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Chapter 9. Frequency and Phase Modulation Figure 9-1. General block diagrams for transmitters of amplitude, frequency, and phase modulation. 1 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Modulation A frequency-modulated signal is any periodic signal whose instantaneous frequency f is deviated from an average value fc by an information signal m(t). The maximum amount by which the instantaneous frequency is deviated from the carrier frequency fc is called the peak deviation, Dfc(pk). Hence the instantaneous output frequency is expressed as f = fc + kovm(t) (9-1) It is assumed that when vm(t) = 0, Dfc = 0 and f = fc. Otherwise, Dfc = kovm(t) and f = fc + Dfc (9-2) 2 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Modulation Figure 9-2. Frequency variations in FM are directly proportional to the modulating voltage. 3 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Modulation Index In general a signal with angle modulation is expressed simply as s(t) = A.cosq(t) (9-3) = A.cos(wt+f), (9-4) where either the angle f or the angle wt is varied. When f is varied by the information signal, f(t) a m(t), the result is called Phase Modulation (PM). If f is held constant and the angle wt is modulated by the information signal deviating the carrier frequency, the result is called Frequency Modulation (FM). 4 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Modulation Index Angular frequency w is the rate of change of phase, that is, w = dq/dt Since w = 2pf, then dq/dt = 2pfc + 2pkovm(t) (9-6) This differential equation is solved for q(t) (9-7) where qo is the arbitrary starting phase at t = 0. (9-8) 5 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Modulation Index For vm(t) = Vpkcos2pfmt, the integral of the third term is evaluated as follows: (9-9) The arbitrary constant phase fo at t = 0, can be combined with qo in Equ. (9-8) and set to zero q(t) = 2pfct + (koVpk/fm).sin2pfmt = 2pfct + [Dfc(pk)/fm].sin2pfmt (9-10) 6 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Modulation Index The cosine-modulated FM signal is written, using Equ. (9-3), as sFM(t) = Acos(2pfct + mf sin2pfmt) (9-11) where mf = Dfc(pk)/fm is the peak of the sinusoidally varying carrier phase angle. The modulation index, in units of radians, for sinusoidal FM is calculated from mf = Dfc(pk)/fm (9-12) 7 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Modulation Index EXAMPLE 9-1: A 1-MHz VCO with a measured sensitivity of ko = 3 kHz/V is modulated with a 2-Vpk, 4-kHz sinusoid. Determine the following: 1. The maximum frequency deviation of the carrier. 2. The peak phase deviation of the carrier and therefore the modulation index. 3. mf if the modulation voltage is doubled. 4. mf for Vm(t) = 2cos[2p(8kHz)t] volts. 5. Express the FM signal mathematically for a cosine carrier and the cosine-modulating signal of part 4. The carrier amplitude is 10-Vpk. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 8 Modulation Index Solution: 1. Dfc(pk) = koVm(pk) = (3 kHz/V)(2Vpk) = 6 kHz. 2. From Equ. (9-12) the peak phase deviation by modulating sinusoid is Dfc(pk)/fm = 6 kHz/4 kHz = 1.5 rad = mf . 3. For a linear modulator Df a DVm so mf = 3.0 rad is double the answer of part 2. Of course this can be computed from Equ. (9-12). Since Vm increases to 4V, then mf = Dfc(pk)/fm = (4V x 3kHz)/4kHz = 3 (radians). 4. The modulation is sinusoidal with 2Vpk so the carrier frequency deviation is the same as part 1 --- Dfc(pk) = 6 kHz. Since fm = 8kHz, the peak carrier phase deviation and modulation index is mf = 6 kHz/8 kHz = 0.75. 5. The modulating signal is a cosine (so is the carrier). Therefore using the value of mf = 0.75, we have vFM = 10cos(2p106t + 0.75sin2p8x103t). Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 9 Sidebands and Spectrum For low deviation (mf < 0.25), called narrowband FM (NBFM), there is a carrier and a set of sidebands much like AM. The two sidebands are shifted 90° relative to the carrier (as compared to AM.). This comparison is illustrated in Fig.9-3. Figure 9-3. Comparison of phasor relationships in AM and FM/PM. 10 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Sidebands and Spectrum Figure 9-4 shows a comparison of AM and NBFM phasor variations. Notice that the amplitude for the NBFM seems to vary slightly in the illustration. This is because the angle drawn for the illustration is greater than 14.3o (0.25 rad). 11 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Sidebands and Spectrum Figure 9-4. Comparison of instantaneous transmitted phasors –sAM(t) and sPM(t). Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 12 Sidebands and Spectrum The spectrum for angle-modulated signals (FM and PM) may be determined from vFM = Acos(wct + mf sinwmt) (9-13) = Acos(mf sinwmt)coswct - Asin(mf sinwmt)sinwct. (9-14) For low-deviation FM (mf < 0.25) the angular variations of the cosine and the sine in the brackets of Equ. (9-14) can be approximated as cosDf = 1 and sinDf = Df. Therefore narrowband FM can be approximated as vNBFM = Acoswct – A(mf sinwmt) sinwct (9-15) Also, Amf sin(wmt).sin(wct) = (Amf/2)cos(wc-wm)t – (Amf/2)cos(wc+wm)t. This shows clearly the constant amplitude carrier and the quadrature sidebands as illustrated in Figure 9-4. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 13 Wideband FM The amplitudes of the frequency components [sinusoid (mfsinwmt)] cannot be evaluated in closed form but are tabulated in the table of Bessel functions of the 1st kind of order n (Table 9-1). The continuous Bessel functions Jn(mf) are shown in Figure 9-5, where Vo(pk) = AJo(mf) (9-16) gives the peak voltage amplitude of the carrier for any modulation index mf up to and including mf = 10. The peak voltage of sidebands on either side of the carrier is given by Vn(pk) = AJn(mf) (9-17) where n = 1, 2, …is the order of the sideband -- that is, the number of the particular sideband pair. 14 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Wideband FM Figure 9-5. Amplitudes of carrier and sidebands for FM (and PM) relative to the unmodulated carrier. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 15 Wideband FM Table 9-1. Bessel Function of the First Kind, Jn(mf). 16 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Wideband FM Fig. 9-6. Frequency spectra for FM signals with constant modulation frequency fm but different carrier-frequency deviations. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 17 Wideband FM Notice that for mf < 0.25, the carrier drops by less than 2% and only one set of sidebands, the 1st-order side-band J1(mf), have a value exceeding 1% (0.01) of the unmodulated carrier. These values justify the approximations used for NBFM. The 1st-order sets of sidebands are separated from the carrier by fm. For mf > 0.25, there are additional sets of sidebands. This is the wideband FM case, and the sideband sets are separated from the carrier by fm, 2fm, 3fm, …, nfm corresponding to the order of the Bessel function Jn(mf). This can be seen in the example FM signal spectra of Figure 9-6. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 18 Wideband FM EXAMPLE 9-2: An FM signal expressed as vFM = 1000cos(2p107t + 0.5cos2p104t) is measured in a 50-W antenna. Determine the following: 1. Total power; 2. Modulation index; 3. Peak frequency deviation; 4. Modulation sensitivity if 200 mVpk is required to achieve part3. 5. Spectrum; 6. Bandwidth (99%) and approximate circuit bandwidth by Carson's rule. 7. Power in the smallest sideband (just one sideband of the 99% bandwidth). 8. Total information power. 19 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Wideband FM Solution: 1. PT = (V(pk))2/2R = 10002/100 = 10 kW. 2. The 0.5cos2p104t term gives the peak phase variation and mf =0.5. 3. mf = Dfc(pk)/fm , hence Dfc(pk) = 0.5 x 104 Hz = 5 kHz. 4. ko = Dfc / DVm = 5kHz/0.2V = 25 kHz/V. 5. The spectrum is determined from Table 9-1 with mf = 0.5, A = 1000Vpk, and fm = 10 kHz. The carrier is AJo(0.5) = 940Vpk at 10 MHz. The 1st sets of sidebands are AJ1(0.5) = 240Vpk at 9.990 MHz and 10.010 MHz, and the 2nd (and least significant) sidebands are AJ2(0.5) = 30Vpk at 9.98 MHz and 10.020 MHz. 20 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Wideband FM 6. From 5, the 99% information bandwidth is 2x20 kHz = 40kHz, and from Carson's rule, BW = 2(10 kHz + 5 kHz) = 30kHz for reasonably low circuit distortion. 7. There are two ways to arrive at the solution: From part 5 each sideband is 30Vpk, so Plsb = (30Vpk)2/100 = 9W. From Equ. (9-18) and PT = 10kW, Plsb = (0.03)2 x 10kW = 9W. 8. The power in the information part of the signal is PT - Pc, where the carrier power is Pc= (940Vpk)2/100 = 8.836kW. PT - Pc = 10 kW – 8.836 kW = 1.164 kW. Low-index modulation is not efficient (11.64% in this case). Incidentally, the total power must add up as follows: PT = Pc+P1+P2 = 8.836 kW + 2[(2402/100)+9W] = 10.006 kW, a 0.06% round-off error. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 21 Linear Phase Modulation Linear phase modulation is expressed by vPM = Acos[wct + kpvm(t)] (9-23) where the proportionality constant kp, in rad/V, is called the modulator sensitivity. For a sinusoidal modulation signal of vm(t) = Vpkcos2pfmt, the PM signal becomes sPM(t) = Acos[2pfct + kpVpkcos2pfmt] (9-25) = Acos[2pfct + mpcos2pfmt] (9-26) where mp = kpVpk = Df(pk) (9-28) is the peak phase deviation of the sinusoidally varying carrier phase angle or the modulation index in units of radians. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 22 Linear Phase Modulation When Equ. (9-26) is expanded by Trigonometric identity and the narrow-band approximation (Df < 0.25 rad) applied, the result becomes vNBPM(t) = Acoswct – A(mpcoswmt).sinwct (9-29) Figure 9-8. NBPM using a balanced modulator and sideband phase shift. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 23 Diode Modulator for PM Figure 9-9 shows the most straightforward method of producing a PM signal. A voltage-variable capacitance diode (called Varicap, Varactor, and tuning diode) is used to vary the tuning of a parallel-resonant circuit. The stable crystal-controlled-frequency carrier signal will thus be phase-shifted in accordance with the modulation voltage. Figure 9-9. Phase modulation system. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 24 Diode Modulator for PM The vertical scale in the diode voltage-capacitance curve of Figure 9-10 will yield the fractional change DCd/Cd in diode capacitance for small input voltage changes. The capacitance changes can be estimated by determining the slope of the C(V) curve at the bias point V. Thus dCd/dV = (d/dV).[Co(1+2|V|)-1/2] = -Cd/(1+2|V|) (9-30a) or DCd/Cd = -DV/(1+2|VR|) (9-30b) where VR is the reverse bias across Cd, and DV is the peak value of Vm, the modulation signal voltage. Equ. (9-30) is then used to determine the tuned-circuit frequency shift Dfo/fo = -0.5(DCd/Cd) (9-31) Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 25 Diode Modulator for PM Figure 9-10. Ratio of junction capacitance at reverse-bias to junction capacitance at 6V reverse-bias versus applied reverse-bias voltage. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 26 Diode Modulator for PM For sinusoid angle modulation mp = Df(pk), and knowing the peak input voltage swing, the modulator sensitivity can be calculated from kp = Df/DVm (9-32) An exact formula for the phase shift is derived from the phase-versus-frequency curve for the parallel tank impedance, where Z = Rp/(1+jQr) (9-33a) r = f/fo – fo/f = [1+(Dfo/fo)] - [1+(Dfo/fo)]-1 (9-33b) 27 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Diode Modulator for PM Dfo is a small deviation, less than 10% from center frequency fo. The phase-versus-frequency curve, f(f), follows a tangent curve that is the phase of impedance Z; that is, f(f) = -tan-1Qr for which the magnitude provides the FM modulation index given by Df = mp = tan-1Qr (9-34) 28 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Derivation of Modulator Equations The modulator sensitivity can be calculated from kp = df/dVm = (dC/dV).(df/dC).(df/df) In general, Cd(V) = Co(1 + 2|VR|)-p (9-35) (9-38) The rate of diode capacitance change with voltage is computed by differentiating Equ. (9-38) as dCd/dVR = -2pCd/(1+2VR) (9-39) where, for various diode junction doping profiles, 0.3 < p < 0.6. For convenience, let the power be p = 0.5, resulting in dCd/dVR = -Cd/(1+2|VR|) Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan (9-40) 29 Derivation of Modulator Equations The second term in Equ. (9-35), df/dC, is df/dC = df/dCd = -0.5fo/Cd (9-41) For small frequency deviations, the phase change is also small, so that (9-42a) (9-42b) which, when combined with f = -tan-1Qr = -Qr, yields the approximation df/df = -2Q/fo (9-43) 30 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Derivation of Modulator Equations Substituting Eqs. (9-40), (9-41), and (9-43) into Equ. (9-35) yields df/dVm = [-Cd/(1+2|VR|)].[-fo/2Cd].[-2Q/fo] = -Q/(1+2|VR|) (9-44) and the magnitude of Equ. (9-44) gives the phase modulator sensitivity at the circuit bias point VR as kp = Q/(1 + 2|VR|) (9-45) 31 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Derivation of Modulator Equations The modulation index mp resulting from a change in voltage, dVm, applied to the tuning diode is, from Equ. (9-28), mp = QDVm(pk)/(1 + 2|VR|) (9-46) for the circuit of Figure 9-9. Eqs. (9-45) and (9-46) are exact slope equations for a simple phase modulator. 32 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Indirect FM from PM Figure 9-16. High-index FM from a phase modulator preceded an integrator (Amstrong method). 33 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Indirect FM from PM Equ. (9-8) for FM, when substituted into the general angle modulation expression (Equ. (9-3)), yields vFM(t) = Acos[2pfct + qo + 2pko∫vm(t)dt] (9-60) This equation reveals the trick needed for using a phase modulator to produce FM: integrate the modulating signal vm(t) before modulating the phase modulator. To achieve the high-modulation indices required by the FCC, frequency multipliers follow the FM modulator. As an example, NBPM and FM have an index less than 0.25 rad; in order to produce 5.0 rad (Dfc(pk)/fm = 75kHz/15kHz = 5.0), a multiplication of at least 20 is required. 