Angle Modulation ELEN 3024 - Communication Fundamentals School of - PowerPoint PPT Presentation

Angle Modulation ELEN 3024 - Communication Fundamentals School of Electrical and Information Engineering, University of the Witwatersrand July 15, 2013

Angle Modulation Proakis and Salehi, “Communication Systems Engineering” (2nd Ed.), Chapter 3

Overview

3.3.Angle Modulation Amplitude-modulation methods → linear-modulation methods (AM DSB-FC not linear) FM and PM other analogue modulation techniques. FM → frequency of carrier f c changed by message PM → phase of carrier is changed by variations in message signal FM and PM → angle-modulation methods → nonlinear

3.3.Angle Modulation Angle-modulation → due to nonlinearity • complex to implement • Difficult to analyse Many cases only approximate analysis. Bandwidth-expansion of angle modulation → effective bandwidth of modulated signal >> bandwidth of message signal ⇒ Trade-off bandwidth for high noise immunity

3.3.1. Representation of FM and PM signals Angle-modulated signal: u ( t ) = A c cos( θ ( t )) θ ( t ) → phase of the signal Instantaneous frequency f i ( t ): f i ( t ) = 1 d dt θ ( t ) 2 π

3.3.1. Representation of FM and PM signals Since u ( t ) bandpass signal: u ( t ) = A c cos(2 π f c t + φ ( t )) Therefore, f i ( t ) =

3.3.1. Representation of FM and PM signals Since u ( t ) bandpass signal: u ( t ) = A c cos(2 π f c t + φ ( t )) Therefore, f i ( t ) = f c + 1 d dt φ ( t ) 2 π

3.3.1. Representation of FM and PM signals PM → message m ( t ) → φ ( t ) = k p m ( t ) FM: f i ( t ) − f c = k f m ( t ) = 1 d dt φ ( t ) 2 π k p and k f → phase and frequency deviation constants � k p m ( t ) , PM φ ( t ) = � t 2 π k f −∞ m ( τ ) d τ, FM

3.3.1. Representation of FM and PM signals Observations: FM → phase modulate carrier with integral of a message. Or � k p d d dt m ( t ) , PM dt φ ( t ) = 2 π m ( t ) , FM PM → frequency modulate carrier with derivative of message m ( t ) Fig 3.25: Important. Fig 3.26: Important.

3.3.1. Representation of FM and PM signals Demodulation of FM signal → finding instantaneous frequency of the modulated signal and subtracting the carrier frequency from it. Demodulation of PM signal → finding the phase of the signal and then recovering m ( t ) Maximum phase deviation in PM system → ∆ φ max = k p max [ | m ( t ) | ] Maximum frequency-deviation in FM → ∆ f max = k f max [ | m ( t ) | ]

Example 3.3.1. Message signal → m ( t ) = a cos(2 π f m t ) Modulate FM system and PM system Find the modulated signal in each case. For PM φ ( t ) =

Example 3.3.1. Message signal → m ( t ) = a cos(2 π f m t ) Modulate FM system and PM system Find the modulated signal in each case. For PM φ ( t ) = k p m ( t ) = k p a cos(2 π f m t )

Example 3.3.1. Message signal → m ( t ) = a cos(2 π f m t ) Modulate FM system and PM system Find the modulated signal in each case. For PM φ ( t ) = k p m ( t ) = k p a cos(2 π f m t ) For FM φ ( t ) =

Example 3.3.1. Message signal → m ( t ) = a cos(2 π f m t ) Modulate FM system and PM system Find the modulated signal in each case. For PM φ ( t ) = k p m ( t ) = k p a cos(2 π f m t ) For FM � t m ( τ ) d τ = k f a φ ( t ) = 2 π k f sin(2 π f m t ) f m −∞

Example 3.3.1. Modulated signals: � u ( t ) =

Example 3.3.1. Modulated signals: � A c cos(2 π f c t + k p a cos(2 π f m t )) , PM u ( t ) = A c cos(2 π f c t + k f a f m sin(2 π f m t )) , FM Define β p = k p a and β f = k f a f m we have � A c cos(2 π f c t + β p cos(2 π f m t )) , PM u ( t ) = A c cos(2 π f c t + β f sin(2 π f m t )) , FM β p and β f → modulation indices

3.3.1. Representation of FM and PM signals We can extend the definition of the modulation index for a general nonsinusoidal signal m ( t ) as β p = k p max [ | m ( t ) | ] β f = k f max [ | m ( t ) | ] W In terms of the maximum phase and frequency deviation: β p = ∆ φ max β f = ∆ f max W

3.3.1.1 Narrowband Angle Modulation If k p or k f and m ( t ) such that φ ( t ) ≪ 1 ∀ t : u ( t ) = A c cos(2 π f c t ) cos( φ ( t )) − A c sin(2 π f c t ) sin( φ ( t )) ≈ A c cos(2 π f c t ) − A c φ ( t ) sin(2 π f c t ) Modulated signal very similar to conventional AM signal (AM DSB FC) Sine wave modulated by m ( t ) instead of cosine Bandwidth ≈

3.3.1.1 Narrowband Angle Modulation If k p or k f and m ( t ) such that φ ( t ) ≪ 1 ∀ t : u ( t ) = A c cos(2 π f c t ) cos( φ ( t )) − A c sin(2 π f c t ) sin( φ ( t )) ≈ A c cos(2 π f c t ) − A c φ ( t ) sin(2 π f c t ) Modulated signal very similar to conventional AM signal (AM DSB FC) Sine wave modulated by m ( t ) instead of cosine Bandwidth ≈ Bandwidth(AM) → 2 × bandwidth( m ( t ))

3.3.1.1 Narrowband Angle Modulation Fig. 3.27 → phasor diagrams for narrowband angle modulation and AM Narrowband angle modulation far less amplitude variations than AM Narrowband angle modulation → constant amplitude Slight amplitude variations due to approximation Narrowband angle-modulation does not provide better noise immunity compared to AM DSB FC.

