[4.1] ANGLE MODULATION U1. PHASE AND FREQUENCY MODULATION For angle modulation, the modulated carrier is represented by xx cc (tt) = AA cccccc[ωω cc tt + φφ(tt)] (1.1) Where A ω c are constants the phase angle φφ(tt) is a function of the message signal m(t). Equation (1.1) can be written as xx cc (tt) = AA cccccc θθ(tt) where θθ(tt) = ωω cc tt + φφ(tt) The instantaneous radian frequency of x c (t), denoted by ω i is ωω ii = dd θθ(tt) φφ(tt) = instantaneous phase deviation. dd φφ(tt) = ωω cc + = instantaneous frequancy deviation. dd φφ(tt) (1.2) The maximum (or peak) radian frequency deviation of the angle-modulated signal ( ω) is given by Δωω = ωω ii ωω cc mmmmmm (1.3) In phase modulation (PM) the instantaneous phase deviation of the carrier is proportional to the message signal; that is, φφ(tt) = kk pp mm(tt) (1.4) where k p is the phase deviation constant, expressed in radians per unit of m(t). In frequency modulation (FM), the instantaneous frequency deviation of the carrier is proportional to the message signal; that is, dd φφ(tt) = kk ff mm(tt) (1.5aa) tt or φφ(tt) = kk ff mm(λλ) (1.5bb) thus, we can express the angle-modulated signal as From Eq.(1.2),we have xx PPPP (tt) = AA cccccc [ωω cc tt + kk PP mm(tt)] (1.6) xx FFFF (tt) = AA cccccc ωω cc tt + kk ff tt mm(λλ) (1.7)
ωω ii = ωω cc + kk pp dd mm(tt) [4.2] ffffff PPPP (1.8) ωω ii = ωω cc + kk ff mm(tt) ffffff FFFF (1.9) Thus, in PM, the instantaneous frequency ωω ii varies linearly with the derivative of the modulating signal, in FM, ωω ii varies linearly with the modulating signal. Figure (1.1) illustrates AM, FM, PM waveforms produced by a sinusoidal message waveform. Fig.(1.1) U2. FOURIER SPECTRA OF ANGLE-MODULATED SIGNALS An angle-modulated carrier can be represented in exponential form by writing Eq.(1.1) as xx cc (tt) = RRRR AA ee jj ωωcctt+φφ(tt) = RRRR AA ee jj ωωcctt ee jjjj (tt) (2.1) Exping ee jjjj (tt) in a power series yields xx cc (tt) = RRRR AAee jj ωωcctt 1 + jjjj(tt) φφ 2 (tt) + jj nn φφnn (tt) + 2! nn! = AA cccccc ωω cc tt φφ(tt)ssssssωω cc tt φφ2 (tt) ccccccωω 2! cc tt + φφ3 (tt) ssssssωω 3! cc tt + (2.2) Thus the angle-modulated signal consists of an unmodulated carrier plus various amplitude-modulated terms, such as φφ(tt)ssssss ω cc tt, φφ 2 (tt)cccccc ω cc tt, φφ 3 (tt)ssssss ω cc tt..., etc. Hence its Fourier spectrum consists of an unmodulated carrier plus spectra of φφ(tt), φφ 2 (tt), φφ 3 (tt),..., etc., centered at ω c. It is clear that the Fourier spectrum of an angle-modulated signal is not related to the message signal spectrum in any simple way, as was the case in AM.
