Traditional Analog Modulation Techniques

Transcription

1 Chapter 5 Traditional Analog Modulation Techniques Mikael Olosson Modulation techniques are mainly used to transmit inormation in a given requency band. The reason or that may be that the channel is band-limited, or that we are assigned a certain requency band and requencies outside that band is supposed to be used by others. Thereore, we are interested in the spectral properties o various modulation techniques. The modulation techniques described here have a long history in radio applications. The inormation to be transmitted is normally an analog so called baseband signal. By that we understand a signal with the main part o its spectrum around zero. Especially, that means that the main part o the spectrum is below some requency W, called the bandwidth o the signal. We also consider methods to demodulate the modulated signals, i.e. to regain the original signal rom the modulated one. Noise added by the channel will necessarily aect the demodulated signal. We separate the analysis o those demodulation methods into one part where we assume an ideal channel that does not add any noise, and another part where we assume that the channel adds white Gaussian noise. 5.1 Amplitude Modulation Amplitude modulation, normally abbreviated AM, was the irst modulation technique. The irst radio broadcasts were done using this technique. The reason or that is that AM signals can be detected very easily. Essentially, all you need is a nonlinearity. Actually, almost any nonlinearity will suice to detect AM signals. There have even been reports o people hearing some nearby radio station rom their stainless steel kitchen sink. And some 67

2 68 Chapter 5. Traditional Analog Modulation Techniques (including the author) have experienced that with a guitar ampliier. The crystal receiver is a demodulator or AM that can be manuactured at a low cost, which helped making radio broadcasts popular. In Chapter 3, Theorem 9, we noted that a convolution in the time domain corresonds to a multiplication in the requency domain. In act, the opposite is also true. Theorem 10 (Fourier transorm o a multiplication) Let a(t) and b(t) be signals with Fourier transorms A() and B(). Then we have F {a(t)b(t)} = (A B)(). Proo: The proo is along the same line as the proo o Theorem 8, but starting with the inverse transorm o the suggested spectrum. Based on our deinitions, we have F 1 {(A B)()} = (A B)()e j2πt d = A(φ)B( φ)dφ e j2πt d. We can rewrite the expression above as F 1 {(A B)()} = A(φ)B( φ)e j2πt dφ d. Now, set λ = φ, and we get F 1 {(A B)()} = A(φ)B(λ)e j2π(λ+φ)t dτ dλ = A(φ)e j2πφt dφ B(λ)e j2πλt dλ. Finally, we identiy the last two integrals as the inverse Fourier transorms o A() and B(), and we get F 1 {(A B)()} = a(t)b(t). So, multiplying in the time domain corresponds to a convolution in the requency domain.

3 5.1. Amplitude Modulation antenna diode BP ilter LP ilter earphone (a) (b) (c) Figure 5.1: (a) A standard AM signal or the message m(t) = cos(2πt) with C = 2 and A = 1. The dark line is C + m(t). (b) Principle o an envelope detector. (c) The corresponding output rom an envelope detector. Standard AM An AM signal, x(t), corresponding to the message signal, m(t), is given by the equation x(t) = A(C + m(t)) cos (2π c t), where c is reerred to as the carrier requency, A is some non-zero constant, and where the constant C is chosen such that m(t) < C holds or all t. In Figure 5.1a a standard AM signal is presented together with the message, which in this particular example is a cosine signal. We mentioned that AM signals can be detected using a nonlinearity. The irst AM receiver was the so called crystal receiver. It consists o an antenna, a resonance circuit (bandpass ilter), a diode and a simple low-pass ilter. It extracts the envelope C + m(t) rom x(t), and is thereore oten called an envelope detector. The diode in Figure 5.1b is the nonlinearity that makes the detection possible. The ew simple components makes it possible to manuacture the receiver at a low cost. In addition to that, it doesn t even need a power source o its own. The power is taken directly rom the antenna. The output power is o course very small, and only one listener could use the small earphone that was used. In Figure 5.1c, the AM signal is presented together with the output o an envelope detector. Note that the output is very similar to the original message. The mechanical parts in the earphone, and the ear will urther low-pass ilter the output, so the listener will hear almost the same signal as the one transmitted. Modern envelope detectors have ampliiers in various places and may be implemented digitally, but the basic construction is still a bandpass ilter, a diode (or some other rectiier) ollowed by a lowpass ilter. An advantage o envelope detectors is that the BP ilter that ilters out everything except the intended