3.3.2 Spectral Characteristics of Angle-Modulated Signals Due to inherent nonlinearity of angle-modulation → difficult to characterise spectral properties Study simple modulation signals and certain approximations Generalized to more complicated messages Study 3 cases for m ( t ): • sinusoidal signal • periodic signal • nonperiodic signal

3.3.2.1 Angle Modulation by a Sinusoidal Signal For both PM and FM u ( t ) = A c cos(2 π f c t + β sin(2 π f m t )) β → modulation index � A c e j 2 π f c t e j β sin (2 π f m t ) � u ( t ) = Re

3.3.2.1 Angle Modulation by a Sinusoidal Signal Since sin(2 π f m t ) periodic with period T m = 1 f m , same true for complex exponential signal e j β sin(2 π f m t ) Therefore, can be expanded in Fourier series representation: 1 e j β sin(2 π f m t ) e − jn 2 π f m t dt � = fm c n f m 0 � 2 π e j β sin u − nu du 1 = ( u = 2 π f m t ) 2 π 0 Last integral → Bessel function of the first kind of order n → J n ( β )

3.3.2.1 Angle Modulation by a Sinusoidal Signal Therefore, Fourier series for complex exponential ∞ e j β sin 2 π f m t = � J n ( β ) e j 2 π nf m t n = −∞ Substituting into complex baseband representation � ∞ n = −∞ J n ( β ) e j 2 π nf m t e j 2 π f c t � � u ( t ) = Re A c � ∞ = n = −∞ A c J n ( β ) cos(2 π ( f c + nf m ) t ) Even for single sinusoidal modulating signal, angle-modulated signal contains all frequencies of the form f c + nf m for n = 0 , ± 1 , ± 2 , . . . Actual bandwidth → infinite

3.3.2.1 Angle Modulation by a Sinusoidal Signal Amplitude of sinusoidal components of frequencies f c + nf m , n large → very small Therefore define finite effective bandwidth of modulated wave Series expansion of Bessel function: � n +2 k ( − 1) k � β ∞ 2 � J n ( β ) = k !( k + n )! k =0

3.3.2.1 Angle Modulation by a Sinusoidal Signal For small β , can use following approximation J n ( β ) ≈ β n 2 n n ! Thus for small β → only first sideband corresponding to n = 1 of importance

3.3.2.1 Angle Modulation by a Sinusoidal Signal Properties of Bessel function (verified by expansion): � J n ( β ) , n even J − n ( β ) = − J n ( β ) , n odd Fig. 3.28 → Plots of J n ( β ) for various values of n Table 3.1. → Table of the values of the Bessel function

Example 3.3.2 carrier → c ( t ) = 10 cos(2 π f c t ) message → cos(20 π t ) message used to frequency modulate carrier with k f = 50 Find expression for the modulated signal and determine how many harmonics should be selected to contain 99% of the modulated signal power

3.3.2.1 Angle Modulation by a Sinusoidal Signal In general, effective bandwidth of an angle-modulated signal which contains at least 98 % of the signal power: B c = 2 ( β + 1) f m β → modulation index f m frequency of sinusoidal message signal.

3.3.2.1 Angle Modulation by a Sinusoidal Signal Consider effect of amplitude and frequency of sinusoidal m ( t ) on bandwidth and number of harmonics in modulated signal m ( t ) = a cos(2 π f m t ) bandwidth (effective) is given by: � 2( k p a + 1) f m , PM B c = 2 ( β + 1) f m = � � k f a 2 f m + 1 f m , FM or, � 2( k p a + 1) f m , PM B c = 2( k f a + f m ) , FM

3.3.2.1 Angle Modulation by a Sinusoidal Signal Increasing a → in PM and FM almost same effect on increasing bandwidth B c Increasing f m : • PM → increase in B c is proportional to increase in f m • FM → increase in B c is additive (for large β not substantial)

3.3.2.1 Angle Modulation by a Sinusoidal Signal Consider Harmonics: � 2 ⌊ k p a ⌋ + 3 , PM M c = 2 ⌊ β ⌋ + 3 = � � k f a 2 + 3 , FM f m Increasing a → increases the number of harmonics Increasing f m • No effect on PM • Almost linear decrease in number of harmonics for FM

3.3.2.2 Angle Modulation by a Periodic Message Signal Consider periodic message signal m ( t ) For PM u ( t ) = A c cos(2 π f c t + β m ( t )) rewrite as � e j 2 π f c t e j β m ( t ) � u ( t ) = A c Re