' 2.4 Frequency and Phase Modulation
FIGURE2.18 Phase shift due to
noise
phase shift in radians. It would seem that the ratio of signal voltage to noise voltage at the output would be proportional to mƒ, and this is approximately true under strong-signal conditions.
EXAMPLE2.12
Y
An FM signal has a frequency deviation of 5 kHz and a modulating frequency of 1 kHz. The signal-to-noise ratio at the input to the receiver detector is 20 dB.
Calculate the approximate signal-to-noise ratio at the detector output.
SOLUTION
First, notice the word “approximate.” Our analysis is obviously a little sim-plistic, since noise exists at more than one frequency. We are also going to as-sume that the detector is completely unresponsive to amplitude variations and that it adds no noise of its own. Our results will not be precise but they will show the process that is involved.
First, let us convert 20 dB to a voltage ratio.
E
Remembering that the receiver will interpret the noise as an FM signal with a modulation index equal toφN, we find
mƒN = 0.1
This can be converted into an equivalent frequency deviationδNdue to the noise.
δN = mƒƒm
= 0.1×1 kHz
= 100 Hz
The frequency deviation due to the signal is given as 5 kHz, and the re-ceiver output voltage is proportional to the deviation. Therefore, the output S/N as a voltage ratio will be equal to the ratio between the deviation due to the signal and that due to the noise.
E
Since S/N is nearly always expressed in decibels, change this to dB.
(S/N)o(dB) = 20 log 50
= 34 dB
This is an improvement of 14 dB over the S/N at the input.
X
Threshold Effect and Capture Effect
An FM signal can produce a better signal-to-noise ratio at the output of a receiver than an AM signal with a similar input S/N, but this is not always the case. The superior noise performance of FM depends on there being a sufficient input S/N ratio. There exists a threshold S/N below which the per-formance is no better than AM. In fact, it is worse, because the greater band-width of the FM signal requires a wider receiver noise bandband-width. When the signal strength is above the threshold, the improvement in noise perfor-mance for FM can be more than 20 dB compared with AM.
The noise-rejection characteristic of FM applies equally well to ence. As long as the desired signal is considerably stronger than the interfer-ence, the ratio of desired to interfering signal strength will be greater at the output of the detector than at the input. We could say that the stronger sig-nal “captures” the receiver, and in fact this property of FM is usually called the capture effect. It is very easy to demonstrate with any FM system. For ex-ample, it is the reason that there is less interference between cordless tele-phones, which share a few channels in the 46- and 49-MHz bands, than one might expect.
Pre-emphasis and De-emphasis
An FM receiver interprets the phase shifts due to noise as frequency modula-tion. Phase and frequency deviation are related by Equation (2.19):
mƒ = ƒ
δ
m
which can be restated δ = mƒƒm
The modulation index mƒis simply the peak phase deviation in radians.
The frequency deviation is proportional to the modulating frequency. This tells us, if the phase deviation due to thermal noise is randomly distributed over the baseband spectrum, the amplitude of the demodulated noise will be proportional to frequency. This relationship between noise voltage and frequency is shown in Figure 2.19. Since power is proportional to the square of voltage, the noise power will have the parabolic spectrum shown in Figure 2.19. An improvement in S/N can be made by boosting (pre-emphasizing) the higher frequencies in the baseband signal before modula-tion, with a corresponding cut in the receiver after demodulation. Obviously it is necessary to use similar filter characteristics for pre- and de-emphasis.
Usually simple first-order filters are used.
Note that pre-emphasis and de-emphasis are unnecessary with phase modulation. Since the phase deviation due to noise is converted directly into baseband noise output without the intermediate step of conversion into an equivalent frequency deviation, the output noise will have a flat spectrum, assuming thermal noise at the input to the demodulator.
FIGURE2.19
Spectrum of demodulated noise
Summary
'
The main points to remember from this chapter are:( In the time domain, the process of amplitude modulation creates a signal with an envelope that closely resembles the original information signal.