4 70 Chapter 5. Traditional Analog Modulation Techniques Message M() W W Carrier 1/2 F {cos(2π c t)} 1/2 c c Standard AM X() upper sideband lower sideband lower sideband upper sideband c c Figure 5.2: Spectrum or standard AM signals. requency band is not critical. It is enough i its center requency is approximately correct. In other words, it does not need to know the carrier requency exactly, or the carrier phase or that matter. We wish to study the spectrum o AM signals. Since an AM signal is the product o a message and a carrier, that is easiest done based on Theorem 10. Thus, we need to ind the Fourier transorm o cos(2π c t). First consider F 1 {δ( c )} = δ( c )e j2πt d = e j2πct, where the irst equality is given by the deinition o the inverse Fourier transorm, and where the last equality is given by the deinition o the unit impulse. So, we have { } e j2π ct + e j2πct F {cos (2π c t)} = F = (δ( c) + δ( + c )). Now we are ready to apply Theorem 10 on x(t). Let M() be the spectrum o m(t) and let X() be the spectrum o x(t). Then we get X() = F {AC cos (2π c t)} + F {Am(t) cos (2π c t)} = AC 2 [δ( c) + δ( + c )] + A 2 [M( c) + M( + c )].

5 5.1. Amplitude Modulation 71 It is let as an exercise to veriy that the last equality holds. The involved spectra are displayed in Figure 5.2. Here AC (δ( 2 c) + δ( + c )) is reerred to as the carrier, since that term corresponds to AC cos (2π c t). The other part o the spectrum, A (M( 2 c) + M( + c )), is reerred to as the sidebands. Those sidebands are the only parts o the spectrum that depend on the message m(t). The sidebands are called upper and lower sidebands based on where they are compared to the carrier requency, according to the ollowing. Upper sideband: A 2 (M( c) + M( + c )) or > c. Lower sideband: A 2 (M( c) + M( + c )) or < c. Because o those two sidebands, this type o AM modulation is oten called double sideband AM, abbreviated AM-DSB. There are also other versions o AM, but they cannot be detected using an envelope detector. Suppressed Carrier Modulation All the inormation about the message m(t) in standard AM is in the sidebands. The carrier itsel does not carry any inormation, and in that respect the carrier corresponds to unnecessary power dissipation. One version o AM that cannot be detected using an envelope detector is called AM-SC or AM-DSB-SC, where SC should be interpreted as Suppressed Carrier. For this type o modulation, the constant C is simply set to zero, i.e. we have x(t) = Am(t) cos (2π c t), and the corresponding spectrum is X() = A 2 (M( c) + M( + c )). So the carrier is removed rom the spectrum, as the name suggests. In Figure 5.3, an AM-SC signal is presented together with the message and the corresponding envelope detector output, as well as the absolute value o the message. Note that the output rom an envelope detector in this case is close to the absolute value o the message. Thus, an envelope detector cannot be used to receive AM-SC. Demodulation o AM-SC can instead be done by modulating once more. Let y(t) with spectrum Y () be the output o that modulation. Then we have, similarily as above, y(t) = x(t) cos (2π c t) = Am(t) cos 2 (2π c t) = A 2 m(t) (1 + cos(4π ct)),

6 72 Chapter 5. Traditional Analog Modulation Techniques (a) (b) (c) Figure 5.3: (a) An AM-SC signal or the message m(t) = cos(2πt) with A = 1. The thick line is m(t). (b) The corresponding output rom an envelope detector. (c) The AM-SC signal together with m(t) or comparison. Message M() W W AM-SC X() c c Demodulated Y () 2 c 2 c Figure 5.4: Modulation o AM-SC and demodulation by modulating again.

