AC cos[ C t sin( m t )] PM Am FM Am f / f m Remember: cos( ) cos( )cos( ) sin( )sin( ) Therefore: xC (t ) AC cos( sin( m t ))cos( C t ) AC sin( sin( m t ))sin( C t ) cos( sin( m t )) JO ( ) n even 2 J n ( )cos(n m t ) sin( sin( m t )) n odd 2 J n ( )sin( n m t ) 17

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Jn is the first kind, order n Bessel function Wideband FM and PM spectra After simplifications we can write: xC (t ) AC n Jn ( )cos( C n m )t Am , PM Am f / f m , FM note: J n ( ) ( 1) n J n ( ) 18 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Determination of transmission bandwidth

The goal is to determine the number of significant sidebands Thus consider again how Bessel functions behave as the function of , e.g. we consider Am 1, fm W Significant sidebands: Jn ( ) Minimum bandwidth includes 2 sidebands (why?): BT ,min 2 fm Generally: BT 2 M ( ) fm , M ( ) 1 M( ) 2 J M ( ) 0.01 J M ( ) 0.1 A , PM A f / f , FM m

19 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen m m Assignment 20 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Assignment Solution M() is determined from the condition |Jn()| > , therefore the curve on the previous page can be obtained for instance by fixing a value for and then adjusting n until the condition is met. Note: n = 1 equals the case with two sidebands (magnitude spectra the same as for AM) 21

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Transmission bandwidth and deviation D Tone modulation is extrapolated into arbitrary modulating signal by defining deviation by A f / f m m Am 1, f m W f / W D Therefore transmission BW is also a function of deviation BT 2 M ( D)W For very large D and small D with M ( D ) D 2

BT 2( D 2) f m D 1, f m W 2( D 2)W 2 DW , D 1 BT 2 M ( D)W 2W , D 1 (a single pair of sidebands) that can be combined into BT 2 D 1 W , D 1,and D 1 A , PM A f / f , FM m 22

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen m m Example: Commercial FM bandwidth Following commercial FM specifications f 75 kHz, W 15 kHz D f / W 5 BT 2( D 2)W 210 kHz,(D > 2) High-quality FM radios RF bandwidth is about T Note that

B 200 kHz underBTestimates slightly 2 D 1the W bandwidth 180 kHz, D >> 1 23 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen A practical FM modulator circuit 24 A tuned circuit oscillator biased varactor diode capacitance directly proportional to x(t) other parts: input transformer RF-choke DC-block

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Generating FM/PM De-tuned tank circuit Capacitance of a resonant circuit can be made to be a function of modulation voltage. fCC 1 / (2 LC ) Resonance frequency fCC [ x (t )] 1 / {2 LC[ x (t )]} De-tuned resonance frequency C[ x (t )] C0 Cx (t ) Capacitance diode fCC [ x (t )] fC (1 Cx (t ) / C0 ) 1/ 2 , fC 1 / (2 LC0 ) That can be simplified by the series expansion kx 3k 2 x 2 1/ 2 Note that this applies for a (1 - kx ) 1 ... kx 1 2

8 relatively small modulation 1/ 2 index fCC [ x (t )] fC (1 Cx (t ) / C0 ) L 1 Cx (t ) O 1 d (t ) M P 2 dt N2 C Q C (t ) 2f t 2 f z x ( ) d 2C fC 1 0 C C

0 25 t f Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Remember that the instantaneous frequency is the derivative of the phase Frequency detection Methods of frequency detection FM-AM conversion followed by envelope detector Phase-shift discriminator Zero-crossing detection (tutorials) PLL-detector (tutorials)

FM-AM conversion is produced by a transfer function having magnitude distortion, as the time derivative (other possibilities?): xC (t ) AC cos( 0 t (t )) 27 dxC (t ) AC sin[ C t (t )]( C d (t ) / dt ) dt d (t ) / dt 2f (t ) 2 [ fC f x (t )] FM As for example Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 28 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 29 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

30 25 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Assignment 31 Show that from figure above follows: Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 32 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 33 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 34

Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Assignment Consider the following RC-filter transfer function 1 1 H( f ) 1 j 2 fRC 1 j ( f / B3dB ) Inspect the relating group delay and state what is a frequency range of low linear distortion 35 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 36 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Frequency range of low linear distortion? 37 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Indirect FM transmitter Mathematica-expressions FM/PM modulator with high linearity and modulation index difficult to realize One can first generate a small modulation index signal that is then applied into a nonlinear circuit Therefore applying FM/PM wave into non-linearity increases modulation index should be filtered away, how? 38 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Indirect FM transmitter:circuit realization

The frequency multiplier produces n-fold multiplication of instantaneous frequency f 2 (t ) nf1 (t ) FM: n 2 f x( )d t Frequency multiplication of tone modulation increases modulation index but the line spacing remains the same nf c1 n In[14]:= Out[14]= 39 @ @ @

D D D [email protected]@ DD L TrigReduce Cos w0 t + A m Sin wm t 1 2 1 + Cos 2 Sin t wm Am + 2 t w0 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen 2 Annex Narrowband tone modulation Tone modulation with PM and FM: modulation index

Remember the FM and PM waveforms: x PM (t ) AC cos[ C t x (t )] (t ) Assume tone modulation A sin( t ), PM R x ( t ) S TA cos( t ), FM m m z xFM (t ) AC cos[ C t 2f x ( )d ] t (t )

m m Then x(t ) Am sin( mt ), PM (t ) x( )d ( Am f / f m )sin( mt ), FM 2 f t 41 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Tone modulation in frequency domain: Phasors and spectra for narrowband case

Remember the quadrature presentation: xC (t ) AC cos[ C t (t )] xC (t ) xci (t )cos( C t ) xcq (t )sin( C t ) xci (t ) AC cos (t ) AC [1 (1 / 2 !) 2 (t )...] xcq (t ) AC sin (t ) AC [ (t ) (1 / 3!) 3 (t )...] For narrowband assume 1, ( t ) sin( mt ), FM, PM xC (t ) AC cos( C t ) AC sin( mt )sin( C t ) AC cos( C t ) AC cos( C m )t 2 AC cos( C m )t 2 42 PM Am

FM Am f / f m 1 sin( ) sin( ) cos( ) cos( ) 2 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen Narrow band tone modulation: spectra and phasors Phasors and spectra resemble AM: AC xC (t ) AC cos( C t ) cos( C m )t 2 AC cos( C m )t 2 43 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

Assignment AM xC (t ) AC Am cos(mt )cos(C t ) AC cos(C t ) AC Am A A cos(C m )t C m cos(C m )t 2 2 AC cos(C t ) NB-FM xC (t ) AC cos( C t ) AC cos( C m )t 2 AC cos( C m )t

2 (i) Discuss the phasor diagrams and explain phasor positions based on analytical expressions (ii) Comment amplitude and phase modulation in both cases 44 Helsinki University of Technology,Communications Laboratory, Timo O. Korhonen

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