Frequency Modulation

Transcription

1 10/22/2010 Frequency Modulation.doc 1/11 Frequency Modulation The problem in electrical communication systems is this: 1. The signals that represent useful information are difficult to transmit/propagate. 2. The signals that are easy to propagate contain no useful information! For example, a signal that is an analog representation of audio (voice, music, etc.) provides useful information (at least ideally). Say we now need to quickly transfer this audio information to a distant site. Directly propagating this signal in either a bounded channel (i.e., through transmission line) or an unbounded channel (using antennas) would result in dismal failure.

2 10/22/2010 Frequency Modulation.doc 2/11 The problem is the frequency spectrum of this signal. Humans can hear acoustic pressure waves (i.e., sound) from a frequency of about 20 Hz to a frequency between 10 khz and 20 khz (depending on your age). Thus, audio signals typically exhibit this spectrum. With respect to efficient transmission, this audio signal spectrum has three fundamental problems: 1. It is very low frequency. 2. It is a very wide percentage bandwidth (i.e., 1,000- to-1). 3. It is the same spectrum as every other audio signal!

3 10/22/2010 Frequency Modulation.doc 3/11 In contrast, a signal that can easily be propagated across a great distance has these characteristics. 1. It has spectrum across high frequencies (e.g., > 80 MHz) 2. It has a very narrow percentage bandwidth (e.g., < 1 %). 3. It occupies a different spectrum from every other audio signal! For example, your average FM radio station has a broadcast frequency of around 100 MHz, a bandwidth of about 200 khz (i.e., 0.2 %), and spectral location on the FM dial that is distinct and unique with respect to all other stations in the general vicinity. Q: But FM radio stations do convey audio information! How do they accomplish this?? A: We need to effectively glue our useful information onto a signal that is suitable for propagation. We do this by modulating a Radio Frequency (RF) signal.

4 10/22/2010 Frequency Modulation.doc 4/11 For example, consider an RF signal operating at some frequency ω c : vrf ( t) = a cos ωc t This sinusoidal signal is parameterized by two variables its amplitude and its phase/frequency. In the static case shown above, both the amplitude a and the frequency ω c are constant with respect to time this signal conveys no information! However, the amplitude and/or the frequency can be made to vary with respect to time. This variation is relatively slow much slower than one period of the sinusoidal oscillation. v ( t) = a ( t) cos ω + ω( t) t rf c We call this process modulation. If the amplitude varies then the signal is Amplitude Modulated (AM), and if the frequency varies, the signal is Frequency Modulated (FM). Q: So what does this have to do with transmitting information? A: The slow variations of amplitude or frequency can be in direct response to some information rich signal. For example, the amplitude/frequency variation can be directly proportional to an audio signal! This is exactly how

5 10/22/2010 Frequency Modulation.doc 5/11 your favorite FM station transmits information to your FM receiver. An example of AM An example of FM From Q: But 101 The Fox claims to broadcast at the precise frequency of MHz. If it is frequency modulated, won t the signal now be at some other frequency? A: Yes, but it will always be very close to a frequency of MHz! The modulating audio signal will cause the frequency of the RF carrier signal to either increase slightly or decrease slightly, as the AC audio signal changes with time. If the AC audio signal is negative, the RF frequency will be slightly less than the carrier frequency of MHz. If

6 10/22/2010 Frequency Modulation.doc 6/11 the AC audio signal is positive, the RF frequency will be slightly more. Q: Just what do you mean by slightly more? A: It depends. In the case of FM radio, the frequency deviation can be no more that +/- 100 khz; otherwise the signal starts to overlap with adjacent stations (e.g., at MHz), and the FCC gets quite miffed. Q: But doesn t this frequency deviation of +/- 100 khz result in a signal bandwidth of 200 khz? A: Yes, it absolutely does! A modulated signal will always occupy some non-zero bandwidth in the frequency spectrum. This bandwidth is sometimes referred to as the modulation bandwidth. Q: Wait a second! The audio signal had a bandwidth of only 20 khz, and you said that was bad. This modulated FM signal has a bandwidth of 10 times that! A: True. But the problem with the audio signal was its percentage bandwidth. The upper end of the audio bandwidth (20kHz) is one thousand times higher than the low end (20 Hz). It s this ratio that makes this audio signal problematic (from the standpoint of direct propagation).