7 5.1. Amplitude Modulation 73 and the corresponding spectrum is Y () = A 2 (X( c) + X( + c )) = A 2 M() + A 4 (M( 2 c) + M( + 2 c )). So, we have regained M(), but we also have copies o M() centered around ±2 c. The involved spectra are displayed in Figure 5.4. I W, the bandwidth o the message m(t), is smaller than c, which normally is the case, then those copies do not overlap with the original spectrum. Thus, we can use a suitable low-pass ilter to remove the unwanted copies. The urther away the unwanted copies are in the requency domain, the simpler that ilter can be. It should be noted that standard AM can also be demodulated using this method. Single Sideband Modulation Since there is a one-to-one relation between M() and M( ), there is also a one-to-one relation between the two sidebands, at least i W is smaller than c. So, in both standard AM and AM-SC, we actually transmit our data twice in the requency domain. No inormation is lost i we only transmit one o the sidebands. This type o AM is reerred to as SSB, which should be interpreted as Single SideBand. There are SSB versions o both standard AM and AM-SC, and they can be obtained by irst generating standard AM or AM-SC, and then using a suitable band-pass ilter to remove the unwanted sideband. Spectra o AM-SSB and AM-SSB-SC are displayed in Figure 5.5. Obviously, SSB modulation only needs hal the bandwidth compared to original AM or AM-SC. SSB-modulated signals can also be demodulated by modulating again using AM-SC, and we still get copies near ±2 c, that has to be removed by a low-pass ilter. However these copies now contain only one sideband. Synchronization or AM Demodulation Demodulation by remodulation as described above is a method that can be used or detection o all variants o AM. However, that demands that we have a correct carrier available in the demodulator, with both correct requency and at least approximately correct phase. For standard AM and or AM-SSB, where the carrier is available in the signal, it can easily be extracted rom the received signal using a narrow BP-ilter with center requency c. For AM-SC, and or AM-SSB-SC, the absence o a carrier makes it impossible to extract the carrier in that way. One way or the receiver to extract a carrier signal rom an AM-SC signal x(t) = Am(t) cos (2π c t),

8 74 Chapter 5. Traditional Analog Modulation Techniques AM-SSB X() upper sideband c c AM-SSB-SC X() upper sideband c c Figure 5.5: Spectra or AM-SSB and AM-SSB-SC. cos(2π 0 t) x(t) Narrow LP ilter cos(φ) VCO 2 cos(2π 0 t + φ) Figure 5.6: A phase-locked loop or generation o a well-deined carrier signal. The signal x(t) is given by x(t) = cos(2π 0 t)cos(2π 0 t + φ) = 1 2 (cos(4π 0t + φ) + cos(φ)). The device labelled VCO is a voltage controlled oscillator. Am(t) (cos(4π 0 t + φ) + cos(φ)) LP ilter Am(t) cos(φ) Am(t) cos(2π 0 t) 2 cos(2π 0 t + φ) VCO sin(2φ) Narrow LP ilter A 2 m 2 (t) sin(2φ) phase shit 2 sin(2π 0 t + φ) Am(t) (sin(4π 0 t + φ) + sin(φ)) LP ilter Am(t) sin(φ) Figure 5.7: A Costas loop or detection o AM-SC.