34 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Indirect FM from PM Fig. 9.5. (a) Generation of AM; (b) Generation of NBPM; (c) Generation of NBFM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 35 Indirect Generation of FM: The Armstrong Method 1). Integrate the modulating signal s(t) to produce the signal 2). Phase-modulate a carrier with z(t) to produce NBFM. 3). Use a system of frequency multipliers to convert NBFM wideband FM. Figure 7.6-1. Method of generating indirect FM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 36 Indirect Generation of FM: The Armstrong Method A block diagram of an IFM system is shown in Fig. 7.6-1. At point A the output is (7.6-1) and at point B the output is proportional to (7.6-2) The frequency multiplier produces a signal with frequency given by w = N.dq1(t)/dt, where q1(t) = wc1t + b1sinwmt. Hence w = Nwc1 + Nb1wmcoswmt. (7.6-3) The new phase is proportional to the integral of N.dq1(t)/dt, so at C we have x(t) = cos[Nwc1t + Nb1sinwmt] = cos(wct + b sinwmt), (7.6-4) where b is now a value typical of wideband FM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 37 Indirect Generation of FM: The Armstrong Method Example Let fm range from 50 to 5000 Hz, and let the maximum frequency deviation, Df, at the output be 75 kHz. Then bmin = (75x103)/(5x103) = 15 bmax = (75x103)/50 = 1500. If [b1]max = 0.5, then the required frequency multiplication is N = bmax/[b1]max = 1500/0.5 = 3000 The maximum allowed frequency deviation at the input is Df1 = (75x103)/3000 = 25 Hz. 38 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Indirect Generation of FM: The Armstrong Method If ihe maximum specified Df at the output is 75 kHz. If the initial carrier frequency fc1 were, say, 200 kHz, then the final frequency would be fc = 600 MHz. This figure is too high for standard FM broadcasting, amd frequecncy converters are used to reduce fc to the desired band. For example, if we heterodyne x(t) with a carrier wave of frequency, fLO, the modulation index Nb1 remains unaffected but the wave is shifted to the new carrier frequency, Nf1 - fLO. 39 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Indirect Generation of FM: The Armstrong Method A block diagram of an Armstrong-type indirect FM transmitter is shown in Fig. 7.6-2. The largest modulation index furnished by the NBFM system is determined by the lowest modulating frequency (≒50 Hz), and this determines the required frequency multiplication. For example, if b1 = 0.5 for a 50-Hz tone, then the 75-kHz frequency deviation requires a b = 1500. The required multiplication is 1500/0.5 = 3000. The largest tone frequency determines the bandwidth. The value of b1 for the 15-kHz signal is approximately 1.7x10-3. The extremely small initial values of b1 are required to prevent distortion due to amplitude-modulation effects that can occur in the generation of NBFM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 40 Indirect Generation of FM: The Armstrong Method Figure 7.6-2. Amstrong-type indirect FM transmitter. 41 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Stability and AFC High-frequency deviation is not practical with crystal-controlled VCOs even with multiplication. And the stability is reduced by the multiplication factor N. LC-tuned VCOs have good deviation sensitivity but poor stability with respect to frequency drifts due to aging effects and temperature changes. One solution to these dilemmas is to modulate the LC VCO, then lock the average value of the VCO frequency to a crystal reference through a narrowbandwidth phase-locked loop. This technique is shown in Figure 9-20. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 42 Frequency Stability and AFC The frequency divider of Figure 9-20 is useful for minimizing interference from the crystal oscillator (XO). The low-pass filter (LPF) prevents feedback of modulation frequencies (FM feedback) and eliminates the possibility of the loop locking to a sideband. Figure 9-20. Phase-locked FM generator. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 43 Frequency Stability and AFC Fig. 9-21. Transmitter automatic frequency control loop. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 44 Frequency Stability and AFC One more technique for stabilizing an oscillator frequency is the automatic frequency control (AFC) system shown in Figure 9-21. This system is used for stabilizing the carrier center frequency against drift in FM transmitters. AFC without the multiplier and down-converter (mixer and XO) is used in receivers for stabilizing the first local oscillator. In many television receivers it is called AFT-automatic fine-tuning. The Crosby AFC loop of Figure 9-21 is an excellent system to analyze for understanding feedback. Suppose the VCO frequency fo drifts by an amount dfo. Then with switch S1 open (position 2), there will be no feedback and the transmit frequency will be fxmt = (N1 x N2 x N3)( fo + dfo) Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 45 Frequency Stability and AFC To meet FCC specifications, dfxmt = (N1N2N3)dfo < 2 kHz (9-71) With the generator set exactly to f* = N1N2 fo the frequency at the frequency discriminator input is fd = fXO – f*. The frequency discriminator is a special LC tunedcircuit that changes frequency variations to amplitude variations (FM to AM) and is followed by peak AM detectors in a balanced configuration. When properly aligned the circuit is tuned to fd and the output voltage is zero volt. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 46 Frequency Stability and AFC The vo(t)/fd(t) transfer characteristic of the discriminator is linear over a wide frequency range, with a slope (gain) of Dvo/Dfd = kd An input frequency shift of +Dfd will result in an output voltage change of +Dvo = kdDfd = +Vo. The purpose of the low-pass filter (LPF) in Figure 9-21 is to prevent any modulation signals from being fed back (FM feedback), thereby reducing the modulation index. The very slow dc changes in Vo due to Dfd-drifts detected by the frequency discriminator are fed back to the VCO with the correct polarity for forcing the VCO back to nearly fo. 47 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Stability and AFC Let us go back to switch S1. The signal at pin 1 of S1 has an open-loop frequency error of dfol* = N1N2dfo. When the switch is closed to pin 1, the system with feedback will reduce this frequency discriminator input to be Dfd = -dffb*. The discriminator output voltage will be (Vo)fb= -kddffb*, so the VCO frequency will be corrected to (dfo)fb = ko(Vo)fb = -kokddffb*. This frequency correction, when multiplied by N1 and N2, is the correction signal which reduces the open-loop error of f*. That is, dffb* = dfol* - kokdN1N2dffb* (9-73) 48 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Stability and AFC Solving for in Equation 9-73 yields the amount of drift at f* for the system with feedback: dffb* = dfol*/[1+ kokdN1N2] (9-74) The error in the transmitted carrier frequency will be dfxmt(fb) = N3 x dffb* (9-75) Equation (9-75) shows the expected result for a system with feedback, the open-loop drift is reduced 1+(loop gain), where the magnitude of the gain once-aroundthe-loop is loop gain = kokdN1N2 49 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Frequency Stability and AFC Figure 9-21. Transmitter automatic frequency control loop. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 50 Frequency Stability and AFC Example 9-8: A VCO with a long-term frequency stability of dfo/fo = 200 ppm is used in the AFC system of Figure 9-21. The transmit carrier frequency is to be 108 MHz, and the circuit component gains are as follows: ko = 10 kHz/V, N1 = 2, N2 = 2, N3 = 3, kd = 2V/kHz, also fXO = 37 MHz. All circuits except the VCO are considered to be perfectly stable. Determine the following: 1. fo, f*, and fd . 2. dfxmt open-loop; is it within FCC specs? 3. Transmitted carrier frequency and frequency error close-loop. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 51 Frequency Stability and AFC Solution: 1. fo = 108MHz/12 = 9MHz; f* = 4fo = 36 MHz; fd = fXO - f* = 1MHz. 2. Without feedback, the VCO will drift by (dfo/fo) x fo = 200 Hz/MHz x 9 MHz = 1800 Hz. At the transmitter output this will be dfxmt(ol) = 12 x 1800 Hz = 21.6 kHz. This is also the same as 200 ppm x 108 MHz = 21.6 kHz. In any case, it is way beyond the specified +2 kHz and fxmt(ol) = 108.0216 MHz. 3. dfol*= 4dfo = 4x1800Hz = 7200Hz. The loop gain is (10kHz/V).(2x2).(2V/kHz) = 80. The drift at f* is reduced by 81 so that dffb* = 7200Hz/81 = 88.9Hz, which at the transmitter output becomes dfxmt(fb)= 88.9Hz x 3 = 266.7Hz. The transmitted carrier, instead of being 108MHz, is fxmt(fb) = 108.0002667MHz---well within +2kHz. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 52 FM RECEIVERS Receivers for FM and PM are super-heterodyne receivers like their AM counterparts. However, the AGC of AM receivers is not used for FM and PM because there is no information in the amplitude of the transmitted signal. Constant amplitude into the FM detector is still desirable, so additional IF gain is used and the final IF amplifiers are allowed to saturate. The IF amplifiers used in the output stages require special design consideration for good saturation characteristics and are called limiters, or passband limiters when the output is filtered to preserve the information bandwidth. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 53 FM Demodulators There are three general categories of FM demodulator circuits: 1. Phase-locked loop (PLL) demodulator; 2. Slope detection/FM discriminator; 3. Quadrature detector: They all produce an output voltage proportional to the instantaneous input frequency. 54 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Phase-Locked Loop Demodulator Figure 9-22. Frequency discriminator circuit symbol. 55 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Phase-Locked Loop Demodulator Figure 9-23. Phase-locked loop FM detector. 