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Chapter 4 : Angle Modulation Transmission and Reception. 4.1 Introduction to Angle Modulation 4.2 Mathematical Analysis 4.3 FM and PM Waveform 4.4 Modulation Index and Percent Modulation 4.5 Frequency and Bandwidth Analysis of Angle-Modulated Waves 4.6 Deviation Ratio 4.7 FM / PM Modulators

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## Chapter 4 : Angle Modulation Transmission and Reception

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**Chapter 4 : Angle Modulation Transmission and Reception**• 4.1 Introduction to Angle Modulation • 4.2 Mathematical Analysis • 4.3 FM and PM Waveform • 4.4 Modulation Index and Percent Modulation • 4.5 Frequency and Bandwidth Analysis of Angle-Modulated Waves • 4.6 Deviation Ratio • 4.7 FM / PM Modulators • 4.8 Frequency-up Conversion in modulators • 4.9 FM Transmitters • 4.10 FM Receivers • 4.11 FM Demodulators • 4.12 FM Stereo • 4.13 Average Power of an Angle-modulated wave • 4.14 Angle Modulation vs Amplitude Modulation • 4.15 Noise and Angle Modulation • 4.16 Pre-emphasis & De-emphasis BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1 : Introduction to Angle Modulation**• 2 forms of angle modulation : • Frequency modulation (FM) • Phase modulation (PM) • Several advantages over AM – noise reduction, improved system fidelity and more efficient use of power • Several disadvantages over AM – wider bandwidth requirement, utilization of more complex circuits. • used extensively for commercial radio broadcasting, television sound transmission, cellular radio, microwave and satellite communications systems BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1 : Introduction to Angle Modulation**• angle modulation results whenever the phase angle, θof a sinusoidal wave is varied with respect to time and can be expressed as (1) where m(t) = angle-modulated wave Vc = peak carrier amplitude ωc = carrier radian frequency θ(t) = instantaneous phase deviation where θ(t) is a function of the modulating signal given by (2) where ωc = modulating signal radian frequency Vm = peak amplitude of the modulating signal BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1 : Introduction to Angle Modulation**• the difference between FM and PM lies in which property of the carrier is directly varied by the modulating signal and which property is indirectly varied. • FM results when the frequency of the carrier is varied directly by the modulating signal • PM results when the phase of the carrier is varied directly by the modulating signal • Frequency Modulation (FM) • variation of the frequency of the modulating signal with constant amplitude • frequency variation is directly proportional to the amplitude of the modulating signal • rate of variation equal to the frequency of the modulating signal BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1 : Introduction to Angle Modulation**• Phase Modulation (PM) • variation of the phase of the modulating signal with constant amplitude • phase variation is directly proportional to the amplitude of the modulating signal • rate of variation equal to the frequency of the modulating signal BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1.1 : Angle Modulation Representation in Frequency and**Time Domain • An angle modulated signal in the frequency domain : • the carrier frequency, fc is changed when acted on by the modulating signal. • the magnitude and direction of the frequency deviation, Δf is proportional to the amplitude and polarity of the modulating signal. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.1.1 : Angle Modulation Representation in Frequency and**Time Domain • An angle modulated signal in the time domain : • the phase of the carrier is changing proportional to the amplitude of the modulating signal. • the phase shift is called phase deviation Δθ. This shift is also produces a corresponding change in the frequency, known as frequency deviation Δf. • peak-to-peak frequency deviation is determine by (as shown in figure (b)), (3) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.2 : Mathematical Analysis**• to differentiate between FM and PM, the following terms need to be defined : • 1. Instantaneous Phase Deviation • the instantaneous change in the phase of the carrier at a given instant of time. Instantaneous phase deviation = θ(t) rad (4) • 2. Instantaneous phase • the precise phase of the carrier at a given instant of time. Instantaneous phase = ωct + θ(t) rad (5) • 3.Instantaneous frequency deviation • the instantaneous change in the frequency of the carrier and is defined as the first time derivative of the instantaneous phase deviation. Instantaneous frequency deviation = θ’(t) rad/s (6) • 4. Instantaneous frequency • the precise frequency of the carrier at a given instant of time and is defined as the first time derivative of the instantaneous phase. Instantaneous frequency = ωi = ωc + θ’(t) rad/s (7) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.2 : Mathematical Analysis**• from the previous 4 terms, (3) ~ (7), PM and FM can be defined as : • PM : an angle modulation in which θ(t) is proportional to the amplitude of the modulating signal. • FM : an angle modulation in which θ’(t) is proportional to the amplitude of the modulating signal. • For a modulating signal vm(t), θ(t) = Kvm(t) rad (8) θ’(t) = K1vm(t) rad/s(9) where K and K1 are constants and are the deviation sensitivities of the phase and frequency modulators, respectively. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.2 : Mathematical Analysis**• substituting a modulating signal vm(t) = Vmcos(ωmt), equation (8) and (9) into equation (1) yields PM : (10) FM : as (11) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.2 : Mathematical Analysis**• Summarized table : BENG 2413 Communication Principles Faculty of Electrical Engineering**4.3 : FM and PM Waveforms**• Waveforms of FM and PM of a sinusoidal carrier by a single-frequency modulating signal. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.3 : FM and PM Waveforms**• FM and PM waveforms are identical except for their time relationship. • for FM, the maximum frequency deviation occurs during the maximum positive and negative peaks of the modulating signal. • for PM, the maximum frequency deviation occurs during the zero crossings of the modulating signal (i.e. the frequency deviation is proportional to the slope of first derivative of the modulating signal). BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4 : Modulation Index and Percent Modulation**• comparing equation (10) and (11), equation (1) can be rewritten in general form as (12) where m is called the modulation index. 4.4.1 : Modulation Index and Percent Modulation for PM • for PM, the modulation index is also known as peak phase deviationΔθ, and is proportional to the amplitude of the modulating signal and is expressed as (13) where m = modulation index K = deviation sensitivity (radians/volt) Vm = peak modulating signal amplitude (volt) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4.1 : Modulation Index and Percent Modulation for PM**• therefore, for PM : (14) 4.4.2 : Modulation Index and Percent Modulation for FM • for FM, the modulation index is directly proportional to the amplitude of the modulating signal and inversely proportional to the frequency of the modulating signal. (15) where K1 = deviation sensitivities (radians/second per volt or cycles/second per vol Vm = peak modulating signal amplitude (volt) ωm = radian frequency (radians/second) fm = cyclic frequency (cycles/second or hertz) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4.2 : Modulation Index and Percent Modulation for FM**• also for FM, the peak frequency deviation Δf is simply the product of the deviation sensitivity and the peak modulating signal voltage. I.e. (16) • therefore, for FM, equation (11) can be rewritten as (17) • figure 7.4 & table 7.2 BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4.2 : Modulation Index and Percent Modulation for FM**• Relationship between modulation index, frequency deviation and phase deviation in respect to the modulation signal amplitude and frequency : (a) modulation index vs amplitude (b) frequency deviation vs modulating frequency (c) phase deviation vs amplitude (d) frequency deviation vs amplitude BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4.2 : Modulation Index and Percent Modulation for FM**• Summarized : BENG 2413 Communication Principles Faculty of Electrical Engineering**4.4.3 : Percent Modulation**• percent modulation for angle modulation is determined in different manner than for amplitude modulation. • with angle modulation, percent modulation is the ratio of frequency deviation actually produced to the maximum frequency deviation allowed, stated in percent form Percent modulation (18) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5 : Frequency and Bandwidth Analysis of Angle-Modulated**Waves • frequency analysis of the angle-modulated wave is much more complex compared to the amplitude modulation analysis. • in phase/frequency modulator, a modulating signal produces an infinite number of side frequencies pairs (i.e. it has infinite bandwidth), where each side frequency is displaced from the carrier by an integral multiple of the modulating frequency. 4.5.1 : Bessel Function • from equation (12), the angle-modulated wave is expressed as • based on the above equation, the individual frequency components of the angle-modulated wave is not obvious. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.1 : Bessel Function**• Bessel function identities can be used to determine the side frequencies components (19) where Jn(m) is the Bessel function of the first kind. • applying equation (19) to equation (12) yields, (20) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.1 : Bessel Function**• expanding (20), where m(t) = angle modulated wave m = modulation index Vc = peak carrier ampitude J0(m) = carrier component J1(m) = first set of side frequencies displaced from carrier by ωm J2(m) = second set of side frequencies displaced from carrier by 2ωm Jn(m) = nth set of side frequencies displaced from carrier by n ωm BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.1 : Bessel Function**• in other words, angle modulation produces infinite number of sidebands, called as first-order sidebands, second-order sidebands, and so on. Also their magnitude are determined by the coefficients J1(m), J2(m),...Jn(m). • Bessel function of the first kind for several values of modulation index. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.1 : Bessel Function**• Curves for the relative amplitudes of the carrier and several sets of side frequencies for values of m up to 10. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.1 : Bessel Function**• Conclusion from the table & graph : • modulation index m of 0 produces zero side frequencies. • the larger the m, the more sets of side frequencies are produced. • values shown for Jn are relative to the amplitude of the unmodulated carrier. • as the m decreases below unity, the amplitude of the higher-order side frequencies rapidly becomes insignificant. • as the m increases from 0, the magnitude of the carrier J0(m) decreases. • the negative values for Jn simply indicate the relative phase of that side frequency set • a side frequency is not considered significant unless its amplitude is equal or greater that 1% of the unmodulated carrier amplitude (Jn ≥ 0.01). • as m increases, the number of significant side frequencies increases. I.e. the bandwidth of an angle-modulated wave is a function of the modulation index. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.2 : Bandwidth Requirement**• angle-modulated wave consumes larger bandwidth than an amplitude-modulated wave. • bandwidth of an angle-modulated wave is a function of the modulating signal and the modulation index. • the actual bandwidth required to pass all the significant sidebands for an angle-modulated wave is equal to 2 times the product of the highest modulating signal frequency and the number of significant sidebands determined from the table of Bessel function. • I.e. the minimum bandwidth for angle-modulated wave using the Bessel table, (21) • Carson’s Rule • it is a general rule to estimate the bandwidth for all angle-modulated systems regardless of the modulation index. • the Carson’s Rule states that the bandwidth necessary to transmit an angle-modulated wave as twice the sum of the peak frequency deviation and the highest modulating signal frequency. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.5.2 : Bandwidth Requirement**• Carson’s Rule (22) • for a low modulation index ( fm is much larger than Δf ), (23) • for a high modulation index (Δf is much larger than fm ) (24) • Carson’s Rule approximate and gives a narrower bandwidth than the bandwidth determined using Bessel function. Therefore, a system designed using Carson’s Rule would have a narrower bandwidth but a poorer performance than system designed using the Bessel table. • for modulation index above 5, Carson’s Rule is a close approximation to the actual bandwidth required. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.6 : Deviation Ratio**Deviation ratio DRis the worst case modulation index and is equal to the maximum peak frequency deviation divided by the maximum modulating-signal frequency – producing the widest frequency spectrum. where DR = deviation ratio (unitless) Δf(max) = maximum peak frequency deviation (Hertz) fm(max) = maximum modulating-signal frequency (Hertz) BENG 2413 Communication Principles Faculty of Electrical Engineering BENG 2413 Communication Principles Faculty of Electrical Engineering Chapter 4 : Angle Modulation 28**4.6 : Deviation Ratio**Ex : a. Determine the deviation ratio and bandwidth for the worst-case (widest bandwidth) modulation index for an FM broadcast-band transmitter with a maximum frequency deviation of 75 kHz and a maximum modulating-signal frequency of 15 kHz. From Bessel Table, a modulation index of 5 produces 8 significant sidebands Thus, the bandwidth is B = 2(8 x 15000) = 240 kHz Ex : b. For a 37.5 kHz frequency deviation and a modulating-signal frequency fm = 7.5 kHz, the modulation index is and the bandwidth is B = 2(8 x 7500) = 120 kHz BENG 2413 Communication Principles Faculty of Electrical Engineering BENG 2413 Communication Principles Faculty of Electrical Engineering Chapter 4 : Angle Modulation 29**4.6 : Deviation Ratio**From Ex. a & b, although the same modulation index (5) was achieved with 2 different modulating-signal frequencies and amplitudes, 2 different bandwidths were produced. The widest bandwidth will only be produced from the maximum modulating-signal frequency and maximum frequency deviation. The same condition applies in the case of using the Carson’s rule : BENG 2413 Communication Principles Faculty of Electrical