9 5.1. Amplitude Modulation 75 is to produce the square x 2 (t) = A 2 m 2 (t) cos 2 (2π c t) = A2 m 2 (t) 2 (1 + cos (4π c t)) When we send inormation, the average o m 2 (t) is non-zero, which means that a scaled version o cos (4π c t) can be extracted rom x 2 (t), again using a narrow BP-ilter, but with center requency 2 c. Extracting carriers in those ways will produce signals with a requency that is the correct carrier requency in the AM or AM-SSB case and twice the carrier requency in the AM-SC case, but the amplitude can vary rom time to time depending on the actual behaviour o the channel or the statistics o the inormation. Also, the extracted signal may include noise and parts o the sideband(s). A clean carrier with both the correct requency and a well deined amplitude, without any noise or residues rom the sidebands, can be obtained rom the extracted signal using a phase-locked loop (PLL). There are several variants o phase-locked loops in use, and a simple variant is displayed in Figure 5.6. The signals given in Figure 5.6 assume that the input is already a clean sinusoid. In practice, the input is an extracted approximate carrier which, as noted above, is polluted with noise and residues rom the sidebands. A phase-locked loop is a contol loop that produces a sinusoid with constant amplitude, the correct requency 0 and an approximate phase φ. The voltage-controlled oscillator (VCO) is chosen such that the wanted carrier requency corresponds to zero input. The requency VCO (V in ) o the output o the VCO is a unction o the input voltage V in, such that the derivative o that unction is positive, i.e. we have d dv in VCO (V in ) > 0. The carrier requency in use may dier slightly rom the wanted carrier requency, and the phase-locked loop ollows the carrier requency, which means that the input to the VCO may dier slightly rom zero. In Figure 5.6 that means that the signal produced by the PLL has phase φ or which cos(φ) 0 holds, in a point where cos(φ) has positive derivative. In other words, we have φ π + k 2π or some integer k. We may assume any integer 2 value o k, since the produced signal is the same or dierent values o k. So, we can or instance say that the signal has the phase φ π. 2 For AM-SC, where we have squared the signal, and where the requency o the extracted signal is twice the carrier requency, the eedback is equipped with a requency doubler which can or instance be a squarer. The resulting output is then a sinusoid with the correct carrier requency and phase approximately π 4. A special type o phase-locked loop that is especially well suited or detection o AM-SC signals is the Costas loop, given in Figure 5.7. It extracts the carrier directly rom the signal. Again, the VCO is chosen such that the wanted carrier requency corresponds to zero input. Thus, the loop produces a sinusoid whose requency is the carrier requency with phase φ or which sin(2φ) 0 holds, in a point where sin(φ) has positive derivative. The resulting phase is thereore φ 0. The output o the Costas loop is the message m(t) scaled by cos(φ), but since we have φ 0, we also have cos(φ) 1.

10 76 Chapter 5. Traditional Analog Modulation Techniques 2 cos(2π 0 t) x(t) + noise BP ilter x(t) + n(t) LP ilter Figure 5.8: Demodulation o AM signals in the presence o noise. The reason that the Costas loop works is the presence o both sidebands, and the one-toone mapping between the two sidebands. The two sidebands point at the carrier requency, and the phase inormation in the sidebands gives us the carrier phase. The Costas loop can thereore not be used or AM-SSB-SC. There is simply no way to extract the carrier rom an AM-SSB-SC signal, due to the act that the carrier is not available in the signal, and nothing in the spectrum gives any hint about the carrier requency. That is the price we have to pay or suppressing both the carrier and one o the sidebands. Thereore, there are a number o modiications o AM-SSB-SC, that makes it possible to extract a carrier anyway. The most simple method is not to suppress the carrier completely. Then the however weak carrier can be extracted rom the signal. Another method is to send a short carrier burst, and let a PLL lock on to that burst. Ater the burst, the oscillator continues producing an internal carrier based on that burst. O course, the oscillator will most probably diverge rom the used carrier eventually. Thereore the carrier burst is repeated regularly. A third possibility is to keep a small part o the removed sideband, and in the receiver ilter the other sideband similarily, and then extract a carrier using one o the methods above. Impact o Noise in AM Demodulation We would like to analyze the impact o noise on demodulation o AM signals. For this analysis we need to make some assumptions about the noise and about the demodulation. The irst assumption is that the noise is dominated by thermal noise, and that it is independent o the message. As mentioned in Section 4.1, such noise can be modeled as white Gaussian noise. We assume that the received signal is iltered by an ideal BP ilter that exactly matches the bandwidth o the AM signal beore demodulation. We assume that the demodulation is done by remodulation by 2 cos(2π 0 t) as in the Costas loop. We also assume that the demodulated signal is iltered by an ideal LP ilter that exactly matches the bandwidth o the message. See Figure 5.8. We need to introduce some notation. Let W denote the bandwidth o the message. Let N 0 denote the one-sided power spectral density o the assumed white Gaussian noise, and let n(t) denote the noise ater the BP ilter. Also, introduce the ollowing notation or the involved powers.

