4.1 REPRESENTATION OF FM AND PM SIGNALS An angle-modulated signal generally can be written as
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2 In frequency-modulation (FM) systems, the frequency of the carrier f c is changed by the message signal; in phase modulation (PM) systems, the phase of the carrier is changed according to the variations in the message signal. Frequency and phase modulation are nonlinear, and often they are jointly called angle-modulation methods. 4.1 REPRESENTATION OF FM AND PM SIGNALS An angle-modulated signal generally can be written as where f c denotes the carrier frequency and φ(t) denotes a time-varying phase. The instantaneous frequency of this signal is given by If m(t) is the message signal, then in a PM system and in a FM system where k p and k f are phase and frequency deviation constants (kayma sabitleri). 2
3 On the other hand, this relation can be expressed as which shows that if we frequency modulate the carrier with the derivative of a message, the result is equivalent to the phase modulation of the carrier with the message itself. Figure 4.1 shows the above relation between FM and PM. Figure 4.2 illustrates a square-wave signal and its integral, a sawtooth signal, and their corresponding FM and PM signals. Figure 4.1 A comparison of frequency and phase modulators. 3
4 Figure 4.2 Frequency and phase modulation of square and sawtooth waves. 4
5 The maximum phase deviation in a PM system is given by and the maximum frequency deviation in an FM system is given by An angle-modulated signal generally can be written as The instantaneous frequency of this signal is given by 5
6 The demodulation of an FM signal involves finding the instantaneous frequency of the modulated signal and then subtracting the carrier frequency from it. In the demodulation of PM, the demodulation process is done by finding the phase of the signal and then recovering m(t). Example The message signal is used to either frequency modulate or phase modulate the carrier A c cos(2πf c t). Find the modulated signal in each case. Solution In PM, we have 6
7 and in FM, we have Therefore, the modulated signals will be By defining and 7
8 we have The parameters β p and β f are called the modulation indices of the PM and FM systems, respectively. We can extend the definition of the modulation index for a general nonsinusoidal signal m(t) as where W denotes the bandwidth of the message signal m(t). In terms of the maximum phase and frequency deviation φmax and fmax, we have 8
9 Narrowband Angle Modulation Consider an angle modulation system in which the deviation constants k p and k f and the message signal m(t) are such that for all t, we have φ(t) 1. Then we can use a simple approximation to expand u(t) in Equation (4.1.1) as where we have used the approximations cos φ(t) 1 and sinφ(t) φ(t) for φ(t) 1. Equation (4.1.19) shows that in this case, the modulated signal is very similar to a conventional-am signal given in Equation (3.2.5). The bandwidth of this signal is similar to the bandwidth of a conventional AM signal, which is twice the bandwidth of the message signal. 9
10 Compared to conventional AM, the narrowband angle-modulation scheme has far less amplitude variations. As we will see later, the narrowband angle-modulation method does not provide better noise immunity than a conventional AM system. Therefore, narrowband angle-modulation is seldom used in practice for communication purposes. 4.2 SPECTRAL CHARACTERISTICS OF ANGLE-MODULATED SIGNALS Angle Modulation by a Sinusoidal Signal Consider the case where the message signal is a sinusoidal signal (to be more precise, sine in PM and cosine in FM). As we have seen in Example 4.1.1, in this case for both FM and PM we have where β is the modulation index that can be either β p or β f and in PM sin2πf m t is substituted by cos 2πf m t. Using Euler s relation, the modulated signal can be written as 10
11 Since sin 2πf m t is periodic with period T m = 1/f m, the same is true for the complex exponential signal Therefore, it can be expanded in a Fourier-series representation. The Fourier-series coefficients are obtained from the integral This latter expression is a well-known integral called the Bessel function of the first kind of order n (n. dereceden birinci tip Bessel Fonksiyonu) and is denoted by J n (β). Therefore, we have the Fourier series for the complex exponential as 11
12 By substituting Equation (4.2.4) in to Equation (4.2.2), we obtain The preceding relation shows that, even in this very simple case where the modulating signal is a sinusoid of frequency fm, the angle-modulated signal contains all frequencies of the form f c +nf m for n = 0,±1,±2,.... Therefore, the actual bandwidth of the modulated signal is infinite. However, the amplitude of the sinusoidal components of frequencies f c ±nf m for large n is very small. Hence, we can define a finite effective bandwidth for the modulated signal. For small β, we can use the approximation 12
13 Thus, for a small modulation index β, only the first sideband corresponding to n = 1 is important. Also, we can easily verify the following symmetry properties of the Bessel function: Plots of J n (β) for various values of n are given in Figure 4.4. The values of the Bessel function are given in Table