56 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Phase-Locked Loop Demodulator As seen in Figure 9-23, the PLL consists of a phase detector (PD), low-pass filter (LPF), and voltage-controlled oscillator (VCO). The PLL VCO, with voltage-to-frequency conversion sensitivity of ko, is tuned to the FM carrier frequency with Vo = 0. The loop produces a voltage Vo which forces the VCO to track the input frequency. Hence, vo will be directly proportional to the input frequency variations of the FM carrier: fc(t) = ko(xmt) x vm(t) With limiter ahead of demodulator and a properly compensated loop, the circuit gain will be constant, so that vo(t) = kdfc(t) = kd[ko(xmt) x vm(t)] = [kdko(xmt)]vm(t) Therefore, vo(t) a vm(t) (9-83) where vm(t) is the transmitted information signal. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 57 Phase-Locked Loop Demodulator A sinusoidal information signal vm(t) = Vpk coswmt (9-84) frequency modulated on a carrier of wc results in sFM(t) = Acos(wct + mfsinwmt) (9-85) where mf = Dfc(pk)/fm = 2pkoVpk/wm (9-86) To recover vm(t) at the demodulator output, the off-resonant slope of the tuned circuit differentiates sFM(t) as follows: (9-87) Equation (9-87) has high-frequency variations around the carrier and amplitude (magnitude) variations of -Awc - Amfwmcoswmt. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 58 Phase-Locked Loop Demodulator The output of the envelope detector is the magnitude of vd(t), with polarity depending on the diode(s) direction. Hence, Vo(t) a Awc + Amfwmcoswmt = Vdc + A2pkoVpk coswmt (9-88) using Equ. (9-88) and noting that the dc voltage is proportional to carrier amplitude and frequency. After coupling through a series capacitor, the information signal is vo(t) = kdVpk coswmt (9-89) where kd = 2pkoA plus any circuit losses is the demodulator sensitivity. 59 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Slope Detection/FM Discriminator In Figure 9-24a, the center frequency fo of the tuned circuit is set such that the input carrier signal falls on the slope of the resonance curve. With fc on the low side of fo and the diode in the direction shown in Figure 9-24a, a positive S curve transfer characteristic is realized. As frequency varied from fc, the RF amplitude varied. These RF amplitude variations are peak-detected with the diode and RF low-pass filter. Figure 9-24b illustrates the balanced slope detector. It provides better linearity than the single slope detector, but it still gas a somewhat limited frequency range, and the two variable capacitors increase the complexity. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 60 Slope Detection/FM Discriminator Figure 9-24. (a) Slope detector. (b) Balanced slope detector. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 61 Foster-Seeley Phase-Shift Discriminator In Figure 9-25, the input signal voltage across the transformer primary winding vp is coupled by capacitor Cc directly across the RF choke. The secondary induced-voltage vs is divided equally between v1 and v2 by a secondary-winding center-tap. The secondary (as well as the primary) circuit is tuned to resonance at the carrier frequency. L2C2 from a high-Q resonant circuit which at resonance circulates a current is with the same phase as vs. 62 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Foster-Seeley Phase-Shift Discriminator Figure 9-25. Foster-Seeley phase-shift discriminator. 63 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Foster-Seeley Phase-Shift Discriminator The vector vs = v1 + v2 is in phase quadrature (90o) to vp at resonance. The resultant vector diagram is shown in Figure 9-26, where va and vb are the phasor signals applied to the upper and lower peak detectors, respectively. The detector output voltage Vo will be Vo = Vao – Vbo where Vao = √2 |va(rms)| and Vbo = √2 |vb(rms)| (9-90) 64 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Foster-Seeley Phase-Shift Discriminator Figure 9-26. Vector diagram for phasors in FosterSeeley discriminator for fin at circuit resonance. 65 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Foster-Seeley Phase-Shift Discriminator When the FM signal frequency reaches its peak deviation f = fc + Dfc(pk), then one calculates r(max) and the phase shift of v1 from the argument of 1/(1+jQsr); that is, Dq = -tan-1Qsr (9-94) Then the vector diagram of Figure 9-26 is modified as illustrated in Figure 9-27 (for f > fo). The peak output Vo(pk) and the detector sensitivity kd = DVo/Df for this phase-shift discriminator can then be determined graphically or by calculation using the law of cosines. 66 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Foster-Seeley Phase-Shift Discriminator Fig. 9-27. Identical vector diagrams for discriminator with fin above circuit resonance. (a) corresponds to Fig. 9-26. 