11 5.1. Amplitude Modulation 77 P: The (expected) power o the message m(t). P m mod : The (expected) power o the received modulated signal x(t). Note that this means that A includes impacts o the channel. P m : The (expected) power o the message ater demodulation and LP ilter. P n mod : The expected power o the ideally BP-iltered noise n(t) beore demodulation. P n : The expected power o the demodulated and LP-iltered noise. We deine the signal-to-noise ratio P m /P n ater demodulation. We will compare this signalto-noise ratio or DSB and SSB modulation using the same sent power P m mod transmitted over a channel with the same N 0. First we consider AM-SC. Then we have the signal x(t) = Am(t) cos (2π c t) with bandwidth 2W and expected power P m mod = A 2 P/2 since the carrier cos (2π c t) has average power 1/2. Ater demodulation, we regain Am(t), which means that we have P m = A 2 P. For the noise, we have P n mod = 2WN 0. The demodulated noise n(t) 2 cos (2π c t) has expected power 2P n mod, since the carrier 2 cos (2π c t) has average power 2. Hal o that expected power is in the requency interval < W, while the other hal is in the requency interval 2 0 W < < W. The latter part is removed by the LP ilter, leaving us with P n = P n mod. Finally, that gives us the signal-to-noise ratio P m P n = A2 P 2WN 0 For AM-SSB-SC, one o the sidebands rom AM-SC is removed, which means that the power P m mod is reduced to hal that o AM-SC. To produce an SSB signal with the same power as in the DSB case, we thereore need to ampliy the signal by 2. So, we start with x(t) = 2Am(t) cos (2π c t), and ilter out one o the sidebands. Then we have the same sent power P m mod = A 2 P/2. Ater demodulation, we get a scaled version o the message. More precisely, the output is A 2 m(t), which has power P m = A 2 P/2. For the noise, we have P n mod = WN 0, since the bandwidth is W. The demodulated noise n(t) 2 cos (2π c t)

12 78 Chapter 5. Traditional Analog Modulation Techniques still has expected power 2P n mod, since the carrier 2 cos (2π c t) has average power 2. Hal o that expected power is in the requency interval < W, while the other hal is in the requency interval 2 0 W < < W. Actually, the other hal o the power is in the interval 2 0 W < < 2 0 i the lower sideband is used, or in the interval 2 0 < < W i the upper sideband is used. In any case, the part o the spectrum that is near 2 0 is removed by the LP ilter, leaving us with P n = P n mod. Finally, that gives us the signal-to-noise ratio P m = A2 P, P n 2WN 0 i.e. the same signal-to-noise ratio as or DSB. 5.2 Angle Modulation Angle modulation is the common name or a class o modulation techniques, with that in common that the bandwidth o the modulated signal is not given only by the bandwidth o the message, but also by a parameter called the modulation index. By setting this modulation index to a suitable number, we can decide what bandwidth to use, and the larger that index is, the better is the obtained quality. These methods, and combinations o them are used in radio broadcasts in the FM-band ( MHz). The idea that angle modulation is based on is to let a unction φ(m(t)) o the message m(t) be the phase o a carrier, i.e. the sent signal is x(t) = A cos (2π c t + φ(m(t))), where A is some non-zero constant, and where c again is reerred to as the carrier requency. We say that φ(m(t)) is the momentary phase o x(t). Then the phase deviation φ d (t) is the dierence between the momentary phase and the average o the momentary phase. Typically, m(t) has average zero, and the unction is chosen such that φ(m(t)) also has average 0. Then we have and the peak phase deviation is deined as φ d (t) = φ(m(t)), φ d,max = max φ d (t). The peak phase deviation is also called the phase modulation index, and is denoted µ p. An alternative interpretation o the varying phase o the signal x(t), is to say that x(t) has varying requency. We deine the momentary requency as mom (t) = 1 2π d dt (2π ct + φ(m(t))) = c + 1 2π d dt φ(m(t)).