[4.3] U3. NARROWBAND ANGLE MODULATION If φφ(tt) mmmmmm 1, then Eq. (2.2) can be approximated by [neglecting all higherpower terms of φφ(tt)] xx cc (tt) AAAAAAAAωω cc tt AAAA(tt)ssssssωω cc tt (3.1) xx cc (tt) in Eq.(3.1) is called the narrowb (NB) angle-modulated signal. Thus, xx NNNNNNNN (tt) AAAAAAAAωω cc tt AAkk pp mm(tt)ssssssωω cc tt (3.2) xx NNNNNNNN (tt) AAAAAAAAωω cc tt AA kk ff tt mm(λλ) ssssssωω cc tt (3.3) Equation (3.1) indicates that a narrowb angle-modulated signal contains an unmodulated carrier plus a term in which φφ(tt) [a function of m(t)] multiplies a π/2 (rad) phase-shifted carrier. This multiplication generates a pair of sidebs, if φφ(tt) has a-bwidth W B, the bwidth of an NB angle-modulated signal is 2W B. U4. SINUSOIDAL (OR TONE), MODULATION If the message signal m(t) is a pure sinusoid, that is, mm(tt) = aa mmssssssωω mm tt ffffff PPPP aa mm ccccccωω mm tt ffffff FFFF then Eqs. (1.4) (1.5b) both give φφ(tt) = ββ ssssssωω mm tt (4.1) Where kk pp aa mm ffffff PPPP ββ = kk ff aa mm ffffff FFFF (4.2) ωω mm The parameter β is known as the Umodulation indexu for angle modulation is the maximum value of phase deviation for both PM FM. Note that β is defined only for sinusoidal modulation it can be expressed as ββ = Δωω (4.3) ωω mm Substituting Eq. (4.1) into Eq. (1.1), we obtain xx cc (tt) = AA cccccc[ωω cc tt + ββ ssssssωω mm tt] (4.4) which can be expressed as xx cc (tt) = AA RRRR(ee jj ωωcctt ee jjjjjjjjjj ωω mm tt ) (4.5) The function ee jjjjjjjjjj ωω mm tt is clearly a periodic function with period TT mm = 2ππ ωω mm. It therefore has a Fourier series representation ee jjjjjjjjjj ωω mm tt = cc nn ee jjjj ωω mm tt nn= The Fourier coefficients c n can be found to be cc nn = ωω mm 2ππ Setting ωω mm tt = xx, we have ππ ωω mm eejjjjjjjjjj ωω mm tt ππ ωω mm ee jjjj ωω mm tt
ππ (ββββββββββ nnnn ) eejj ππ [4.4] cc nn = 1 = JJ 2ππ nn (ββ) Where JJ nn (ββ)is the Bessel function of the first kind of order n argument β. These functions are plotted in Fig.(4.1) as a function of n for various values of β. Note that : 1. JJ nn (ββ) = ( 1) nn JJ nn (ββ) 2. JJ nn 1 (ββ) + JJ nn+1 (ββ) = 2nn JJ ββ nn(ββ) 3. nn= JJ 2 nn (ββ) = 1 Thus ee jjjjjjjjjj ωω mm tt = JJ nn (ββ)ee jjjj ωω mm tt nn= Substituting Eq.(4.6) into Eq.(4.5), we obtain Taking the real part yield xx cc (tt) = AA RRRR ee jj ωω cctt JJ nn (ββ)ee jjjj ωω mm tt nn= = AA RRRR JJ nn (ββ)ee jj (ωω cc +nnωω mm )tt nn= xx cc (tt) = AA JJ nn (ββ)cccccc(ωω cc + nnωω mm )tt nn= (4.6) (4.7) Fig.(4.1)
[4.5] We observe that; 1. The spectrum consists of a carrier-frequency component plus an infinite number of sideb components at frequencies ω c ± nω m (n = 1, 2, 3,... ). 2. The relative amplitudes of the spectral lines depend on the value of J n (β), the value of J n (β) becomes very small for large values of n. 3. The number of significant spectral lines (that is, having appreciable relative amplitude) is a function of the modulation index β. With β 1, only J 0 J 1 are significant, so the spectrum will consist of carrier two sideb lines. But if β 1, there will be many sideb lines. Figure (4.2) shows the amplitude spectra of angle-modulated signals for several values of β. Fig.