67 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Ratio Detector The ratio detector is a circuit very similar to the FosterSeeley discriminator, except for the balanced peakdetector configuration that makes it much less sensitive to AM. As seen in Figure 9-29, diode Db is reversed, a large-C (~10mF) capacitor has been connected from diode Da to Db, and the demodulated output information is taken from between the resistors as shown. Reversing the direction of Db allows for a current path, which keeps the voltage Vc across the large-capacity smoothing capacitor constant; that is, capacitor C absorbs AM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 68 Ratio Detector Figure 9-29. Ratio detector. 69 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Ratio Detector If the FM signal frequency increases, vs remains constant, but the vectors shift as in Figure 9-27. Then Da will conduct more heavily and Vao will increase, while Db conducts less and Vbo decreases -the same as for the Foster-Seeley circuit. Thus, the sum of Vao and Vbo stays constant, but the ratio varies with input frequency. 70 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Quadrature FM Detectors Quadrature FM detectors use a high-reactance capacitor (C1 in Figure 9-30) to produce two signals with a 90o phase difference. The phase-shifted signal is then applied to an LC-tuned circuit resonant at the carrier frequency. Frequency changes will then produce an additional leading or lagging phase shift that is detected by comparing zero-crossings (coincidence detector) or by analog phase detection. The quadrature FM detectors of Figures 9-30 and 9-32 are the coincidence and analog phase-shift detectors, respectively. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 71 Quadrature FM Detectors Figure 9-30. FM quadrature (coincidence) detector. 72 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Quadrature FM Detectors The coincidence detector of Figure 9-30 provides two equal-frequency, phase-quadrature signals to a digital AND gate. As seen in Figure 9-31 the pulse width of the AND gate output varies as a function of the resonant-circuit phase-shifts due to frequency changes at pin B. The RC output circuit provides the pulse integration for deriving the average output voltage. The AND gate can also be thought of as a digital multiplier producing the product AB. 73 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Quadrature FM Detectors Fig. 9-31. Coincidence detector. Input B is shown with two different phase shifts relative to input A. The AND gate outputs are also shown. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 74 Quadrature FM Detectors Figure 9-32. Analog phase-shift type of Quadrature FM detector. 75 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Quadrature FM Detectors Figure 9-32 shows the same circuit as Figure 9-30 except for the analog product detector. The multiplication of two periodic signals with the same frequency produces a dc voltage that is directly proportional to the signal phase difference. For small phase shifts (narrowband FM), the output will be reasonably linear and expressed approximately by Vo = (V1Vin/2).sinQr a Qr = KvcQr (9-96) where Q is the effective resonant-circuit factor, r = f/fo – fo/f is the fractional deviation frequency from resonance, and the phase-to-voltage constant for the circuit is Kvc. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 76 IF Amplifiers and Limiters Parasitics are common in medium-frequency RF work. The most commonly applied preventative is a low-Q ferrite bead (Z-bead), usually placed on the base lead. Another practical design consideration is a resistor placed in series with the collector circuit. Their main function is to maintain at least 100 ohms of impedance across the tank when the collector is in saturation. Too large a value of resistance will unnecessarily reduce the output. Too low a value will allow the circuit Q to drop excessively. 77 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan IF Amplifiers and Limiters Figure 9-34 illustrates a typical bipolar BPL stage. Note that R2 may be eliminated entirely. This would result in class C operation. Class C operation is appropriate in this application because the input signal is rectified and the remaining input circuit will provide a self-adjusting bias for constant amplifier gain. Care must be taken with the RC time constant and the transformer/transistor selection to ensure that the reverse bias on the base does not exceed the baseemitter breakdown voltage. 78 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan IF Amplifiers and Limiters Figure 9-34. Bandpass limiter/amplifier. 79 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan IF Amplifiers and Limiters Figure 6-35. Amplitude limiter. 