7 10/22/2010 Frequency Modulation.doc 7/11 In contrast, the signal for 101 The Fox (with the 200 khz bandwidth), occupies a spot in the frequency spectrum from MHz to MHz a ratio of just (a 0.2% bandwidth). If this FM signal had the same percentage bandwidth as the audio signal, its spectrum would extend from 100 MHz to 100 GHz!!! Q: What about AM? In Amplitude Modulation, only the signal amplitude is varied (i.e., not the signal frequency) Does that mean that an AM signal has zero bandwidth? A: Nope. It turns out that the bandwidth of an AM signal is twice the bandwidth of the modulating signal. Thus, if a carrier is Amplitude Modulated with an audio signal bandwidth with a bandwidth of 20 khz, the resulting signal will exhibit a 40 khz bandwidth! Q: So does that mean that AM radio stations have 40 khz of bandwidth? A: No, AM stations band-limit the modulating audio signal to 10 khz or less. As a result, the bandwidth of an AM station is no more than 20 khz this is one (but only one!) reason why AM stations don t always sound so great.

8 10/22/2010 Frequency Modulation.doc 8/11 Q: So, FM stations broadcast a signal that is 10 times wider than AM stations is this a good thing? A: It s a very good thing! Frequency Modulation provides the opportunity to spread a 20kHz audio signal over a bandwidth of much larger value. This essentially adds a redundancy to the information kind of like repeating the same message 10 times! kind of like repeating the same message 10 times! kind of like repeating the same message 10 times! kind of like repeating the same message 10 times! kind of like repeating the same message 10 times! kind of like repeating the same message 10 times!... As a result, the information can be recovered (demodulated) with very little error. This is one reason why FM radio sounds so much better than AM. Edwin Howard Armstrong is (in my opinion) the greatest electrical engineer in history. In addition to inventing the electronic feedback amplifier, the electronic oscillator, and the superheterodyne radio receiver, Armstrong also invented Frequency Modulation!

9 10/22/2010 Frequency Modulation.doc 9/11 His idea for FM was initially greeted with skepticism and derision (especially by RCA, who both made AM radios and owned AM radio stations). However, the wisdom and efficacy of Howard Armstrong s FM invention soon became apparent, and (RCA) then tried to claim the idea as its own (shameful!). A distraught Armstrong tragically took his own life. But, electrical engineers (and eventually the courts) recognized Howard Armstrong as the rightful inventor of Frequency Modulation. His widow became a very wealthy woman. Q: But how is this accomplished? How is frequency modulation achieved? A: There are many methods, but perhaps the simplest is via a Voltage Controlled Oscillator (VCO). Q: I see! We simply use the audio signal as the control voltage, right?? vc ( t) = va( t) v ( t) cos ω t vco = vco A: Not exactly; it s a bit more complicated than that!

10 10/22/2010 Frequency Modulation.doc 10/11 The control voltage of the VCO must consist of both a DC and an AC (the audio signal) component: v ( t) = V + v ( t) C C a The DC component V C is simply a bias that sets the carrier frequency. The AC audio signal thus increases or decreases the VCO output frequency (as a function of time) above and below this carrier frequency. For example, say the VCO transfer function has the simple form: ω = Kv + ω vco v C 0 The VCO frequency is thus: ( C ) 0 ( KV ω ) Kv( t) ω ( t) = K V + v ( t) + ω vco v a = + + v C 0 v a By inspection, we can conclude that the carrier frequency of this Frequency Modulated signal is: ω = KV + ω c v C 0 And the time-varying deviation from this carrier is: ω ( t) = K v ( t) v a So that:

11 10/22/2010 Frequency Modulation.doc 11/11 ωvco ( t ) = ωc + ω ( t ) = KV + ω + Kv t ( ) ( ) v C 0 v a Note that if the audio signal is zero the audio goes silent (an unlikely event if the audio is generated by your professor) the VCO output simply has a constant frequency of carrier ω. c The modulation bandwidth B m is therefore: Kv v Bm = 2π ( ) a p p where va( p p) is the peak-to-peak voltage of the AC audio signal. Note the modulation bandwidth can be increased or decreased by changing this value! Q: But how can a DC and AC signal be added together? A: There are several ways. Use an AC coupling capacitor, or an op-amp summing network. Q: So how is a signal demodulated? How is the audio signal recovered? A: There are many ways to demodulate an FM signal. One way of course is to use a Phase-Locked Loop!