(4.2) UExample 4.1:U For the angle-modulated signal, xx cc (tt) = 10 cccccc [2ππ(10 6 )tt + 10 ssssss 2ππ(10 3 )tt] Find m(t) if; (a) xx cc (tt) is a PM signal with kk pp = 10. (b) xx cc (tt) is a FM signal with kk ff = 10π. USol. (a) kk pp mm(tt) = 10 ssssss 2ππ(10 3 )tt mm(tt) = ssssss 2ππ(10 3 )tt tt (b) kk ff mm(λλ) = 10 ssssss 2ππ(10 3 )tt tt mm(λλ) = 10 10π ssssss 2ππ(103 )tt mm(tt) = dd 1 sin 2ππ(103 ) tt = 2(10 3 )cccccc 2ππ(10 3 )tt π Note that θθ(tt) = ωω cc tt + φφ(tt) = 2ππ(10 6 )tt + 10 ssssss 2ππ(10 3 )tt
[4.6] φφ(tt) = 10 ssssss 2ππ(10 3 )tt now φφ (tt) = 20 (ππ10 3 ) cccccc 2ππ(10 3 )tt thus, the maximum phase deviation is φφ(tt) mmmmmm = 10 rrrrrr And the maximum frequency deviation is Δωω = φφ (tt) mmmmmm = 20 (ππ10 3 ) rrrrrr/ss = 10 KKKKKK U5. BANDWIDTH OF ANGLE MODULATED SIGNALS U5.1 Sinusoidal Modulation The bwidth of angle-modulated signal with sinusoidal modulation depends on β ω m. From mathematic relations; JJ nn 2 (ββ) = JJ 0 2 (ββ) + 2(JJ 1 2 (ββ) + JJ 2 2 (ββ) + ) = 1 nn= This property will help us to find the power of angle modulated signal as PP = AA JJ 0 (ββ) 2 2 + 2 AA JJ 1 (ββ) 2 2 + AA JJ 2 (ββ) 2 2 + = AA2 2 [JJ 0 2 (ββ) + 2(JJ 2 1 (ββ) + JJ 2 2 (ββ) + )] = AA2 2 1 = AA2 2 Now we can define the bwidth of angle modulated signal as the b of frequencies or harmonics which consists about of 98% of the normalized total signal power is given by; WW BB 2(ββ + 1)ωω mm (5.1) When β 1, the signal is an NB angle-modulated signal its bwidth is approximately equal 2ω m. Usually a value of β < 0.2 is taken to be sufficient to satisfy this condition. Let us consider β =1, then JJ 0 (1) = 0.7652, JJ 1 (1) = 0.44, JJ 2 (1) = 0.1149, JJ 3 (1) = 0.002477, then the power considered in the terms of n = 0,1, 2 PP = 1 JJ 2 0 2 (1) + 2 JJ 2 1 (1) + JJ 2 2 (1) = 1 2 [0.76522 + 2(0.44 2 + 0.1149 2 )] = 0.495 The sum of P for n=2 is 99% of the total power, which is 0.5. If the amplitude JJ nn (ββ) < 0.1, it can be neglected. U5.2 Arbitrary Modulation: For an angle-modulated signal with an arbitrary modulating signal m(t) blimited to ω M rad/s, we define the deviation ratio D as mmmmmmmmmmmmmm ffffffffffffffffff dd DD = bbbbbbbbbbbbbbbbh oooo mm(tt) The deviation ratio D plays the same role for arbitrary modulation as the modulation index β plays for sinusoidal modulation. Replacing β by D ω m by ω M in Eq. (5.1), we have WW BB 2(DD + 1)ωω MM 2( ωω + ωω MM ) (5.3) = ωω ωω MM (5.2)
[4.7] This expression for bwidth is generally referred to as Carson's rule. If DD 1, the bwidth is approximately 2ω M, the signal is known as a narrowb (NB) angle-modulated signal. If DD 1, the bwidth is approximately 2Dω M = 2 ω, which is twice the peak frequency deviation. Such a signal is called a wideb (WB) angle-modulated signal. UExample 5.1: (a) Estimate B FM B PM for the modulating signal m(t) for k f = 2π 10 5 k p = 5π. (b) Repeat the problem if the amplitude of m(t) is doubled. USol. (a) The peak amplitude of m(t) is unity. Hence, mm(tt) mmmmmm = 1. We now determine the essential bwidth B of m(t). The Fourier series for this periodic signal is given by 2ππ mm(tt) = aa nn cccccc nnωω oo tt ωω oo = 2 10 4 = 104 ππ nn Where 8 aa nn = ππ 2 nn oooooo nn2 