As an example of a digital TDM system, we consider a multiplexing scheme common to many telephone systems. The sampling format is illustrated in Figure 3.31(a). A voice signal is sampled at 8000 samples per second, and each sample is quantized into seven binary digits. An additional binary digit, known as asignaling bit, is added to the basic seven bits that represent the sample value. The signaling bit is used in establishing calls and for synchronization. Thus, eight bits are transmitted for each sample value, yielding a bit rate of 64,000 bit/s (64 kbps).
Twenty-four of these 64-kbps voice channels are grouped together to yield a T1 carrier. The T1 frame consists of24(8) + 1 = 193bits. The extra bit is used for frame synchronization. The
frame duration is the reciprocal of the fundamental sampling frequency, or 0.125 ms. Since the frame rate is 8000 frames per second, with 193 bits per frame, the T1 data rate is 1.544 Mbps.
As shown in Figure 3.31(b), four T1 carriers can be multiplexed to yield a T2 carrier, which consists of 96 voice channels. Seven T2 carriers yield a T3 carrier, and six T3 carriers yield a T4 carrier. The bit rate of a T4 channel, consisting of 4032 voice channels with signaling bits and framing bits, is 274.176 Mbps. A T1 link is typically used for short transmission distances in areas of heavy usage. T4 and T5 channels are used for long transmission distances.
Further Reading
One can find basic treatments of modulation theory at about the same technical level of this text in a wide variety of books. Several selected examples are Carlson and Crilly (2009), Haykin and Moher (2009), Lathi and Ding (2009), and Couch (2013).
Summary
1. Modulation is the process by which a parameter of a carrier is varied in one-to-one correspondence with an information-bearing signal usually referred to as themes- sage. Several uses of modulation are to achieve efficient transmission, to allocate channels, and for multiplexing.
2. If the carrier is continuous, the modulation is continuous-wave modulation. If the carrier is a sequence of pulses, the modulation is pulse modulation.
3. There are two basic types of continuous-wave mod- ulation: linear modulation and angle modulation.
4. Assume that a general modulated carrier is given by 𝑥𝑐(𝑡) = 𝐴(𝑡) cos[2𝜋𝑓𝑐𝑡 + 𝜙(𝑡)]
If𝐴(𝑡)is proportional to the message signal, the result is linear modulation. If𝜙(𝑡)is proportional to the message signal, the result is PM. If the time derivative of𝜙(𝑡)is proportional to the message signal, the result is FM. Both PM and FM are examples of angle modulation. Angle modulation is a nonlinear process.
5. The simplest example of linear modulation is DSB.
Double sideband is implemented as a simple product de- vice, and coherent demodulation must be used, where co- herent demodulation means that a local reference at the receiver that is of the same frequency and phase as the incoming carrier is used in demodulation.
6. An AM signal is formed by adding a carrier com- ponent to a DSB signal. This is a useful modulation tech- nique because it allows simple envelope detection to be used for implementation very simple, and inexpensive, receivers.
7. The efficiency of a modulation process is defined as the percentage of total power that conveys information.
For AM, this is given by 𝐸 = 𝑎2⟨𝑚2𝑛(𝑡)⟩
1 + 𝑎2⟨𝑚2𝑛(𝑡)⟩(100%)
where the parameter𝑎is themodulation index and𝑚𝑛(𝑡) is𝑚(𝑡)normalized so that the negative peak value is unity.
If envelope demodulation is used, the index must be less than unity.
8. The modulation trapezoid provides a simple tech- nique for monitoring the modulation index of an AM sig- nal. It also provides a visual indication of the linearity of the modulator and transmitter.
9. An SSB signal is generated by transmitting only one of the sidebands in a DSB signal. Single-sideband signals are generated either by sideband filtering a DSB signal or by using a phase-shift modulator. Single-sideband signals can be written as
𝑥𝑐(𝑡) = 1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) ±1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) in which the plus sign is used for lower-sideband SSB and the minus sign is used for upper-sideband SSB. These sig- nals can be demodulated either through the use of coherent demodulation or through the use of carrier reinsertion.
10. Vestigial sideband results when a vestige of one side- band appears on an otherwise SSB signal. Vestigial side- band is easier to generate than SSB. Either coherent de- modulation or carrier reinsertion can be used for message recovery.
11. Frequency translation is accomplished by multiply- ing a signal by a carrier and filtering. These systems are known as mixers.
12. The concept of mixing has many applications in- cluding the implementation of superheterodyne receivers.
Mixing results inimage frequencies, which can be trou- blesome if not removed by filtering.
13. Interference, the presence of undesired signal com- ponents, can be a problem in demodulation. Interference at the input of a demodulator results in undesired components at the demodulator output. If the interference is large and if the demodulator is nonlinear, thresholding can occur.
The result of this is a drastic loss of the signal component.
14. Pulse-amplitude modulation results when the ampli- tude of each carrier pulse is proportional to the value of the message signal at each sampling instant. Pulse-amplitude modulation is essentially a sample-and-hold operation. De- modulation of PAM is accomplished by lowpass filtering.