80 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan IF Amplifiers and Limiters Figure 6-36. Amplitude limiter transfer characteristic. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 81 IF Amplifiers and Limiters Figure 6-37. Typical limiter response characteristic. 82 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan Preemphasis for FM The main difference between FM and PM is in the relationship between frequency and phase. f = (1/2p).dq/dt. A PM detector has a flat noise power (and voltage) output versus frequency (power spectral density). This is illustrated in Figure 9-38a. However, an FM detector has a parabolic noise power spectrum, as shown in Figure 9-38b. The output noise voltage increases linearly with frequency. If no compensation is used for FM, the higher audio signals would suffer a greater S/N degradation than the lower frequencies. For this reason compensation, called emphasis, is used for broadcast FM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 83 Preemphasis for FM Figure 9-38. Detector noise output spectra for (a). PM and (b). FM. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 84 Preemphasis for FM A preemphasis network at the modulator input provides a constant increase of modulation index mf for high-frequency audio signals. Such a network and its frequency response are illustrated in Figure 9-39. Fig. 9-39. (a)Premphasis network, and (b) Frequency response. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 85 Preemphasis for FM With the RC network chosen to give t = R1C = 75ms in North America (150ms in Europe), a constant input audio signal will result in a nearly constant rise in the VCO input voltage for frequencies above 2.12 kHz. The larger-than-normal carrier deviations and mf will preemphasize high-audio frequencies. At the receiver demodulator output, a low-pass RC network with t = RC = 75ms will not only decrease noise at higher audio frequencies but also deemphasize the high-frequency information signals and return them to normal amplitudes relative to the low frequencies. The overall result will be nearly constant S/N across the 15kHz audio baseband and a noise performance improvement of about 12dB over no preemphasis. Phase modulation systems do not require emphasis. 86 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM In Figure 9-40, audio signals from both left and right mircrophones are combined in an linear matrixing network to produce an L+R signal and an L-R signal. Both L+R and L-R are signals in the audio band and must be separated before modulating the carrier for transmission. This is accomplished by translating the L-R audio signal up in the spectrum. As seen in Figure 9-40, the frequency translation is achieved by amplitude-modulating a 38-kHz subsidiary carrier in a balanced modulator to produce DSB-SC. 87 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM Figure 9-40. FM stereo generation block diagram. 88 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM The stereo receiver will need a frequency-coherent 38kHz reference signal to demodulate the DSB-SC. To simplify the receiver, a frequency- and phasecoherent signal is derived from the subcarrier oscillator by frequency division (÷2) to produce a pilot. The 19-kHz pilot fits nicely between the L+R and DSBSC L-R signals in the baseband frequency spectrum. 89 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM As indicated by its relative amplitude in the baseband composite signal, the pilot is made small enough so that its FM deviation of the carrier is only about 10% of the total 75-kHz maximum deviation. After the FM stereo signal is received and demodulated to baseband, the 19-kHz pilot is used to phase-lock an oscillator, which provides the 38-kHz subcarrier for demodulation of the L-R signal. A simple example using equal frequency but unequal amplitude audio toned in the L and R microphones is used to illustrate the formation of the composite stereo (without pilot) in Figure 9-41. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 90 STEREO FM Figure 9-41. Development of composite stereo signal. The 38 kHz alternately multiplies L-R signal by +1 and –1 to produce the DSB-SC in the balanced AM modulator (part d). The adder output (shown in e without piot) will be filtered 91 to reduce higher harmonics before FM modulation. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM Fig. 9-45. Stereo FM multiplex generator with optional SCA. 92 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM Fig. 9-46. Stereo FM multiplex demodulation with optional SCA output. 93 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM Figure 9-47. Spectrum of stereo FM signal. 94 Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan STEREO FM Figure 9-48. Stereo FM decoder using the CA3090 IC with PLL pilot synchronizing. Prof. J.F. Huang, Fiber-Optic Communication Lab. National Cheng Kung University, Taiwan 95