0 nn eeeeeeee It can be seen that the harmonic amplitudes decrease rapidly with n. The third fifth harmonic powers are 1.21 0.16%, respectively, of the fundamental component power. Hence, we are justified in assuming the essential bwidth of m(t) as the frequency of the third harmonic, that is, 3(10 4 /2) Hz. Thus, f M =15 khz UFor FM: ff = ωω 2ππ = 1 2ππ kk ffmm(tt) mmmmmm = 1 2ππ (2π 105 ) 1 = 100 kkkkkk BB FFFF = 2( ff + ff MM ) = 230 kkkkkk The deviation ratio D is given by DD = ff = 100 ff MM 15 UFor PM:U the peak amplitude of mm (tt) is 2 10 4 ff = 1 2ππ kk ppmm (tt) mmmmmm = 50 kkkkkk Hence, BB PPPP = 2( ff + ff MM ) = 130 kkkkkk
DD = ff = 50 ff MM 15 (b) Doubling m(t) doubles its peak value. Hence, mm(tt) mmmmmm =2. But its bwidth is unchanged (ff MM = 15 kkkkkk). UFor FM: ff = 1 2ππ kk ffmm(tt) mmmmmm = 1 2ππ (2π 105 ) 2 = 200 kkkkkk BB FFFF = 2( ff + ff MM ) = 430 kkkkkk The deviation ratio D is given by DD = ff = 200 ff MM 15 UFor PM:U Doubling m(t) doubles its derivative so that now mm (tt) mmmmmm = 4 10 4, ff = 1 2ππ kk ppmm (tt) mmmmmm = 100kkkkkk Hence, BB PPPP = 2( ff + ff MM ) = 230 kkkkkk DD = ff = 100 ff MM 15 Observe that doubling the signal amplitude roughly doubles the bwidth of both FM PM waveforms. [4.8] If m(t) is time-exped by a factor of 2; that is, if the period of m(t) is 4 10-4,then the signal spectral width (bwidth) reduces by a factor of 2. We can verify this by observing that the fundamental frequency is now 2.5 khz, its third harmonic is 7.5 khz. Hence, f M = 7.5 khz, which is half the previous bwidth. Moreover, time expansion does not affect the peak amplitude so that mm(tt) mmmmmm = 1. However, mm (tt) mmmmmm is halved, that is, mm (tt) mmmmmm = 10, 000. UFor FM: ff = 1 2ππ kk ffmm(tt) mmmmmm = 100 kkkkkk BB FFFF = 2( ff + ff MM ) = 2(100 + 7.5) = 215 kkkkkk UFor PM: Hence, ff = 1 2ππ kk ppmm (tt) mmmmmm = 25 kkkkkk BB PPPP = 2( ff + ff MM ) = 65 kkkkkk
Note that time expansion of mm(tt) has very little effect on the FM b width, but it halves the PM bwidth. This verifies that the PM spectrum is strongly dependent on the spectrum of mm(tt). [4.9] U6. GENERATION OF ANGLE-MODULUED SIGNALS U6.1 Narrowb Angle-Modulated Signals: The generation of narrowb angle-modulated signals is easily accomplished in view of Eq.(3.2) (3.3). This is illustrated in Fig.(6.1) x NBPM (t) A sinω c t A cosω c t x NBFM (t) A sinω c t A cosω c t Fig.(6.1) U6.2 Wideb Angle-Modulated Signals: There are two methods of generating wideb (WB) angle-modulated signals; the indirect method the direct method. U6.2.1 Indirect Method of Armstrong In this method, an NB angle-modulated signal is produced first then converted to a WB angle-modulated signal by using frequency multipliers. The frequency multiplier multiplies the argument of the input sinusoid by n. Thus, if the input of a frequency multiplier is xx(tt) = AA cccccc[ωω cc tt + φφ(tt)] Then the output of the frequency multiplier is yy(tt) = AA cccccc[nnnn cc tt + nnnn(tt)] Use of frequency multiplication normally increases the carrier frequency to an impractically high value. To avoid this, a frequency conversion (using a mixer or DSB modulator) is necessary to shift the spectrum.
