ERG 2310: Principles of Communications 3. FM Chapter 3 Angle Modulation 3.1 Introduction Besides using the amplitude of a carrier to carrier information, one can also use the angle of a carrier to carrier information. This approach is called angle modulation, and includes frequency modulation (FM) and phase modulation (PM). The amplitude of the carrier is maintained constant. The major advantage of this approach is that it allows the trade-off between bandwidth and noise performance. An angle modulated signal can be written as where is usually of the form and is the carrier frequency. The signal is derived from the message signal ! . If "#%$! (3.1) for some constant #&$ , the resulting modulation is called phase modulation. The parameter #'$ is called the phase sensitivity. If "#)(*,- + ! ./10. (3.2) for some constant #( , the resulting modulation is called frequency modulation. The parameter #( is called the frequency sensitivity. Notice that PM and FM differ only in the interpretation of message 3.1ERG 2310: Principles of Communications 3. FM 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.50 0.510 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.50 0.51 Figure 3.1: Frequency modulation signals. The instantaneous frequency of an angle modulated signal is defined by 00 (3.3) For PM and FM, +! for PM #)(! for FM The maximum phase deviation in PM is called the modulation index $ of PM. $ #%$! (3.4) Suppose that the message signal is a pure sinusoid or ! #" . Then the maximum frequency deviation $ divided by the message frequency is called the modulation index ( of FM. ( #)( (3.5) In general, the maximum frequency deviation $ divided by the message bandwidth % is called of deviation ratio & of FM. & #)('! % (3.6) 3.2ERG 2310: Principles of Communications 3. FM 3.2 Frequency Modulation Narrowband FM Consider that the message signal is the pure sinusoid &. Then it is easy to check that &! #"&! #"& ! #"! #"! #" Suppose that . Then ! #"! #"! #"! #" Therefore, ! #"! #"(3.7) Notice that the result is very similar to AM modulation. The main difference is that the message is now modulated onto the quadrature carrier instead of the in-phase carrier. Narrowband FM signals can be approximately generated with this approximation. The envelope of the resultant signal is given by ! #" which is not constant. However, if is small, it would be close to a constant. Notice that for the same modulation index, the amplitude variation of an AM signal is much larger than that of a narrowband FM signal. General FM Consider that the message signal is the pure sinusoid. Then the FM signal is of the form ! #" Re(+(+Consider the term (+. Since ! #" is periodic with period "! , #$(+is periodic with period . We find its Fourier series representation. The Fourier coefficient %$& is given by %& ( '*) ( ,+ -(,+(+-&(+0') -/. -&.010 3.3ERG 2310: Principles of Communications 3. FM 0 1 2 3 4 5 6 7 8 9 10 −0.50 0.51 Figure 3.2: Bessel functions The last integral is the Bessel function of the first kind of order n evaluated at , and is usually denoted by & . Notice that 1& is real. With the Fourier series representation, (+&-& &(+ Therefore, &-& (3.8) Notice that contains an infinite number of frequency components of the form ). The Bessel Functions There are various equivalent definitions of the Bessel functions. We take the following definition. The Bessel function of the first kind of order is defined by & * --&0(3.9) We note the following properties of the Bessel functions: 1. & is real. Notice that /-&&! #" ! #"! #" Since ! #" ! #" ! #"! #" &! #" ! #" 3.4ERG 2310: Principles of Communications 3. FM is an odd function of , its integral from to is zero. Notice that ! #" is an even function of . Therefore, 1& can also be expressed as & * - ! #" 102. -& && . Using the last result, -&* - ! #" 10Put . - & ' ) - &! #" ,10 ' ) - ! #" & ! #"! #" ! #" 10 && 3. & has the following Taylor series expansion (proof omitted): & - !&! ! For small , the series is dominated by its first term. Therefore, & & "& When the values of 1& are compared across different for small , - '& for Towards the other extreme, for large , & & which a decaying sinusoid. 4. For all , & - &(proof omitted). A consequence is that for any fixed , the infinite sum is dominated by a finite number of terms. 3.5ERG 2310: Principles of Communications 3. FM Observations on FM signals With knowledge about the Bessel functions, we can have the following observations on tone-modulated FM signals. Recall that &-& 1. contains an infinite number of frequency components of the form , where ! 2. The power of is distributed among the components. The power carried by the component )is &!. The total power is &!!, which is reasonable since the overall FM signal has a constant amplitude . 3. Most of the power is carried by the components /for . (Since &", we know that most of the power is carried by a finite number of components. The value is determined empirically.) 