( In the frequency domain, an amplitude-modulated signal consists of the carrier, which is unchanged from its unmodulated state, and two side-bands. The total bandwidth of the signal is twice the maximum modulat-ing frequency.
( An amplitumodulated signal can be demodulated by an envelope de-tector, which consists of a diode followed by a lowpass filter.
( The peak voltage of an amplitude-modulated signal varies with the modu-lation index, becoming twice that of the unmodulated carrier for the maximum modulation index of 1.
( The power in an amplitude-modulated signal increases with modulation.
The extra power goes into the sidebands. At maximum modulation, the total power is 50% greater than the power in the unmodulated carrier.
( Angle modulation includes frequency and phase modulation, which are closely related.
( Frequency modulation is widely used for analog communication, while phase modulation sees greatest application in data communication.
( The power of an angle-modulation signal does not change with modula-tion, but the bandwidth increases due to the generation of multiple sets of sidebands.
( The voltage and power of each sideband can be calculated using Bessel functions. An approximate bandwidth is given by Carson’s rule.
( Frequency modulation has a significant advantage compared with AM in the presence of noise or interference, provided the deviation is relatively large and the signal is reasonably strong.
( The signal-to-noise ratio for FM can be improved considerably by using pre-emphasis and de-emphasis. This involves greater gain for the higher baseband frequencies before modulation, with a corresponding reduc-tion after demodulareduc-tion.
( Equation List
v(t)=(Ec+Emsinωmt) sinωct (2.2)
m=Em/Ec (2.3)
v(t)=Ec(1+m sinωmt) sinωct (2.4)
mT = m1 +m + ⋅ ⋅ ⋅
B≅2[δ(max)+ ƒm(max)] (2.22) φN
N
S
E
≈ E (2.24)
( Key Terms
angle modulation term that applies to both frequency modulation (FM) and phase modulation (PM) of a transmitted signal
capture effect tendency of an FM receiver to receive the strongest signal and reject others
deviation in FM, the peak amount by which the instantaneous signal frequency differs from the carrier frequency in each deviation envelope imaginary pattern formed by connecting the peaks of
individual RF waveforms in an amplitude-modulated signal frequency modulation modulation scheme in which the transmitted
frequency varies in accordance with the instantaneous amplitude of the information signal
frequency modulation index peak phase shift in a frequency-modulated signal, in radians
modulation index number indicating the degree to which a signal is modulated
overmodulation modulation to an extent greater than that allowed for either technical or regulatory reasons
phase modulation communication system in which the phase of a high-frequency carrier is varied according to the amplitude of the baseband (information) signal
side frequencies frequency components produced above and below the carrier frequency by the process of modulation
sideband a group of side frequencies above or below the carrier frequency
splatter frequency components produced by a transmitter that fall outside its assigned channel
( Questions
1. What is meant by the “envelope” of an AM waveform, and what is its significance?
2. Although amplitude modulation certainly involves changing the am-plitude of the signal; it is not true to say that the amam-plitude of the carrier is modulated. Explain this statement.
3. Why is it desirable to have the modulation index of an AM signal as large as possible, without overmodulating?
4. Describe what happens when a typical AM modulator is over-modulated, and explain why overmodulation is undesirable.
5. How does the bandwidth of an AM signal relate to the information signal?
6. Describe two ways in which the modulation index of an AM signal can be measured.
7. By how much does the power in an AM signal increase with modula-tion, compared to the power of the unmodulated carrier?
8. What two types of modulation are included in the term “angle modu-lation”?
9. Compare, in general terms, the bandwidth and signal-to-noise ratio of FM and AM.
10. Describe and compare two ways to determine the practical bandwidth of an FM signal.
11. What is pre-emphasis and how is it used to improve the signal-to-noise ratio of FM transmissions?
12. For FM, what characteristic of the modulating signal determines the in-stantaneous frequency deviation?
13. What is the capture effect?
14. Where is phase modulation used?
15. Explain why the signal-to-noise ratio of FM can increase with the band-width. Is this always true for FM? Compare with the situation for AM.
