NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGY Amafel Bldg. Aguinaldo Highway Dasmariñas City, Cavite ASSIGNMENT # 3 “FREQUENCY MODULATION”Tagasa, Jerald A. July 11, 2011Communications 1 / BSECE 41A1 Score: Eng’r. Grace Ramones Instructor
FREQUENCY MODULATION Frequency modulation (FM) is a method of impressing data onto an alternating-current (AC)wave by varying the instantaneous frequency of the wave. This scheme can be used with analog ordigital data . In analog FM, the frequency of the AC signal wave, also called the carrier, varies in a continuousmanner. Thus, there are infinitely many possible carrier frequencies. In narrowband FM, commonly usedin two-way wireless communications, the instantaneous carrier frequency varies by up to 5 kilohertz(kHz, where 1 kHz = 1000 hertz or alternating cycles per second) above and below the frequency of thecarrier with no modulation. In wideband FM, used in wireless broadcasting, the instantaneous frequencyvaries by up to several megahertz (MHz, where 1 MHz = 1,000,000 Hz). When the instantaneous inputwave has positive polarity, the carrier frequency shifts in one direction; when the instantaneous inputwave has negative polarity, the carrier frequency shifts in the opposite direction. At every instant intime, the extent of carrier-frequency shift (the deviation) is directly proportional to the extent to whichthe signal amplitude is positive or negative. In digital FM, the carrier frequency shifts abruptly, rather than varying continuously. Thenumber of possible carrier frequency states is usually a power of 2. If there are only two possiblefrequency states, the mode is called frequency-shift keying (FSK). In more complex modes, there can befour, eight, or more different frequency states. Each specific carrier frequency represents a specificdigital input data state. Frequency modulation is similar in practice to phase modulation (PM). When the instantaneousfrequency of a carrier is varied, the instantaneous phase changes as well. The converse also holds: Whenthe instantaneous phase is varied, the instantaneous frequency changes. But FM and PM are not exactlyequivalent, especially in analog applications. When an FM receiver is used to demodulate a PM signal, orwhen an FM signal is intercepted by a receiver designed for PM, the audio is distorted. This is becausethe relationship between frequency and phase variations is not linear; that is, frequency and phase donot vary in direct proportion.
TYPES OF FREQUENCY MODULATION The bandwidth of an FM signal depends on the deviation Kff(t). When the deviation is high, thebandwidth will be large, and vice-versa. From the equation Δω = Kf f(t) it is clear that deviation iscontrolled by Kff (t) (where Δω = frequency deviation). Thus, for a given f (t), the deviation, and hence,bandwidth will depend on frequency sensitivity Kf. If Kf is too small then the bandwidth will be narrowand vice-versa. Thus depending on the value of Kf, FM can be divided into two categories. (1) Narrow band FM: - When Kf is small, the bandwidth of FM is narrow. (2) Wide band FM: - When Kf has an appreciable value, then the FM signal has a widebandwidth. Ideally it is infinite.Narrow Band Frequency Modulation The general expression for FM in the phasor form is given by ΦFM(t) = A ej*ωct +Kfg(t)] For a narrow band FM, Kf g(t) <<1 for all value of t. Hence ejKfg(t) ≈ 1 + j Kfg(t) And FM phasor expression becomes ΦFM(t) ≈ A *1 + jKfg(t)] ejωct The FM signal is the real part of its phasor representation, ΦFM(t) ≈ Re* ΦFM(t)+ = A cosωct - AKfg(t) sinωct The above expression of narrow band FM is very much similar to the expression of the AM(Amplitude modulation) signal, with only a slight modification. In AM all the three components i.e.,carrier and two sidebands are in phase, but in narrow band FM, the lower sideband is 180 degree out ofthe phase with respect to carrier as well as upper sideband. Thus the bandwidth of a narrow band FM issame as that of the AM. Generation of Narrowband FM: Above equation suggest methods for generating the narrow band FM. The sideband terms areobtained by a balanced modulator, as in Double sideband suppressed carrier amplitude modulation(DSB-SC) systems and then the carrier term is added t sideband terms. The method for generatingnarrow band FM is shown in the figure drawn below. The block diagrams satisfy the correspondingexpression for FM. Carrier generator generates the carrier signal, and then 90 degree phase-shifter provides theshift in the carrier signal. After that, this signal is fed to balance modulator in which another signal g (t) isalso fed which is the output of integrator, whereas f(t) is the input to the integrator. After balancemodulator the signal goes to adder where signal is added with unmodified carrier signal. The output ofthe adder is required signal i.e., FM signal.Wide Band Frequency Modulation : When Kf has larger value, then the signal has a wide bandwidth, ideally infinite and the signal iscalled wideband signal. The general equation of wideband FM is given by
ΦFm (t) = AJ0(mf)cosωct + AJ1(mf)*cos(ωc+ωm)t - cos(ωc-ωm)t] +AJ2(mf)*cos(ωc + 2ωm)t + cos(ωc - 2ωmt)] +AJ3(mf) *cos(ωc + 3ωm)t - cos(ωc - 3ωm)t] + ... Wide Band Frequency Modulation : Frequency Components The FM signal has the following frequency components: Carrier term cosωct with magnitude AJ0 (mf), i.e. the magnitude of the carrier term is reduced bya factor J0 (mf) here, J0 is the Bessel function coefficient and mf is known as modulation index of the FMwave. It is defined as the “ratio of frequency deviation to the modulating frequency. The modulationindex mf decides whether an FM wave is a narrowband or a wideband because it is directly proportionalto the frequency deviation. Normally mf = 0.5 is the transition point between a narrowband and awideband FM. If mf<0.5, then FM is a narrowband, otherwise it is a wideband. As discussed, when m f islarge, the FM produces a large number of sidebands and the bandwidth of FM is quite large. Suchsystems are called wideband FM. Theoretically infinite numbers of sidebands are produced, and the amplitude of each sideband isdecided by the corresponding Bessel function Jn (mf). The presence of infinite number of sidebandsmakes the ideal bandwidth of the FM signal is infinite. However, the sidebands with small amplitudesare ignored. The sidebands having considerable amplitudes are known as significant sidebands. They arefinite in numbers. Power content in FM signal: Since the amplitude of FM remains unchanged, the power of theFM signal is same as that of unmodulated carrier. (i.e.` A^2/2` ) where A is amplitude of signal.
