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**Frequency Modulation ANGLE MODULATION:**

The intelligence of the modulating signal can be conveyed by varying the frequency or phase of the carrier signal. When this is the case, we have angle modulation, which can be subdivided into two categories: frequency modulation (FM), and phase modulation (PM). An inherent problem with AM is its susceptibility to noise superimposed on the modulated carrier signal. If this noise falls within the passband of the receiving system and its amplitude is large enough, it will interfere with the detected intelligence. To improve on this shortcoming, Major Edwin E. Armstrong has been credited with developing in 1936 the first frequency modulation (FM) radio communication system, a system that is much more immune to noise than its AM counterpart. Since its inception, FM has remained one of the most prevalent modulation techniques in the telecommunications industry, being used in applications such as cellular and cordless telephony, paging systems, modem technology, television, commercial FM broadcast, amateur radio, and more. It is the best choice for fidelity and offers a much higher SNR than its AM counterpart. In this chapter, the principles of FM are examined. We do not present an entire analysis of an FM transmitter and receiver; it would take several chapters to do justice to that lengthy subject. A brief comparison of phase modulation (PM) is presented, however, so the student understands that both FM and PM are regarded as angle modulation. Unlike amplitude modulation, FM is difficult to treat mathematically due to the complexity of the sideband behavior resulting from the modulation process. For this reason, mathematical presentations are limited to the conventional treatment, using both charts and table derivations.

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Frequency Modulation. The carrier's instantaneous frequency deviation from its unmodulated value varies in proportion to the instantaneous amplitude of the modulating signal. Phase Modulation. The carrier's instantaneous phase deviation from its unmodulated value varies as a function of the instantaneous amplitude of the modulating signal; eFM = instantaneous voltage of the FM wave epm = instantaneous voltage of the PM wave Ac= peak amplitude of the carrier ωc = angular velocity of the carrier ωm = angular velocity of the modulating signal wct = carrier phase in radians wmt = modulation phase in radians mf = FM modulation index φm = maximum phase deviation in radians caused by the modulating signal (also regarded as the PM modulation index)

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FIGURE The FM and PM waveforms for sine-wave modulation: (a) carrier wave; (b) modulation wave; (c) FM wave; (d) PM wave. (Note: The derivative of the modulating sine wave is the cosine wave shown by the dotted lines. The PM wave appears to be frequency modulated by the cosine wave.) Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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**MODULATION INDEX modulation index for an FM signal**

the waveform alone cannot be used to distinguish between FM and PM. It is their modulation indices, mf and φf. that differ. The modulation index for an FM signal is defined as the ratio of the maximum frequency deviation to the modulating signal's frequency. Note that the modulation index, mf, for FM is proportional to the amplitude of the modulating signal through δ and inversely proportional to the frequency of the modulating signal. Herein lies the subtle difference between FM and PM. Although the modulation index, φm , for a PM signal is proportional to the amplitude of the modulating signal, in contrast to FM it is also dependent on the modulation frequency whereas FM is not. δ = maximum frequency deviation of the carrier caused by the amplitude of the modulating signal fm = frequency of the modulating signal

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**FREQUENCY ANALYSIS OF THE FM WAVE**

Recall that in AM, the frequency components consist of a fixed carrier frequency with upper and lower sidebands equally displaced above and below the carrier frequency. The frequency components of the upper and lower sidebands are mirror images of each other and identical to that of the modulating signal, except that they translate up to the carrier frequency. The frequency spectrum of the FM wave is much more complex, however. In equation (4-1), a single sinusoid used to modulate the FM carrier produces an infinite number of sidebands. Furthermore, the complexity of the sideband activity increases with the frequency complexity in the modulating signal. Analysis of the frequency components and their respective amplitudes in the FM wave requires use of a complex mathematical integral known as the Bessel function of the first kind of the nth order. Evaluating this integral for sine-wave modulation yields AcJO(mf) sin wct = the carrier frequency component Ac{J1 (mf) [sin(wc + ωm)t - sin(w, - (om)tl} = the first-order sideband A,{J2 (mf) [sin(w, + 2oQt - sin(co, - 2wjt]} the second-order sideband A,(J3 (mf) [sin(co, + 3oQt - sin(o), - 3co,)t]} the third-order sideband A,{J,, (mf) [sin((o~ + n(om)t - sin((.t), - n(t)m)t]} the nth-order sideband Where:eFm = the instantaneous amplitude of the modulated FM wave Ac = the peak amplitude of the carrier Jn = solution to the nth order Bessel function for a modulation index mf. mf = FM modulation index, Δf/fm