Engineering BENG 2413 Communication Principles Faculty of Electrical Engineering Chapter 4 : Angle Modulation 30**4.7 : FM/PM Modulators**• a phase modulator is a circuit in which the carrier instantaneous phase is proportional to the modulating signal. • a frequency modulator is a circuit in which the carrier instantaneous phase is proportional to the integral of the modulating signal. PM modulator : FM modulator : • considering the FM modulator, if the modulating signal is v(t) is differentiate before being applied to the FM modulator, the instantaneous phase is now proportional to the modulating signal (i.e. PM modulator). Differentiator + FM modulator = = PM modulator BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7 : FM/PM Modulators**• Meanwhile, if the modulating signal is integrated before being applied to the PM modulator, the instantaneous phase is now proportional to the integral of the modulating signal (i.e. FM modulator). Integrator + PM modulator = = FM modulator 4.7.1 : Direct FM Modulators • with direct FM, the instantaneous frequency deviation is directly proportional to the amplitude of the modulating signal. • schematic diagram of a simple direct FM generator : BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.1 : Direct FM Modulators**• the tank circuit (L and Cm) is the frequency determining section for a standard LC oscillator. • Cm is a capacitor microphone that converts the acoustical energy into a mechanical energy, which is used to vary the distance between the plates of Cm and consequently change its capacitance. • as Cm is varied, the resonant frequency is varied. I.e. the oscillator output frequency varies directly with the external sound forces (i.e. direct FM). BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.1.1 : Varactor diode modulator**• Direct FM generator using varactor diode to deviate the frequency of a crystal oscillator : • R1 and R2 develop a DC voltage that reverse bias the varactor diode VD1 and determine the resonant frequency of the oscillator. • external modulating signal voltage added or subtracted from the DC bias, which changes the capacitance of the diode and consequently changes the frequency of the oscillation. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.1.1 : Varactor diode modulator**• positive alternations of the modulating signal increase the reverse bias of VD1, which decrease its capacitance and increase the frequency of the oscillation. • negative alternations of the modulating signal decrease the reverse bias of VD1, which increase its capacitance and decrease the frequency of the oscillation. • simple to use, stable and reliable but limited peak frequency deviation thus limited use to the low index applications. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.1.2 : VCO FM Modulator**• the use of varactor diode to transform changes in modulating signal amplitude to changes in frequency : • the center frequency for the oscillator : (25) where fc = carrier frequency L = inductance of the primary winding of T1 C = varactor diode capacitance BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.1.2 : VCO FM Modulator**• when a modulating signal is applied, the frequency is (26) where f = new frequency ΔC = change in varactor diode capacitance due to modulating signal • the change in frequency is (27) where Δf = peak frequency deviation (hertz) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.2 : Indirect FM (Direct PM) Modulator**• with indirect FM, the instantaneous phase deviation rather than instantaneous frequency deviation is directly proportional to the modulating signal. • I.e. the indirect FM is accomplished by directly changing the phase of the carrier. • schematic diagram of an indirect FM modulator using a varactor diode : BENG 2413 Communication Principles Faculty of Electrical Engineering**4.7.2 : Indirect FM (Direct PM) Modulator**• varactor diode VD1 placed in series with the inductive network (L1 and R1). • this combined series-parallel network appears as series resonant circuit to the output frequency from the crystal oscillator. • the modulating signal is applied to VD1, which changes its capacitance and subsequently the phase angle of the impedance seen by the carrier also varies, which results in a corresponding phase shift in the carrier. • advantage of using indirect FM modulator is it is more stable than the direct modulator. • However, it has more distortion in the modulated waveform compared to direct FM. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.8 : Frequency Up-conversion**• after the modulation, the frequency of the modulated-wave is up-converted to the desired frequency of transmission. • 2 basic methods of frequency up-conversion : • heterodyning process • frequency multiplication 4.8.1 : Heterodyne Method BENG 2413 Communication Principles Faculty of Electrical Engineering**4.8.1 : Heterodyne Method**• 2 inputs to the balanced modulator : angle-modulated carrier and its side frequencies, an also the unmodulated RF carrier signal. • the 2 inputs mix nonlinearly in the balanced modulator producing the sum and difference frequencies at its output. • the BPF (bandpass filter) is tuned to the sum frequency with a passband wide enough to pass carrier plus the upper and lower side frequencies while the difference frequencies are blocked. • the frequency deviation, rate of change, modulation index, phase deviation and bandwidth are unaffected by the heterodyne process. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.8.2 : Multiplication method**• with multiplication method, the frequency of the modulated carrier is multiplied by a factor of N in the frequency multiplier. • frequency deviation, phase deviation and modulation index are also multiplied. • However, the rate of the deviation is unaffected (i.e. the separation between adjacent side frequencies remains unchanged). • as frequency deviation and modulation index are multiplied, the number of side frequency also increases. Thus, the bandwidth also increases. • For modulation index higher than 10, Carson’s Rule can be applied BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9 : FM Transmitter4.9.1 : Direct FM Transmitter**• Block diagram for a commercial broadcast-band transmitter : • also known as Crosby direct FM transmitter (includes an automatic frequency control –AFC loop) • the carrier frequency is basically the center frequency of the master oscillator fc = 1.5 MHz, which is multiplied by 18 to produce a final transmission carrier frequency ft = 98.1 MHz. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.1 : Direct FM Transmitter**• the frequency and phase deviations at the output of the modulator are also multiplied by 18. To achieve maximum deviation allowed for FM stations at antenna (75 kHz), the deviation at the output of the modulator is BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.1 : Direct FM Transmitter**The modulation index at the output of the modulator, For maximum modulating signal frequency allowed for FM (15 kHz) BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.1.1 : AFC Loop**• for medium and high index FM systems, the oscillator cannot be a crystal type because the frequency at which the crystal oscillates cannot be significantly deviated. • as a result, the stability of the oscillator in the direct FM is low. • to overcome this problem, AFC loop is used. • with AFC, the carrier signal is mixed in a nonlinear device with the signal from a crystal reference oscillator (the output is down-converted in frequency). • the output is then fed back to the input of a frequency discriminator. It is a frequency-selective device whose output voltage is proportional to the difference between the input frequency and its resonant frequency. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.1.1 : AFC Loop**• if there is a master oscillator frequency drift (resulting in a change of carrier center frequency), the discriminator responds by producing a DC correction voltage. • this voltage is added to the modulating signal to automatically adjust the master oscillator’s center frequency. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.2 : Indirect FM Transmitter**• Indirect FM transmitters produce an output waveform in which the phase deviation is directly proportional to the modulating signal. • Consequently, the carrier oscillator is not directly deviated. As a result, the stability of the oscillators can be achieved without using an AFC circuit. • Block diagram for wideband Armstrong indirect FM transmitter : • low frequency sub-carrier fc is phase shifted 90˚ and fed to a balanced modulator. It is mixed with the modulating signal fm. • the output from the balanced modulator is DSBSC wave that is combined with the original carrier in a combining network to produce a low-index, phase-modulated waveform. BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.2 : Indirect FM Transmitter**• Proof : By using trigonometric function : cos (A+B) =cos A cos B – sin A sin B For a small modulation index, Thus, where Vccos(ωct) = original carrier Vcsin(ωct ) = phase-shifted carrier cos(ωmt ) = modulating signal BENG 2413 Communication Principles Faculty of Electrical Engineering**4.9.2 : Indirect FM Transmitter**• Ex : Consider a 200 kHz carrier being phase-modulated with a 15 kHz modulating signal producing modulation index of 0.00096. • the frequency deviation at the output of the combining network : Δf = mfm = 0.00096 x 15000 = 14.4 Hz • in order to achieve the required 75 kHz deviation for the FM broadcast at the antenna, the frequency must be multiplied by approximately 5208. However, this would produce a transmission carrier at the antenna of ft = 5208 x 200 kHz = 1041.6 MHz This value is beyond the limits for the commercial FM broadcast band (30 ~ 300MHz). BENG 2413 Communication Principles Faculty of Electrical Engineering

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