[4.10] U6.2.2 Direct Generation In a voltage-controlled oscillator (VCO), the frequency is controlled by an external voltage. The oscillation frequency varies linearly with the control voltage. We can generate an FM wave by using the modulating signal m(t) as a control signal. ω i (t) = ω c + k f m(t) One can construct a VCO using variable reactive element (C or L) in resonant circuit of an oscillator. In Hartley or Colpitt oscillators, the frequency of oscillation is given by ωω oo = 1 LLLL If the capacitance C is varied by the modulating signal m(t) CC = CC oo kkkk(tt) 1 1 ωω oo = LLCC oo 1 kkkk(tt) CC oo 1 1 + kkkk(tt) LLCC oo 2CC oo = LLCC oo 1 kkkk(tt) CC oo kkkk(tt) CC oo 1 Where (1 + xx) nn 1 + nnnn ffffff xx 1. Thus, ωω oo = ωω cc 1 + kkkk(tt) 2CC oo ωω cc = 1 LLCC oo ωω oo = ωω cc + kk ff mm(tt) kk ff = kkωω cc 2CC oo The main advantage of direct FM is that large frequency deviations are possible thus less frequency multiplication is required. The major disadvantage is that the carrier frequency tends to drift so additional circuitry is required for frequency stabilization. U7. DEMODULATION OF ANGLE-MODULATED SIGNALS Demodulation of an FM signal requires a system that produces an output proportional to the instantaneous frequency deviation of the input signal. Such a system is called a frequency discriminator. If the input to an ideal discriminator is an angle-modulated signal xx cc (tt) = AA cccccc[ωω cc tt + φφ(tt)] 1 2
[4.11] then the output of the discriminator is yy dd (tt) = kk dd (tt) where k d is the discriminator sensitivity. For FM yy dd (tt) = kk dd kk ff mm(tt) The characteristics of an ideal frequency discriminator are shown below The frequency discriminator also can be used to demodulate PM signals. For PM, φφ(tt) is given by φφ(tt) = kk pp mm(tt) Then yy dd (tt) yy dd (tt) = kk dd kk pp dd mm(tt) Integration of the discriminator output yields a signal which is proportional to m(t). A demodulator for PM can therefore be implemented as an FM demodulator followed by an integrator. A simple approximation to the ideal discriminator is an idealdifferentiator followed by an envelope detector. the output of the differentiator is xx cc (tt) = AA ωω cc + (tt) ssssss[ωω cc + φφ(tt)] The signal xx cc (tt) is both amplitude- angle-modulated. The envelope of xx cc (tt)is AA ωω cc + (tt) The output of the envelop detector is yy dd (tt) = ωω ii Which is the instantaneous frequency of the xx cc (tt).
[4.12] U8. NOISE IN ANGLE MODULATION SYSTEMS The transmitted signal XX cc (tt) has the form XX cc (tt) = AA cc cccccc[ωω cc tt + φφ(tt)] (8.1) Where kk pp XX(tt) ffffff PPPP φφ(tt) = tt kk ff XX(λλ) ffffff FFFF (8.2) Figure (8.1) shows a model for the angle demodulation system. The predetection filter bwidth BB TT = 2(DD + 1)BB. The detector input is YY ii (tt) = XX cc (tt) + nn ii (tt) Where Fig.