MODULATION INDEXAs with other modulation indices, this quantity indicates by how much the modulated variable variesaround its unmodulated level. It relates to the variations in the frequency of the carrier signal: where is the highest frequency component present in the modulating signal xm(t), and is the Peak frequency-deviation, i.e. the maximum deviation of the instantaneous frequency from the carrier frequency. If , the modulation is called narrowband FM, and its bandwidth is approximately .If , the modulation is called wideband FM and its bandwidth is approximately . Whilewideband FM uses more bandwidth, it can improve signal-to-noise ratio significantly. For example,doubling the value of while keeping fm constant, results in an eight-fold improvement in the signalto noise ratio. Compare with Chirp spread spectrum, which uses extremely wide frequency deviationsto achieve processing gains comparable to more traditional, better-known spread spectrum modes.With a tone-modulated FM wave, if the modulation frequency is held constant and the modulationindex is increased, the (non-negligible) bandwidth of the FM signal increases, but the spacing betweenspectra stays the same; some spectral components decrease in strength as others increase. If thefrequency deviation is held constant and the modulation frequency increased, the spacing betweenspectra increases.Frequency modulation can be classified as narrow band if the change in the carrier frequency is aboutthe same as the signal frequency, or as wide-band if the change in the carrier frequency is much higher(modulation index >1) than the signal frequency.  For example, narrowband FM is used for two wayradio systems such as Family Radio Service where the carrier is allowed to deviate only 2.5 kHz aboveand below the center frequency, carrying speech signals of no more than 3.5 kHz bandwidth. Wide-bandFM is used for FM broadcasting where music and speech is transmitted with up to 75 kHz deviation fromthe center frequency, carrying audio with up to 20 kHz bandwidth.
POWER IN FREQUENCY MOULATIONFrom the equation for FM vs (t ) Vc J n ( ) cos( c n m )t n thwe see that the peak value of the components is VcJn( ) for the n component. 2 2 2Single normalised average power = V pk then the nth component is Vc J n ( ) 2 Vc J n ( ) (VRMS ) 2 2 2Hence, the total power in the infinite spectrum is (Vc J n ( ))2 Total power PT n 2By this method we would need to carry out an infinite number of calculations to find PT. But, consideringthe waveform, the peak value is Vc, which is constant. 2 V pk VcSince we know that the RMS value of a sine wave is 2 2 2 2 2 Vc Vc2 Vc J n ( )and power = (VRMS) then we may deduce that PT 2 2 n 2Hence, if we know Vc for the FM signal, we can find the total power PT for the infinite spectrum with asimple calculation.Now consider – if we generate an FM signal, it will contain an infinite number of sidebands. However, ifwe wish to transfer this signal, e.g. over a radio or cable, this implies that we require an infinitebandwidth channel. Even if there was an infinite channel bandwidth it would not all be allocated to oneuser. Only a limited bandwidth is available for any particular signal. Thus we have to make the signalspectrum fit into the available channel bandwidth. We can think of the signal spectrum as a ‘train’ andthe channel bandwidth as a tunnel – obviously we make the train slightly less wider than the tunnel ifwe can.However, many signals (e.g. FM, square waves, digital signals) contain an infinite number ofcomponents. If we transfer such a signal via a limited channel bandwidth, we will lose some of thecomponents and the output signal will be distorted. If we put an infinitely wide train through a tunnel,the train would come out distorted, the question is how much distortion can be tolerated? Generallyspeaking, spectral components decrease in amplitude as we move awayfrom the spectrum ‘centre’.
In general distortion may be defined as Power in total spectrum- Power in Bandlimite spectrum d D Power in total spectrum PT PBL D PTWith reference to FM the minimum channel bandwidth required would be just wide enough to pass thespectrum of significant components. For a bandlimited FM spectrum, let a = the number of sidebandpairs, e.g. for = 5, a = 8 pairs (16 components). Hence, power in the bandlimited spectrum PBL is a (Vc J n ( ))2 = carrier power + sideband powers. PBL n a 2 2 V cSince PT 2 Vc2 Vc2 a ( J n ( ))2 aDistortion D 2 2 n a 1 ( J n ( ))2 Vc2 n a 2Also, it is easily seen that the ratio a Power in Bandlimite spectrum PBL d D ( J n ( ))2 = 1 – Distortion Power in total spectrum PT n a ai.e. proportion pf power in band limited spectrum to total power = ( J n ( ))2 n a