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FIGURE Spectral components of a carrier of frequency, fc, frequency modulated by a sine wave with frequency fm. (Source: James Martin, Telecommunications and the Computer, 2nd ed. [Englewood Cliffs, N.J.: Prentice-Hall, 1976], p Reprinted with permission from the publisher.) It is apparent from equation (4-4) that the FM wave contains an infinite number of sideband components whose individual amplitudes are preceded by Jn(mf) coefficients. Each set of upper and lower sidebands is displaced from the carrier frequency by an integral multiple of the modulation frequency. These are the Bessel functions; tabulated Bessel functions to the sixteenth order for modulation indices ranging from 0 to 15 are listed in Table The successive sets of sidebands are referred to as firstorder sidebands, second-order sidebands, and so on. A plot of the Bessel functions, as shown in Figure 4-3, illustrates the relationship between the carrier and sideband amplitudes for sine-wave modulation as a function of modulation index, m. From the curves or the table, we can obtain the amplitudes of the carrier and sideband components in relation to the unmodulated carrier. Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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Find the carrier and sideband amplitudes to the fourth-order sideband for a modulation index of mf = 3. The peak amplitude of the carrier, Ac, from equation (4-4), is 10 V. From Table 4-1 or Figure 4-3, we have JO(m3) = -0.26 J1(m3) = 0.34 J2(m3) = 0.49 J3(m3) = 0.31 J4(m3) = 0.13 and, therefore, Jo = X 10 V = -2.6 V J1 =.0.34 X 10 V = 3.4 V J2 = 0.49 X 10 V = 4.9 V J3 = 0.31 X 10 V = 3.1 V J4 = 0.13 X IOV = 1.3 V Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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FM signal characters The FM wave is comprised of an infinite number of sideband components As the modulation index increases from mf = 0, the spectral energy shifts from the carrier frequency to an increasing number of significant sidebands. Jn(mf) coefficients, decrease in value with increasing order, n. negative Jn(mf) coefficients imply a 180' phase inversion. The carrier component, Jo, and various sidebands, Jn, go to zero amplitude at specific values of modulation index, mf. From Table 4-1 and Figure 4-3, the FM signal is characterized as follows: • The FM wave is comprised of an infinite number of sideband components whose individual amplitudes are preceded by Jn(mf) coefficients. • Each set of upper and lower sidebands are displaced from the carrier frequency by an integral multiple of the modulation frequency. • As the modulation index increases from mf = 0, the spectral energy shifts from the carrier frequency to an increasing number of significant sidebands. This suggests that a wider bandwidth is necessary to recover the FM signal. • The magnitudes of the sideband amplitudes, Jn(mf) coefficients, decrease in value with increasing order, n. • For higher-order sidebands, the magnitudes of the sideband amplitudes, Jn(mf) coefficients, increase in value with increasing modulation index, mf. • Sideband amplitudes with negative Jn(mf) coefficients imply a 180' phase inversion. Because a spectrum analyzer displays only absolute amplitudes, the negative signs have no significance. • The carrier component, Jo, and various sidebands, J,, go to zero amplitude at specific values of modulation index, mf.

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**Carrier Frequency Eigenvalues**

in some cases the carrier frequency component, JO, and the various sidebands, Jn go to zero amplitudes at specific values of m. These values are called eigenvalues. Table 4-2 lists the values of the modulation index for which the carrier amplitude goes to zero, and this implies that one can easily estimate the modulation index for an FM signal with sine-wave modulation by use of a spectrum analyzer. The displayed number of sidebands and their respective amplitudes are simply noted and used in conjunction with Figure 4-3, Table 4-1, and Table 4-2 to determine the modulation index.