16. Compare the effects of modulation on the carrier power and the total signal power in FM and AM.
17. What is the threshold effect?
18. Explain how limiting reduces the effect of noise on FM signals.
19. Explain how noise affects FM signals even after limiting.
20. Explain the fact that there is no simple relationship between modulat-ing frequency and bandwidth for an FM signal.
21. Why does limiting the receiver bandwidth to less than the signal band-width cause more problems with FM than with AM?
( Problems
1. An AM signal has the equation:
v(t)=(15+4 sin 44×103t) sin 46.5×106t volts (a) Find the carrier frequency.
(b) Find the frequency of the modulating signal.
(c) Find the value of m.
(d) Find the peak voltage of the unmodulated carrier.
(e) Sketch the signal in the time domain showing voltage and time scales.
2. An AM signal has a carrier frequency of 3 MHz and an amplitude of 5 V peak. It is modulated by a sine wave with a frequency of 500 Hz and a peak voltage of 2 V. Write the equation for this signal and calculate the modulation index.
3. An AM signal consists of a 10-MHz carrier modulated by a 5-kHz sine wave. It has a maximum positive envelope voltage of 12 V and a mini-mum of 4 V.
(a) Find the peak voltage of the unmodulated carrier.
(b) Find the modulation index and percent.
(c) Sketch the envelope.
(d) Write the equation for the signal voltage as a function of time.
4. An AM transmitter is modulated by two sine waves, at 1 kHz and 2.5 kHz, with a modulation due to each of 25% and 50% respectively.
What is the effective modulation index?
5. For the AM signal sketched in Figure 2.20, calculate:
(a) the modulation index (b) the peak carrier voltage
(c) the peak modulating-signal voltage 6. For the signal of Figure 2.21, calculate:
(a) the index of modulation
(b) the RMS voltage of the carrier without modulation
7. An audio system requires a frequency response from 50 Hz to15 kHz for high fidelity. If this signal were transmitted using AM, what bandwidth would it require?
8. A transmitter operates with a carrier frequency of 7.2 MHz. It is ampli-tude modulated by two tones with frequencies of 1500 Hz and 3000 Hz.
What frequencies are produced at its output?
9. Sketch the signal whose equation is given in Problem 1 in the frequency domain, showing frequency and voltage scales.
10. An AM signal has the following characteristics:
ƒc=150 MHz ƒm=3 kHz Ec=50 V Em=40 V For this signal, find:
(a) the modulation index (b) the bandwidth
(c) the peak voltage of the upper sideband
11. An AM signal observed on a spectrum analyzer shows a carrier at+12 dBm, with each of the sidebands 8 dB below the carrier. Calculate:
(a) the carrier power in milliwatts (b) the modulation index
12. An AM transmitter with a carrier power of 10 W at a frequency of 25 MHz operates into a 50-Ωload. It is modulated at 60% by a 2-kHz sine wave.
(a) Sketch the signal in the frequency domain. Show power and fre-quency scales. The power scale should be in dBm.
(b) What is the total signal power?
(c) What is the RMS voltage of the signal?
FIGURE2.20 FIGURE2.21
13. A 5-MHz carrier is modulated by a 5-kHz sine wave. Sketch the result in both frequency and time domains for each of the following types of modulation. Time and frequency scales are required, but amplitude scales are not.
(a) DSB full-carrier AM (b) DSBSC AM
(c) SSBSC AM (USB)
14. If a transmitter power of 100 W is sufficient for reliable communication over a certain path using SSB, approximately what power level would be required using:
(a) DSBSC
(b) full-carrier AM
15. An AM transmitter has a carrier power of 50 W at a carrier frequency of 12 MHz. It is modulated at 80% by a 1-kHz sine wave.
(a) How much power is contained in the sidebands?