13 5.2. Angle Modulation 79 Note, that this requency is a unction o time, just as the momentary phase also is a unction o time. The requency deviation d (t) is the dierence between the momentary requency and the carrier requency, i.e. d (t) = mom (t) c, and the peak requency deviation is deined as d,max = max d (t). The requency modulation index µ is deined as µ = d,max W, where W is the bandwidth o the message m(t), or rather the highest requency component in m(t). I m(t) is a stationary sine, then the two modulation indices are equal. Spectrum o Angle Modulation The spectra o angle modulated signals are generally hard to determine, due to the act that angle modulation is non-linear. For the simple example x(t) = A cos (2π c t + µ sin(2π m t)), or m c, it is easily shown that µ is both the phase modulation index and the requency modulation index o that signal. It can also be shown that we have x(t) = n= A J n (µ) cos (2π( c + n m )t), where J n (µ) is the Bessel unction o order n. We will not at all try to perorm that proo. The spectrum o that signal is X() = n= A J n (µ) 2 The Bessel unction o order n is given by [δ( + c + n m ) + δ( c n m )]. J n (µ) = k=0 ( 1) k ( µ ) n+2k k!(n + k)! 2 or positive integers n. It can also be written as

14 80 Chapter 5. Traditional Analog Modulation Techniques 1 J 0 (µ) J1 (µ) J2 (µ) J3 (µ) J4(µ) J5(µ) J6(µ) J7(µ) µ Figure 5.9: Bessel unctions J n (µ) or n up to 7. µ = 1 m c µ = 5 c µ = 10 c Figure 5.10: Spectrum o an angle modulated signal with modulation index µ, carrier requency c, a cosine message with requency m. The hights o the arrows denoting impulses represent the amplitude o the corresponding requencies.

15 5.2. Angle Modulation 81 π J n (µ) = 1 cos(µ sin(φ) nφ)dφ, π 0 still or positive integers n. For negative integers n, we have J n (µ) = ( 1) n J n (µ). Formally, the bandwidth o this signal is ininite. However, the coeicients J n (µ) decrease rapidly towards 0 or n > µ, see Figure 5.9. Thus, we can state that the bandwidth o the signal is approximately 2µ m. A common approximation o the bandwidth is 2(µ + 1) m, which is known as Carson s rule. Using the identity µ = d,max / m, we can express the bandwidth as 2 ( µ ) d,max. In Figure 5.10, we have plotted the spectra or three dierent values o the modulation index µ, with a sine shaped message. Note how the bandwidth grows with the modulation index. Phase Modulation Phase modulation is normally abbreviated PM or PhM. The message, m(t), is in this technique used directly to determine the momentary phase, i.e. we have φ(m(t)) = a m(t), where a is some constant. The modulated signal, x(t), is thus given by x(t) = A cos (2π c t + a m(t)). The momentary requency or this modulation is mom (t) = 1 2π d dt (2π ct + a m(t)) = c + a 2π d dt m(t), the requency deviation is d (t) = a 2π d dt m(t). and the peak requency deviation is d,max = a 2π max d dt m(t). Finally, the requency modulation index is given by µ = a 2πW max d dt m(t). We notice that the peak requency deviation depends on a max d dt m(t). Hence, the bandwidth o x(t) depends on the bandwidth o m(t), but also on the amplitude o m(t). In Figure 5.11a, a PM signal is presented together with the corresponding message.