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**Bandwidth Requirements for FM**

The higher the modulation index, the greater the required system bandwidth where n is the highest number of significant sideband components and fm is the highest modulation frequency. In theory, the FM wave contains an infinite number of sidebands, thus suggesting an infinite bandwidth requirement for transmission or reception. In practice, however, the sideband amplitudes become negligible beyond a certain frequency range from the carrier. This range is a function of modulation index, mf, that is, the ratio of carrier frequency deviation to modulating frequency (equation [4-3]). The higher the modulation index, the greater the required system bandwidth. This was shown earlier in the listing of Bessel functions (Table 4-1). Figure 4-5 is a more graphical illustration of how the FM system's bandwidth requirements grow with an increasing modulation index. Here, the modulation frequency, fm, is held constant, whereas the carrier frequency deviation, δ, is increased (and, consequently, mf as well) in proportion to the amplitude of the modulation signal. Based on the Bessel functions listed in Table 4-1, Table 4-3 lists the number of significant sideband components corresponding to various modulation indices. By "significant , " we usually mean all of those sidebands having a voltage of at least 1%, or -40 dB; (20 log 1/100 ), of the voltage of the unmodulated carrier. The bandwidth requirements for an FM signal can be computed by BW=2(n*fm) where n is the highest number of significant sideband components and fm is the highest modulation frequency. Carson's Rule From our previous discussion, it is evident that the bandwidth of an FM signal must be wider than that of an AM signal. In establishing the quality of transmission and reception desired, a limitation must be placed on the number of significant sidebands that the FM system must pass. In 1938, J. R. Carson first stated in an unpublished memorandum that the minimum bandwidth required for the transmission of an angle modulated wave is equal to two times the sum of the peak frequency deviation, δ, plus the highest modulating frequency, fm, to be transmitted. This rule is known as Carson's Rule: Carson's Rule gives results that agree with the bandwidths used in the telecommunications industry. It should be noted, however, that this is only an approximation used to limit the number of significant sidebands for minimal distortion. Carson's Rule

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FIGURE Amplitude versus frequency spectrum for various modulation indices (fm fixed, & varying): (a) mf = 0.25; (b) mf = 1; (c) mf = 2; (d) mf = 5; (e) mf = 10. Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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**Warren Hioki Telecommunications, Fourth Edition**

Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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**Broadcast FM extends from 88 to 108 MHz is divided into 100 channels**

Channels range from 88.1 MHz, where N = 201, to MHz, where N = 300 N=5(f-47.9) Broadcast FIVI The commercial FM broadcast band, as shown in Figure 4-6, extends from 88 to 108 MHz and is divided into 100 channels. The FCC has allocated a bandwidth of 200 kHz and designated a numerical value, N, for each channel. Channels range from 88.1 MHz, where N = 201, to MHz, where N = 300. Thus, FM broadcast stations can only be tuned at odd intervals of 200 kHz (e.g., MHz, MHz, and so on). The channel number, N, or its corresponding broadcast frequency can be computed as where N = the FM broadcast channel number f = the frequency in MHz Guard Band: A range of frequences separating transmitted channels in which no signals should be transmitted.

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**FIGURE 4-6 Commercial FM broadcast band.**