4. For small , using the approximations for the Bessel functions, we have && & ! #" ! #" which is the narrowband FM approximation we have used before. In theory, the FM signal occupies a spectrum of infinite bandwidth. However, since most of the power is carried by a finite number of components, we can define an effective bandwidth. For a tonemodulated FM signal, it is defined by $ For a general message signal with bandwidth % , the effective bandwidth of the corresponding FM signal is defined similarly by $ % & % (3.10) This empirical formula is often referred to as Carson’s rule. 3.6ERG 2310: Principles of Communications 3. FM shift p/2 m(t) Integrator -+ ~ Figure 3.3: Generation of narrowband FM X n Freq. Modulator NBFM BPF ~ Figure 3.4: Generation of wideband FM via frequency multiplication 3.3 Modulators and Demodulators Generation of FM signals An FM signal can be generated with a voltage controlled oscillator (VCO). Given an input signal ! , the instantaneous output frequency of a VCO (with nominal frequency ) is of the form #! which is the desired format for FM. Another approach is to first generate a narrowband FM signal. Notice that a narrowband FM signal can be approximately generated as in Fig. 3.3. Then the frequency of the resulting signal is multiplied to give the desired frequency deviation as shown in Fig. 3.4. A frequency multiplier consists of a non-linear device with input-output relationship of the form "' &&&followed by an appropriate bandpass filter. Ideally, if the instantaneous input frequency is 1, then the instantaneous output frequency is . When the desired frequency deviation is obtained, the signal is translated to the desired carrier frequency with a mixer and a BPF. Example: Suppose that the input narrowband FM signal has a frequency deviation of 2.5 kHz with a carrier frequency of 10 MHz, the desired frequency deviation is 50 kHz, and the desired carrier is 100 MHz. Therefore, should be chosen. As a result, the multiplied carrier frequency is 200 3.7ERG 2310: Principles of Communications 3. FM MHz. We can mix the result with a 100 MHz carrier to obtain the desired carrier frequency. Of course, it is also possible to do frequency translation before frequency multiplication. We can translate the narrowband FM signal to 5 MHz. Then multiply the resultant signal in frequency by 20 times. Demodulation of FM signals One approach to FM demodulation is to generate an AM signal with amplitude proportional to the instantaneous frequency of the FM signal, and then to recover the message signal with an AM demodulator. Ideally, FM to AM conversion can be achieved with a differentiator. Consider the FM signal. "&#)(* - + ! ./10./Its derivative is given by 00 ,#)( ! ! #"#)(*,- + ! ./10./(3.11) Notice that the amplitude varies according to the message signal. The transfer function of the ideal differentiator is given by which is linear. In general, it is hard to realize this linear transfer function over a large range of frequencies. However, since the power of an FM signal concentrates around the carrier frequency , we can try to build a filter with linear transfer function around , i.e., "# for !# ,for ,!where is the bandwidth of the FM signal. The Fourier transform of the output is given by # #",In time domain, the output is # 0 0# ##)( ! ! #"#)(*,- + ! ./10./with its envelope varying according to ! . If a sufficiently large is chosen, the message signal can be recovered with an envelope detector. 3.8ERG 2310: Principles of Communications 3. FM VCO Loop Filter Phase Detector s(t) e(t) r(t) v(t) Figure 3.5: The Phase Locked Loop Phase Locked Loops The phase locked loop (PLL) finds applications in different areas of communications, including carrier phase synchronization and FM demodulation. A typical PLL is shown in Fig. 3.5. We first consider using a PLL to track the phase of a carrier. The operation of the PLL is as follows: The carrier to be “locked” is '. The reference signal is ! #"/. In the simplest case, the phase detector is just a multiplier followed by a low-passed filter (to remove the double frequency term). Therefore the output of the phase detector is the error signal $! #" where $ is the constant gain of the filter over its passband. The loop filter determines the performance of the PLL when noise is present. For simplicity, we assume that it provides a constant gain . Therefore, ". The voltage controlled oscillator adjusts the frequency (and, hence, phase) of its output according to the relations 00 "where is a constant gain. Therefore, 00 &! #" &where the overall gain $. Suppose that is close to . Then, approximately, 00 &3.9ERG 2310: Principles of Communications 3. FM −8 −6 −4 −2 0 2 4 6 8 −2 −1.5 −1 −0.50 0.51 1.52 f − fr e(t) ® ¬ ¬ ® ® ¬ ¬ ® ® ¬ Figure 3.6: Drift of the phase reference to a stable equilibrium The solution of this differential equation is given by & &) Clearly, &tends to . Notice that the overall gain controls the speed of convergence. In case that the initial phase reference is not close to . The situation is depicted in Fig. 3.6. will drift to the vicinity of ,# for some integer #. Then locking begins. Now consider using the PLL for FM demodulation. The FM signal is of the form "&% where #)( ) -+ ! ./10. . Suppose that varies so slowly that it is tracked by the PLL, i.e., & . Then "00 &00 "#)( ! is the desired message signal. 3.10