16 82 Chapter 5. Traditional Analog Modulation Techniques Frequency Modulation Frequency modulation is normally abbreviated FM. As or PM, the message, m(t), determines the phase o the carrier, but not directly. Instead, the derivative o the phase is proportional to m(t), i.e. the phase is a scaled indeinite integral o m(t). More precisely, we have φ(m(t)) = a m(t) dt, where a is some constant. The modulated signal, x(t), is then given by ( x(t) = cos 2π c t + a ) m(t) dt, and the momentary requency is given by mom (t) = 1 2π d ( 2π c t + a dt ) m(t) dt = c + a 2π m(t). Thus, the momentary requency is directly given by the message. Note that any indeinite integral o m(t) can be used as the phase. A natural choice is t φ(m(t)) = a m(τ)dτ, t 0 where t 0 is the time instance when the communication starts. The requency deviation is and the peak requency deviation is d (t) = a 2π m(t). d,max = a max m(t). 2π Finally, the requency modulation index is given by µ = a max m(t). 2πB We notice that the requency deviation depends on max m(t). Hence, the bandwidth o x(t) depends on the amplitude o m(t), but not on the bandwidth o m(t). In Figure 5.11b, an FM signal is presented together with the corresponding message.

17 5.2. Angle Modulation (a) (b) Figure 5.11: (a) A PM signal (thin line) or the message m(t) = cos(2πt) (thick line). (b) An FM signal (thin line) or the message m(t) = cos(2πt) (thick line). The modulation index is in both cases 10 and we have A = 1. Demodulation o PM and FM Recall that the sent signal is x(t) = A cos (2π c t + φ(m(t))). This signal can be demodulated by determining the derivative o the signal, d x(t) = A (2π c + ddt ) dt φ(m(t)) sin (2π c t + φ(m(t))). This gives us a signal or which the amplitude depends on the message m(t) in a way similar to AM-DSB, but its carrier has varying phase. This signal can then be demodulated using an envelope detector, which gives us the envelope A (2π c + ddt ) φ(m(t)). The constant term can be removed using a BP or HP ilter, leaving us with the signal A d φ(m(t)). For PM, we have φ(m(t)) = am(t). This means that we nead to integrate the dt signal A d φ(m(t)) to get the wanted message, i.e. we must produce dt t t 0 A d dτ φ(m(τ))dτ = Aa (m(t) m(t 0)), where t 0 is the time instance when the communication started. For FM, we have φ(m(t)) = a m(t) dt.

18 84 Chapter 5. Traditional Analog Modulation Techniques x(t) d dt Envelope detector BP ilter t t 0 A m(t) Figure 5.12: Demodulation o PM. x(t) d dt Envelope detector BP ilter A m(t) Figure 5.13: Demodulation o FM. x(t) sgn(x) d dt x LP ilter mom (t) Figure 5.14: Detection o momentary requency in angle modulated signals using zero crossings. Then we have the signal A d dt φ(m(t)) = A d dt a m(t)dt = Aa m(t). Thereore, demodulation o PM and FM can be done as indicated in Figures 5.12 and 5.13, respectively. Alternatively, PM and FM can be demodulated by extracting the momentary requency o the modulated signal x(t). That can be done by detecting the zero crossings o x(t). The demodulator in Figure 5.14 is based on this approach. The irst block outputs the sign o its input, i.e. its output is sgn(x(t)). The sgn unction is deined as sgn(x) = 1, x > 0, 0, x = 0, -1, x < 0. In practice, this block is an ampliier with very high gain. The second block produces the derivative o its input. The result is that the output o the second block is a positive (negative) impulse when x(t) passes zero with positive (negative) derivative. Ater the third block, which is a rectiier, all those impulses are positive, with momentary requency that is twice the momentary requency o x(t). All that is let to produce the message is an LP ilter, which is the last block in Figure This produces a signal that is proportional to the momentary requency mom (t) o x(t). For FM this is essentially the message m(t), and or PM this is essentially d m(t). All we have let to do is to remove the DC component dt