The maximum permissible carrier deviation,δ, is ±75 kHz. The transmitter is permitted to modulate its carrier frequency with a band of frequencies ranging from 50 Hz to 15 kHz. Thus, the modulation index can range from as low as 5 for fm = 15 kHz (75 kHz/15 kHz) to as high as 1500 for fm = 50 Hz (75 kHz/50 Hz). The ±75-kHz carrier deviation results in an FM bandwidth requirement of 150 kHz for the receiver. A 25-kHz guard band above and below the upper and lower FM sidebands makes up the remaining 50 kHz of the 200-kHz channel and prevents the sidebands from interfering with adjacent channels. Narrowband FM In contrast to the relatively wide bandwidth of broadcast FM, narrowband FM refers to FM systems in which the FCC has allocated band widths ranging from 10 to 30 kHz. The demand for use of the spectrum has led to the popularity of narrowband FM. Modulation indices are generally kept near unity so that the FM bandwidth can be computed in the same manner as the AM bandwidth. In other words, BW is simply 2 X fm, Examples of narrowband FM include mobile radio systems for police, fire, and taxi services; cellular telephony; amateur radio; and so on. Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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**Commercial FM broadcast band**

The maximum permissible carrier deviation, δ, is ±75 kHz Modulating frequencies is ranging from 50 Hz to 15 kHz The modulation index can range from as low as 5 for fm = 15 kHz (75 kHz/15 kHz) to as high as 1500 for fm = 50 Hz (75 kHz/50 Hz). The ±75-kHz carrier deviation results in an FM bandwidth requirement of 150 kHz for the receiver. A 25-kHz guard band above and below the upper and lower FM sidebands. Total bandwidth of one channel is 200Hz. The maximum permissible carrier deviation,δ, is ±75 kHz. The transmitter is permitted to modulate its carrier frequency with a band of frequencies ranging from 50 Hz to 15 kHz. Thus, the modulation index can range from as low as 5 for fm = 15 kHz (75 kHz/15 kHz) to as high as 1500 for fm = 50 Hz (75 kHz/50 Hz). The ±75-kHz carrier deviation results in an FM bandwidth requirement of 150 kHz for the receiver. A 25-kHz guard band above and below the upper and lower FM sidebands makes up the remaining 50 kHz of the 200-kHz channel and prevents the sidebands from interfering with adjacent channels.

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Narrowband FM NBFM uses low modulation index values, with a much smaller range of modulation index across all values of the modulating signal. An NBFM system restricts the modulating signal to the minimum acceptable value, which is 300 Hz to 3 KHz for intelligible voice. 10 to 15 kHz of spectrum. Used in police, fire, and Taxi radios, GSM, amateur radio, etc. Broadcast FM allows high fidelity and relatively low distortion but requires considerable spectrum handwidth for each station. Bandwidth is always a resource that must be conserved, and many applications that can benefit from the noise resistance of FM do not seed the signal fidelity. For example, police, fire, and Taxi radios need minimum static to make sure that the mssage gets through-but the voice must be only understandable, not necessarily recognizable. This can be achieved with a form of FM called narrowband FM (NBFM), developed for these applications. NBFM uses low modulation index values, with a much smaller range of modulation index across all values of the modulating signal. An NBFM system restricts the modulating signal to the minimum acceptable value, which is 300 Hz to 3 KHz for intelligible voice, (but the voice may not be recognizable, which is acceptable in the application). The user is allocated anywhere from 10 to 15 kHz of spectrum (sometimes less), depending on the frequency band. What is the modulation index for NBFM when the modulating signal varies from 300 to 3000 Hz and the user has 12 kHz of spectrum (±6 kHz deviation)? Solutlon At 300 Hz, mf = 6,000/300 = 20; at 3000 Hz, mf = 6,000/ 3000 = 2. The deviation ratio-the ratio of the maximum carrier frequency deviation to the highest modulating frequency used-is 6 kHz/3 kHz = 2 for this NBFM case. For standard commercial broadcast FM, the deviation ratio is 75 kHz/15 kHz = 5.

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**POWER IN THE FM WAVE power of the unmodulated carrier**

For a modulated carrier The total power in an FM wave is distributed in the carrier and the sideband components. If we sum the power in the carrier and all the sidebands for any given modulation index, it will equal the total power of the unmodulated carrier. Thus, it can be shown that for an unmodulated carrier (mf = 0), Where:PT = the total rms power of the unmodulated wave Vcrms, = the rms voltage of the carrier signal R = resistance of the load For a modulated carrier, where PT = the total rms power of the FM wave PJ0 = rms power in the carrier Pj1 = rms power in the first set of sidebands PJ2 = rms power in the second set of sidebands PJ3 = rms power in the third set of sidebands Pjn = rms power in the nth set of sidebands VJ0, through VJn. = the rms voltage of the carrier through the nth sideband, respectively.