14 Figure 4.4 Bessel functions for various values of n. 14
15 TABLE 4.1 TABLE OF BESSEL FUNCTION VALUES 15
16 Example Let the carrier be given by c(t) = 10 cos(2πf c t), and let the message signal be cos(20πt). Further assume that the message is used to frequency modulate the carrier with k f = 50. Find the expression for the modulated signal and determine how many harmonics should be selected to contain 99% of the modulated-signal power. Solution The power content of the carrier signal is given by The modulated signal is represented by 16
17 The modulation index is given by Equation (4.1.16) as therefore, the FM-modulated signal is The frequency content of the modulated signal is concentrated at frequencies of the form f c +10n for various n. To make sure that at least 99% of the total power is within the effective bandwidth, we must choose a k large enough such that This is a nonlinear equation and its solution (for k) can be found by trial-and-error and by using tables of the Bessel functions. 17
18 In finding the solution to this equation, we must employ the symmetry properties of the Bessel function given in Equation (4.2.7). Using these properties, we have Starting with small values of k and increasing it, we see that the smallest value of k for which the left-hand side exceeds the right-hand side is k = 6. Therefore, taking frequencies f c ±10k for 0 k 6 guarantees that 99% of the power of the modulated signal has been included and only 1% has been left out. This means that if the modulated signal is passed through an ideal bandpass filter centered at fc with a bandwidth of at least 120 Hz, only 1% of the signal power will be eliminated. This gives us a practical way to define the effective bandwidth of theangle-modulated signal as 120 Hz. 18
19 Figure 4.5 shows the frequencies present in the effective bandwidth of the modulated signal. Figure 4.5 The harmonics present inside the effective bandwidth of Example In general, the effective bandwidth of an angle-modulated signal, which contains at least 98% of the signal power, is given by the relation where β is the modulation index and f m is the frequency of the sinusoidal message signal. 19
20 4.2.2 Angle Modulation by an Arbitrary Message Signal The spectral characteristics of an angle-modulated signal for a general message signal m(t) is quite involved due to the nonlinear nature of the modulation process. However, there exists an approximate relation for the effective bandwidth of the modulated signal. This is known as Carson s rule and is given by where β is the modulation index defined as and W is the bandwidth of the message signal m(t). In broadband angle modulation,since β is greater than 1,the bandwidth an angle-modulated signal is much greater than the bandwidth of the GM signals and increases with increasing value of β 20
21 Example Assuming that m(t) = 10 sinc(10 4 t), determine the transmission bandwidth of an FM modulated signal with k f = Solution For FM, we have B c = 2(β+1)W. To find W, we have to find the spectrum of m(t). We have M(f ) = 10 3 Π(10 4 f ), which shows that m(t) has a bandwidth of 5000 Hz. Since the maximum amplitude of m(t) is 10, we have and 4.3 IMPLEMENTATION OF ANGLE MODULATORS AND DEMODULATORS AngleModulators. Angle modulators are generally time-varying and nonlinear systems. 21
22 Direct method for the generation of FM and PM signals One method for directly generating an FM signal is to design an oscillator whose frequency changes with the input voltage. When the input voltage is zero, the oscillator generates a sinusoid with frequency fc; when the input voltage changes, this frequency changes accordingly. There are two approaches to designing such an oscillator, usually called a VCO or voltage-controlled oscillator. One approach is to use a varactor diode. A varactor diode is a capacitor whose capacitance changes with the applied voltage. Figure 4.7 Varactor-diode implementation of an angle modulator. 22
23 Figure 4.7 is L0 and the capacitance of the varactor diode is given by When m(t) = 0, the frequency of the tuned circuit is given by In general, for nonzero m(t), we have As seen in equation,in the oscillator output,instantaneous frequency f i (t) is changing with message signals m(t),thus amplitute is changing frequency (FM modulation). 23
24 Another approach for generating an angle-modulated signal is to generate a narrowband angle-modulated signal and then change it to a wideband signal. This method is usually known as the indirect method for the generation of FM and PM signals. Due to the similarity of conventional AM signals, the generation of narrowband angle-modulated signals is straightforward. In fact, any modulator for conventional AM generation can be easily modified to generate a narrowband angle-modulated signal. Figure 4.8 shows the block diagram of a narrowband angle modulator. Figure 4.8 Generation of a narrowband angle-modulated signal. 24
25 Figure 4.9 Indirect generation of angle-modulated signals. 25