(8.1) nn ii (tt) = nn cc (tt) ccccccω cc tt nn ss (tt) ssssssω cc tt = vv nn (tt) cccccc[ωω cc tt + φφ nn (tt)] The carrier amplitude remains constant, therefore Hence, SS ii = EE[XX cc 2 (tt)] = 1 2 AA cc 2 NN ii = ηηbb TT = AA cc 2 2ηηBB TT (8.3) SS NN ii Which is independent of XX(tt). SS is often called the carrier-to-noise ratio (CNR). NN ii YY ii (tt) = VV(tt) cccccc [ωω cc tt + θθ(tt)] (8.4) Where VV(tt) = {[AA cc cccccccc + vv nn (tt)ccccccφφ nn (tt)] 2 + [AA cc ssssssss + vv nn (tt)ssssssφφ nn (tt)] 2 } 1/2 (8.5) θθ(tt) = tttttt 1 AA ccssssssss(tt) + vv nn (tt)ssssssφφ nn (tt) AA cc cccccccc(tt) + vv nn (tt)ccccccφφ nn (tt) (8.6) The limiter suppresses any amplitude variation VV(tt). Hence, in angle modulation, SNRs are derived from consideration of θθ(tt) only. The detector is assumed to be ideal. The output of the detector is θθ(tt) ffffff PPPP YY oo (tt) = dd θθ(tt) (8.7) ffffff FFFF Let
where [4.13] YY ii (tt) = RRRR YY(tt)ee jj ωω cctt (8.8) YY(tt) = AA cc ee jjφφ(tt) + vv nn ee jj φφ nn (tt) (8.9) For signal dominance case, vv nn AA cc for almost all t. from Fig.(8.2) the length L of arc AB is LL = YY(tt)[θθ(tt) φφ(tt)] (8.10) Fig.(8.2) YY(tt) = AA cc + vv nn (tt) cccccc[φφ nn (tt) φφ(tt)] AA cc (8.11) LL vv nn (tt) ssssss[φφ nn (tt) φφ(tt)] (8.12) Hence, from Eq.(8.10), we obtain θθ(tt) φφ(tt) + vv nn(tt) ssssss[φφ AA nn (tt) φφ(tt)] (8.13) cc Replacing φφ nn (tt) φφ(tt) with φφ nn (tt) will not affect the result. Thus θθ(tt) φφ(tt) + vv nn(tt) AA cc ssssss[φφ nn (tt)] θθ(tt) φφ(tt) + nn ss(tt) AA cc (8.14) From Eqs.(8.7) (8.2) the detector output is YY oo (tt) = θθ(tt) = kk pp XX(tt) + nn ss(tt) AA cc ffffff PPPP (8.15) dd θθ(tt) YY oo (tt) = = kk ff XX(tt) + nn ss(tt) AA cc ffffff FFFF (8.16) U8.1 (S/N) o in PM systems From Eq.(8.15) SS oo = EE kk 2 pp XX 2 (tt) = kk 2 pp EE[XX 2 (tt)] = kk 2 pp SS XX (8.17) Hence, NN oo = EE 1 AA2 nn ss 2 (tt) = 1 cc AA2 EE[nn ss 2 (tt)] = 1 (2ηηηη) (8.18) cc AA2 cc
[4.14] Since, Eq.(8.19) can be expressed as SS NN oo = kk pp 2 AA cc 2 SS XX 2ηηηη γγ = SS ii ηηηη = AA 2 cc 2ηηηη (8.19) (8.20) SS NN oo = kk pp 2 SS XX γγ (8.21) U8.2 (S/N) o in FM systems From Eq.(8.16) SS oo = EE kk ff 2 XX 2 (tt) = kk ff 2 EE[XX 2 (tt)] = kk ff 2 SS XX (8.22) The PSD of nn ss(tt) is given by Then Hence, NN oo = EE 1 AA cc 2 [nn ss(tt)] 2 = 1 AA cc 2 EE[[nn ss(tt)] 2 ] (8.23) SS nnss nn ss (ωω) = ωω 2 SS nnss nn ss (ωω) = ωω2 ηη ffffff ωω < WW (= 2ππππ) 0 ooooheeeeeeeeeeee WW NN oo = 1 1 AA2 cc 2ππ ωω2 ηη = 2 ηη 3 WW AA cc 2 WW 3 2ππ (8.24) (8.25) SS NN = 3AA2 cc (2ππ)kk 2 ff SS XX oo 2ηηWW 3 (8.26) Using Eq.(8.20), we can express Eq.(8.26) as SS NN = 3 kk ff 2 SS XX oo WW 2 AA cc 2ηηηη = 3 kk ff γγ (8.27) WW2 Since Δωω = kk ff XX(tt) = kk mmmmmm ff [ XX(tt) 1], Eq.(8.27) can be rewritten as SS NN = 3 Δωω 2 oo WW SS XX γγ = 3DD 2 SS XX γγ (8.28) Equation (8.25) indicates that the output noise power is inversely proportional to the mean carrier power AA2 cc in FM. This effect is called Unoise quietingu. 2 UExample 8.1:U consider an FM broadcast system with parameter f=75 khz B=15 khz. Assuming SS XX = 1, find the output SNR calculate the improvement (in db) 2 over the baseb system. USol. Substituting the given parameters into Eq.(8.28), we obtain SS NN = 3 75(103 2 ) oo 15(10 3 ) 1 γγ = 37.5γγ 2 Now, 10 log 37.5=15.7 db, which indicates that (SS/NN) oo is about 16 db better than the baseb system. 2 2 SS XX