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FM NOISE increased bandwidth of an FM -- enhance the signal-to-noise ratio (SNR). Advantages of FM over AM. To take this advvantage, large mf is necessary– high order sidebands are important – wider bandwidth is required. Phasor Analysis of FM Noise Noise affects the performance of any communications system, and it must be treated meaningfully. In FoCT, we learned that there are many sources of noise, and that its frequency, amplitude, and occurrence in time can be random or predictable. If the noise falls within the passband of the receiver, it can mix and add with the incoming signal, causing the original intelligence to become distorted. The increased bandwidth of an FM system over an AM system may be used to enhance the signal-to-noise ratio (SNR) performance of the receiver system. This is one of the primary advantages of FM over AM. Recovering the modulation signal has an inherent noise suppression capability that AM does not; however, to take advantage of this, it is necessary to use large indices of modulation, in which case higher-order sidebands become increasingly important. Thus, a wider bandwidth is required for the transmission and reception of an FM signal. In FM, noise added to the carrier signal causes a shift in frequency and phase from its normal state. The potential effect of the noise can be explained through use of phasor diagrams. where α = the maximum phase deviation of the carrier frequency caused by the noise VN = noise voltage Vc= carrier voltage

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FIGURE Phasor addition of noise on an FM signal’s carrier frequency causes a phase shift, whose maximum value is . Recall that a phasor is essentially a vector that portrays both magnitude and direction for vector quantities. In Figure 4-7, V, is a phasor representing the magnitude and direction of an FM signal's carrier rotating at a frequency ωc, For simplicity, we will consider a single noise component represented by the phasor VN, whose rotating frequency is ωN . Ideally, only the modulating voltage phasor would be superimposed on the carrier's phasor. From Figure 4-7, it can be seen that the phasor sum of Vc and VN at any instant in time is represented by a resultant phasor R. It is this resultant phasor that the FM receiver processes. The magnitude of R is its length from tail to head, which can lie anywhere on the circumference of the circle outlined by rotating phasor VN. The resultant phasor's amplitude, R, and phase deviation, α, continuously change with respect to time. The new signal is, therefore, amplitude, phase, and frequency modulated by the noise. The FM receiver is insensitive to the amplitude variations caused by noise, whereas the AM receiver is not. A circuit in the FM receiver, called a limiter, removes any variations in signal amplitude before detection or demodulation of the signal. The noise is essentially "clipped off " by the limiter. The culprit is not the amplitude variations; the phase variation, α, produced by the noise voltage generates the undesirable frequency variation. The angle, a, established between the carrier and resultant phasor represents the maximum phase deviation caused by the noise. The AM receiver will not be affected by the phase changes for voice communications, whereas the FM receiver will not be affected by the amplitude changes caused by the noise. It is the phase deviation, a, that distorts the FM signal. The maximum phase deviation occurs when R and VN are at right angles to each other and can be computed by Warren Hioki Telecommunications, Fourth Edition Copyright ©2001 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved.

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**α represents the equivalent modulation index produced by the noise.**

The ratio of carrier voltage to noise voltage, is the SNR (voltage) Because the modulation index for FM is defined as the ratio of the carrier's peak frequency deviation to the modulation frequency, a represents the equivalent modulation index produced by the noise. Using the following equation, we can compute the peak frequency deviation, δN, produced by the noise if we are given a modulation frequency, fm: where δN ~ peak carrier frequency deviation produced by the noise voltage, VN a = maximum phase deviation of the carrier frequency expressed in radians fm = modulation frequency This degree of frequency deviation, which ideally should be zero under the condition of no noise (VN = 0), may or may not be significant. Equation (4-14) shows that the higher the modulation frequency, fm, the worse the deviation becomes. Also, because fm is directly related to the modulation index (mf = δ/fm, we can also say that for low modulation indices (resulting from increasing fm), the worse the deviation becomes. What must be considered is the relative deviation caused by the noise in comparison to the maximum allowed deviation, δ, of the FM system. For example, because the maximum δ for an FM broadcast system is 75 kHz, then kHz is only a 5% shift in frequency. It is also possible to compute the overall SNR improvement resulting from the FM process alone. The maximum allowed frequency deviation, δ, is caused by the maximum modulation amplitude, and the maximum frequency deviation caused by the noise amplitude is δN. Therefore, the ratio of the two represents the overall FM SNR: α represents the equivalent modulation index produced by the noise.