26 If the narrowband modulated signal is represented by the output of the frequency multiplier (which is the output of the bandpass filter) is given by In general, this is a wideband angle-modulated signal. However, there is no guarantee that the carrier frequency of this signal, nf c, will be the desired carrier frequency. In the last stage, the modulator performs an up/down conversion to shift the modulated signal to the desired center frequency. This stage consists of a mixer and a bandpass filter. If the frequency of the local oscillator of the mixer is f LO and we are using a down converter, the final wideband anglemodulated signal is given by 26
27 The above equation is obtained with the help of the following trigonometric equations. Choosing n and f LO values freely, modulation index can be produced at any carrier frequency 27
28 Ex
29 4.4 FM-RADIO AND TELEVISION BROADCASTING FM-Radio Broadcasting Commercial FM-radio broadcasting utilizes the frequency band MHz for the transmission of voice and music signals. The carrier frequencies are separated by 200 khz and the peak frequency deviation is fixed at 75 khz. The receiver most commonly used in FM-radio broadcast is a superheterodyne Type. The block diagram of such a receiver is shown in Figure
30 Figure 4.16 Block diagram of a superheterodyne FM-radio receiver. 30
31 As in AM-radio reception, common tuning between the RF amplifier and the local oscillator allows the mixer to bring all FM-radio signals to a common IF bandwidth of 200 khz, centered at f IF = 10.7 MHz. Since the message signal m(t) is embedded in the frequency of the carrier, any amplitude variations in the received signal are a result of additive noise and interference. The amplitude limiter removes any amplitude variations in the received signal at the output of the IF amplifier by bandlimiting the signal. A balanced frequency discriminator is used for frequency demodulation. The resulting message signal is then passed to the audio-frequency amplifier, which performs the functions of deemphasis and amplification. The output of the audio amplifier is further filtered by a lowpass filter to remove outof-band noise, and this output is used to drive a loudspeaker. 31
32 Pre-Emphasis (Preemphasis) and the emphasis Lifting (deemphasis) Filters As seen in Figure 6.4, in FM demodulation high frequencies are more susceptible to noise. So the noise is higher than at high frequencies Figure 6.4 Noise power spectrum for at demodulator output a) PM b) FM 32
33 At transmitter,to solve the problem created by this situation, (before being sent to the channel,) high frequencies of the FM modulated signal is raised with the help of a filter and made resistant to noise. At receiver,after demodulation to get back to the original signal, The reverse operation is performed with the help of a filter Figure 6.6 The characteristics of Pre-Emphasis and the emphasis Lifting Filters 33
34 FM-Stereo Broadcasting. Many FM-radio stations transmit music programs in stereo by using the outputs of two microphones placed on two different parts of the stage. Figure 4.17 shows a block diagram of an FM-stereo transmitter. Figure 4.17 FM-stereo transmitter and signal spacing. 34
35 By configuring the baseband signal as an FDM signal, a monophonic FM receiver can recover the sum signal m l (t) + m r (t) by using a conventional FM demodulator. Hence, FM-stereo broadcasting is compatible with conventional FM. In addition, the resulting FM signal does not exceed the allocated 200-kHz bandwidth. Figure 4.18 FM-stereo receiver. 35
36 The FM demodulator for FM stereo is basically the same as a conventional FM demodulator down to the limiter/discriminator. Thus, the received signal is converted to baseband. Following the discriminator, the baseband message signal is separated into the two signals, m l (t)+ m r (t) and m l (t) m r (t), and passed through deemphasis filters, as shown in Figure By taking the sum and difference of the two composite signals, we recover the two signals, m l (t) and m r (t). These audio signals are amplified by audioband amplifiers, and the two outputs drive dual loudspeakers. As indicated, an FM receiver that is not configured to receive the FM stereo sees only the baseband signal m+(t)+mr (t) in the frequency range 0 15 khz. Thus, it produces a monophonic output signal that consists of the sum of the signals at the two microphones. 36
37 4.4.2 Television Broadcasting Commercial TV broadcasting began as black-and-white picture transmission in London in 1936 by the British Broadcasting Corporation (BBC). Color television was demonstrated a few years later, but commercial TV stations were slow to develop the transmission of color-tv signals. To a large extent, this was due to the high cost of color-tv receivers. With the development of the transistor and microelectronic components, the cost of color TV receivers decreased significantly. By the middle 1960s, color TV broadcasting was widely used by the industry. The frequencies allocated for TV broadcasting fall in the VHF and UHF bands. Table 4.2 lists the TV channels allocated in the United States. The channel bandwidth allocated for the transmission of TV signals is 6 MHz. 37
38 TABLE 4.2 ALLOCATION OF VHF AND UHF FREQUENCIES FOR COMMERCIAL TELEVISION 38
39 Figure 4.22 Spectral characteristics of a black-and-white television signal. 39
40 Figure 4.23 Block diagram of a black-and-white TV transmitter. 40
41 Figure 4.24 Block diagram of a black-and-white TV receiver. 41
42 4.1 42
43 4.9 43
44 44
45
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