(b) Suppose the transmitter in part (a) can also be used to transmit a USB signal with an average power level of 50 watts. By how much (in dB) will the signal-to-noise ratio be improved when the trans-mitter is used in this way, compared with the situation in part (a)?
16. An FM signal has a deviation of 10 kHz and a modulating frequency of 2 kHz. Calculate the modulation index.
17. Calculate the frequency deviation for an FM signal with a modulating frequency of 5 kHz and a modulation index of 2.
18. A sine-wave carrier at 100 MHz is modulated by a 1-kHz sine wave. The deviation is 100 kHz. Draw a graph showing the variation of instanta-neous modulated signal frequency with time.
19. A phase-modulated signal has a modulation index of 2 with a modulat-ing signal havmodulat-ing an amplitude of 100 mV and a frequency of 4 kHz.
What would be the effect on the modulation index of:
(a) changing the frequency to 5 kHz (b) changing the voltage to 200 mV
20. An FM signal has a deviation of 10 kHz and is modulated by a sine wave a frequency of 5 kHz. The carrier frequency is 150 MHz, and the signal has a total power of 12.5 W, operating into an impedance of 50Ω.
(a) What is the modulation index?
(b) How much power is present at the carrier frequency?
(c) What is the voltage level of the second sideband below the carrier frequency?
(d) What is the bandwidth of the signal, ignoring all components which have less than 1% of the total signal voltage?
21. An FM transmitter operates with a total power of 10 watts, a deviation of 5 kHz, and a modulation index of 2.
(a) What is the modulating frequency?
(b) How much power is transmitted at the carrier frequency?
(c) If a receiver has a bandwidth sufficient to include the carrier and the first two sets of sidebands, what percentage of the total signal power will it receive?
22. An FM transmitter has a carrier frequency of 220 MHz. Its modulation index is 3 with a modulating frequency of 5 kHz. The total power output is 100 watts into a 50Ωload.
(a) What is the deviation?
(b) Sketch the spectrum of this signal, including all sidebands with more than 1% of the signal voltage.
(c) What is the bandwidth of this signal according to the criterion used in part (b)?
(d) Use Carson’s rule to calculate the bandwidth of this signal, and compare with the result found in part (c).
23. An FM transmitter has a carrier frequency of 160 MHz. The deviation is 10 kHz and the modulation frequency is 2 kHz. A spectrum analyzer shows that the carrier-frequency component of the signal has a power of 5 W. What is the total signal power?
24. Use Carson’s rule to compare the bandwidth that would be required to transmit a baseband signal with a frequency range from 300 Hz to 3 kHz using:
(a) narrowband FM with maximum deviation of 5 kHz (b) wideband FM with maximum deviation of 75 kHz
25. An FM receiver operates with a signal-to-noise ratio of 30 dB at its detec-tor input and is operating with mƒ=10.
(a) If the received signal has a voltage of 10 mV, what is the amplitude of the noise voltage?
(b) Find the maximum phase shift that could be given to the signal by the noise voltage.
(c) Calculate the signal-to-noise ratio at the detector output, assuming the detector is completely insensitive to amplitude variations.
26. A certain full-carrier DSB AM signal has a bandwidth of 20 kHz. What would be the approximate bandwidth required if the same information signal were to be transmitted using:
(a) DSBSC AM (b) SSB
(c) FM with 10 kHz deviation (d) FM with 50 kHz deviation
27. Using the table of Bessel functions, demonstrate that the total power in an FM signal is equal to the power in the unmodulated carrier for m=2.
Compare with the situation for full-carrier AM and for SSBSC AM.
28. Suppose you were called upon to recommend a modulation technique for a new communication system for voice frequencies. State which of the techniques studied so far you would recommend, and why, in each of the following situations:
(a) simple, cheap receiver design is of greatest importance (b) narrow signal bandwidth is of greatest importance
(c) immunity to noise and interference is of greatest importance