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**Increasing fm, degrades the**

The effect of noise on an FM carrier signal is directly proportional to the modulation frequency fm. Increasing fm, degrades the As discussed, the effect of noise on an FM carrier signal is directly proportional to the modulation frequency. Increasing the modulation frequency, f", degrades the SNR. Unfortunately, in most cases, the modulation frequency is not fixed, as in the previous examples; instead, the modulation frequency is continually changing, depending on the nature of the intelligence. For example, voice, data, and music contain many frequencies, which are distributed throughout the given modulation passband. Therefore, the SNR is not uniform throughout. To circumvent this, FM transmitters must boost the signal levels of the higher modulating frequencies before the modulation process. This is to maintain a uniform SNR for the higher modulation frequencies and is called pre-emphasis. Because the original intelligence representing the higher frequencies has been artificially boosted in amplitude to a higher level, -they no longer represent their original amplitudes. The FM receiver must de-emphasize these signals to the same extent, which is called de-emphasis and takes place after the FM signal has been demodulated. A simple high-pass filter is used for pre-emphasis of the signal and a low-pass filter for de-emphasis of the signal. The FCC requires commercial FM broadcast stations to include in their transmitters an inductance-resistance (L/R) pre-emphasis network with a time constant of 75 μs. This implies that the receiver must include a de-emphasis network with the same time constant; an L/R time constant corresponds to a break frequency of 2122 Hz [1/(2πL/R)]. Figure 4-8 illustrates a pre-emphasis and a de-emphasis circuit. Note that the sum of their frequency response curves is unity gain. Pre-eniphasis The process by which FM transmitters boost the signal levels of the higher modulating frequencies before the modulation process to maintain a uniform SINIR for the higher modulation frequencies. De-emphasis The process by which an FM receiver brings back pre-emphasized signals to their original amplitudes. De-emphasis occurs after the FM signal has been demodulated Direct FM One of two basic methods of FM generation. 1 n direct FM generation, the modulation signal is used to directly change the carrier signal's frequency or phase. Indirect FM. indirect FM, the modulation signal is used to change the phase of the carrier signal, which indirectly changes its frequency. Voice, data, and music contain many frequencies, which are distributed throughout the given modulation passband. Therefore, the SNR is not uniform throughout. To maintain a flat SNR, following techniques are employed.

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**Pre-emphasis and De-emphasis**

Pre-emphasis: boost the signal levels of the higher modulating frequencies before the modulation process to maintain a uniform SNR.—high-pass filter De-emphasis: FM receiver brings back pre-emphasized signals to their original amplitudes Low-pass filter. Direct FM: the modulation signal is used to directly change the carrier signal's frequency or phase Indirect FM: the modulation signal is used to change the phase of the carrier signal, which indirectly changes its frequency. Pre-eniphasis The process by which FM transmitters boost the signal levels of the higher modulating frequencies before the modulation process to maintain a uniform SINIR for the higher modulation frequencies. De-emphasis The process by which an FM receiver brings back pre-emphasized signals to their original amplitudes. De-emphasis occurs after the FM signal has been demodulated Direct FM One of two basic methods of FM generation. 1 n direct FM generation, the modulation signal is used to directly change the carrier signal's frequency or phase. Indirect FM. indirect FM, the modulation signal is used to change the phase of the carrier signal, which indirectly changes its frequency.

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