MIT OpenCourseWare http://ocw.mit.edu MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.MAS 160/510 Additional Notes: Modulation From Amplitude Modulation to Frequency Modulation As usually implemented, FM uses much more bandwidth than AM. You’ll note, for instance, that FM radio stations in the US are spaced 200KHz apart, while AM stations are spaced only 10KHz apart. So why would one want to use FM? Among other interesting features, it allows signal-to-noise ratio, or SNR, to be traded off for bandwidth. We’ve already seen that amplitude modulation simply requires us to multiply our input signal f(t) by a “carrier” sinusoid c(t) = cos ωct, where ωc (or more precisely ωc/2π Hertz) is the frequency to which you tune your radio dial in order to receive this signal. In order to s implify receiver design, actual AM broadcasts are of the form s(t) = K[1 + mf(t)] cos ωct, where m is chosen so that |mf(t)| < 1. This causes the “envelope” of the signal to follow the shape of the input, preventing a negative f(t) from flipping the phase of the sinusoid. This pe rmits demodulating the signal very simply: a narrow bandpass filter is tuned to the frequency of the desired station, its output go e s to a nonlinear device called a rectifier which removes the negative portion of the signal, and a lowpass filter then essentially connects the peaks to recover the envelope, which is (1 + mf(t)).1 Such a receiver requires less precision than synchronous demodulation, or multiplying the received signal by a phase-locked sinusoid to shift a spectral replica back down so that it centers on zero frequency (this method is used, however, in some of the digital modulation methods we will examine later). We can rethink the formulation of our sinusoidal carrier c(t). Let θ(t) = ωct. In this case, θ(t) is a linear function of time, and ωc is its derivative. But in other kinds of modulation θ(t) won’t be linear with time, and we can’t think about frequency as we’re accustomed to do. Thus we need to define something called instantaneous frequency ωi as the derivative of the angle: dθ c(t) = cos θ(t), ωi = . dt In FM, we want ωi to vary linearly with the modulating signal f(t). Therefore, ωi = ωc + Kf(t), which implies that � � θ(t) = ωidt = ωct + K f(t)dt. The analysis of FM is far harder than that for AM, as superposition doesn’t hold. It’s typical, nevertheless, to consider what happens when our f(t) is a s inusoid: f(t) = a cos ωmt. Now ωi = ωc + Δω cos ωmt, Δω � ωc. If you’ve ever built an AM receiver, you will justifiably charge us with gross oversimplification, but the basic idea is correct. 1 1� Then s(t) = cos(ωct + β sin ωmt), where we call β the modulation index and define it as the ratio of the maximum frequency deviation to the bandwidth of f (t): Δω .β ≡ Δωm What we’re going to investigate is how the bandwidth of the signal s(t) depends on β. AM:m1m ωc−ωm ωc ωc+ωmNBFM:−β/21β/2 ωc−ωm ωc ωc+ωmFigure 1: Spectra for AM and NBFM, give n a modulating signal that is a single sinusoid. Consider first the e xpansion of the above s(t): s(t) = cos ωct cos(β sin ωmt) − sin ωct sin(β sin ωmt). If β � π/2, which is called narrowband FM, cos(β sin ωmt) ≈ 1, and sin(β sin ωmt) ≈ β sin ωmt so s(t) ≈ cos ωct − β sin ωmt sin ωct. If you consider how we got here, you should be able to see that for small β, for any f(t), sN BF M (t) ≈ cos ωct − K f(t)dt sin ωct. If we recall that sAM (t) = cos ωct + mf(t) cos ωct, we can see that narrowband FM of a sinusoidal f(t) is very similar to AM except that the sidebands are π/2 radians out of phase with the carrier. The bandwidth is essentially the same. If f(t) = cos ωmt the spectra look like the illustration in Figure 1. Possible systems for generating each are shown in Figure 2. But NBFM is more complicated and doesn’t appear to offer us any real advantages. Let’s now consider wideband FM (β > π/2). 2× Σf(t)cosωct++AM:× Σ∫f(t)cosωct–+90ºshiftNBFM:Figure 2: Modulators for AM and NBFM. When β �� π/2 the approximation we did above doesn’t hold. To understand what happens as β increases, it’s usual to expand s(t) into a power series and to retain all the significant terms. See Schwartz’s book, referenced at the end of these notes, for more details. We’ll let an illustration suffice. For a sinusoidal modulating signal again, with β ≈ 2, we get a spectrum as in Figure 3. The bandwidth of FM is, strictly speaking, infinite. But since the terms far away from the carrier are very, very small, they can usually be ignored. A common rule of thumb is to say that if the maximum frequency in f(t) is B, then the approximate FM bandwidth is 2B(1 + β). For broadcast FM radio, β = 5 and B is 15KHz, giving us a bandwidth of 180KHz, w hich corresponds well with the 200KHz channel spacing. Given that an integrator was used in the generation of the signal, you shouldn’t be surprised that we use a differentiator to recover it. Since the gain of a differentiator varies linearly with frequency, the output is the input signal with its amplitude (or envelope) varying as the modulating input f(t). Then we can just use an envelope detector (as in AM) to get back f(t), as in Figure 4. Incidentally, there is an important theorem called Logan’s Theorem2 that applies to FM. It states that if the bandwidth of a signal is les s than an octave, the signal may b e recovered exactly (except for a multiplicative constant) from its zero-crossings.3 Analog laser videodiscs work in this fashion, as the video signal is FM modulated and the spacing of the pits on the disc records the position of the ze ro-cross ings of the FM signal. Digital Modulation – PSK, QAM 2B. F. Logan, “Information in the Zero Crossings of Bandpass Signals,” Bell Sys. Tech. J., 56, pp. 487-510, April 1977. There is also a requirement that the signal have no zeros in common with its Hilbert transform, among other provisos not important here. 3 3ωc−2ωm ωc−ωm ωc ωc+ωm