19 5.2. Angle Modulation 85 that originates rom the carrier requency, using a HP ilter, or by replacing the LP ilter in Figure 5.14 by a BP ilter. For PM, we also have to integrate the output to get the message. Impact o Noise in PM and FM Demodulation We assume that the demodulation is carried out as described above, and we assume that the noise is dominated by white Gaussian noise with one-sided power spectral density N 0. We use the same notation or the involved powers as we did or AM demodulation. P: The (expected) power o the message m(t). P m mod : The average power o the modulated signal x(t). P m : The (expected) power o the message ater demodulation and LP ilter. P n mod : The expected power o the ideally BP-iltered noise beore demodulation. P n : The expected power o the demodulated and LP-iltered noise. The signal x(t) is a cosine with amplitude A. The average power P m mod o the modulated signal x(t) is thereore given by P m mod = A 2 /2. The varying phase - or requency or that matter - is irrelevant in this respect. For both PM and FM, the demodulated signal is Aa m(t), which gives us P m = A 2 a 2 P. The bandwidth o x(t) is approximately 2 d,max. The expected power P n mod o the ideally BP-iltered noise beore demodulation is thereore P n mod = 2 d,max N 0 /2 = d,max N 0. Angle modulation methods are non-linear. That makes the analysis o detection in the presence o noise a lot more complicated than or AM. We simply skip that analysis and state the noise power or the two cases, under the assumption that the signal-to-noise ratio on the channel P m mod /P n mod is high. We start with PM. Then it can be shown that the noise power is given by This gives us the signal-to-noise ratio P n = 2WN 0. P m P n = A2 a 2 P 2WN 0. We would like to express this signal-to-noise ratio using the phase modulation index µ p. We get µ p = max φ(m(t)) = a max m(t),

20 86 Chapter 5. Traditional Analog Modulation Techniques rom which we get a = µ p max m(t). We use this relation to rewrite the signal-to-noise ration as ( ) 2 P m µ p A 2 P =. P n max m(t) 2WN 0 As we can see, we get increased signal-to-noise ratio with increased phase modulation index. Now we turn to FM. It can be shown that the noise power is given by This gives us the signal-to-noise ratio P n = 2W 3 N 0. 3 P m P n = A2 a 2 P 2W 3 N 0 /3. Now we would like to express the signal-to-noise ratio using the requency modulation index µ. We have already noted that we have rom which we get µ = a W = a max m(t), 2πW 2πµ max m(t). We use this relation to rewrite the signal-to-noise ratio as ( ) 2 P m = 12π 2 µ A 2 P. P n max m(t) 2WN 0 Here we get increased signal-to-noise ratio with increased requency modulation index. Pre-emphasized FM The resulting noise ater demodulating PM signals is evenly distributed over requencies rom 0 to W, which resembles the situation or AM. That is not the case or FM, where instead the resulting noise is dominated by high requencies (near W). More precisely, the power spectral density o the resulting noise ater demodulating FM as described above is 2N 0 2, where is requency. Thereore, i we could combine PM and FM in such a way that PM is used or high requencies in the message m(t) and FM is used or low requencies,

21 5.2. Angle Modulation 87 we could hope or reduced noise compared to any o the two methods by themselves. Such a combination exists and is called pre-emphasized FM. Then the message is irst iltered using a pre-emphasis ilter with requency response H 1 () = 1 + j/ 0. The output o that ilter is then requency modulated. In the receiver, ater ordinary demodulation o the FM signal, the result is iltered with an inverse ilter o H 1 () called a de-emphasis ilter. That ilter has requency response H 2 () = 1 H 1 () = j/ 0. The result is that we regain the original signal, and that the resulting noise is smaller than i we would have used ordinary PM or FM. The transmissions on the so called FM band ( MHz) are done using pre-emphasized FM with 0 = 2122 Hz.