15. Digital pulse modulation results when the sample values of the message signal are quantized and encoded prior to transmission.
16. Delta modulation is an easily implemented form of digital pulse modulation. In DM, the message signal is encoded into a sequence of binary symbols. The binary symbols are represented by the polarity of impulse func- tions at the modulator output. Demodulation is ideally accomplished by integration, but lowpass filtering is often a simple and satisfactory substitute.
17. Pulse-code modulation results when the message sig- nal is sampled and quantized, and each quantized sample value is encoded as a sequence of binary symbols. Pulse- code modulation differs from DM in that in PCM each quantized sample value is transmitted but in DM the trans- mitted quantity is the polarity of the change in the message signal from one sample to the next.
18. Multiplexing is a scheme allowing two or more mes- sage signals to be communicated simultaneously using a single system.
19. Frequency-division multiplexing results when si- multaneous transmission is accomplished by translating message spectra, using modulation tononoverlapping lo- cations in a baseband spectrum. The baseband signal is then transmitted using any carrier modulation method.
20. Quadrature multiplexing results when two message signals are translated, using linear modulation with quadra- ture carriers, to the same spectral locations. Demodulation is accomplished coherently using quadrature demodula- tion carriers. A phase error in a demodulation carrier re- sults in serious distortion of the demodulated signal. This distortion has two components: a time-varying attenua- tion of the desired output signal and crosstalk from the quadrature channel.
21. Time-division multiplexing results when samples from two or more data sources are interlaced, using com- mutation, to form a baseband signal. Demultiplexing is accomplished by using a second commutator, which must be synchronous with the multiplexing commutator.
Drill Problems
3.1 A DSB signal has the message signal 𝑚(𝑡) = 3 cos(40𝜋𝑡) + 7 sin(64𝜋𝑡) The unmodulated carrier is given by
𝑐(𝑡) = 40 cos(2000𝜋𝑡)
Determine the frequencies of the upper-sideband compo- nents, the frequencies of the lower-sideband components, and the total transmitted power.
3.2 Using the same message signal and unmodulated carrier as given in the previous problem, and assuming that the modulation technique is AM, determine the modula- tion index and the efficiency.
3.3 An AM system operates with𝐴𝑐= 100 and𝑎 = 0.8. Sketch and fully dimension the modulation trapezoid.
3.4 Sketch and fully dimension the modulation trape- zoid for AM with𝑎 > 1. Write the equation for determing the modulation index in terms of𝐴and𝐵.
3.5 Show that an AM signal can be demodulated us- ing coherent demodulation by assuming a demodulation carrier of the form
2 cos[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)]
where𝜃(𝑡)is the demodulation phase error.
3.6 A message signal is given by 𝑚(𝑡) = 3 cos(40𝜋𝑡) + 7 sin(64𝜋𝑡)
Also𝐴𝑐= 20V and𝑓𝑐= 300Hz. Determine the expres- sion for the upper-sideband SSB signal and the lower- sideband SSB signal. Write these in a way that shows the amplitude and frequency of all transmitted components.
3.7 Equation (3.63) gives the amplitude and phase for the VSB signal components centered about𝑓 = +𝑓𝑐. Give the amplitude and phase of the signal comonents centered about 𝑓 = −𝑓𝑐. Using these values show that the VSB signal is real.
3.8 An AM radio uses the standard IF frequency of 455 kHz and is tuned to receive a signal having a carrier fre- quency of 1020 kHz. Determine the frequency of the local oscillator for both low-side tuning and high-side tuning.
Give the image frequencies for each.
3.9 The input to an AM receiver input consists of both modulated carrier (the message signal is a sin- gle tone) and interference terms. Assuming that 𝐴𝑖= 100 V, 𝐴𝑚= 0.2 V, 𝐴𝑐= 1 V, 𝑓𝑚= 10 Hz, 𝑓𝑐= 300 Hz, and 𝑓𝑖= 320 Hz, approximate the envelope
detector output by giving the amplitudes and fre- quencies of all components at the envelope detector output.
3.10 A PAM signal is formed by sampling an analog sig- nal at 5 kHz. The duty cycle of the generated PAM pulses is to be 5%. Define the transfer function of the holding cir- cuit by giving the value of𝜏in (3.92). Define the transfer function of the equalizing filter.
3.11 Rewrite (3.100) to show that relationship between 𝛿0∕𝐴and𝑇𝑠𝑓1. A signal defined by
𝑚(𝑡) = 𝐴 cos(40𝜋𝑡)
is sampled at 1000 Hz to form a DM signal. Give the minium value of𝛿0∕𝐴to prevent slope overload.
3.12 A TDM signal consists of four signals having band- widths of 1000, 2000, 4000, and 6000 Hz. What is the total bandwidth of the composite TDM signal. What is the low- est possible sampling frequency for the TDM signal?
Problems
Section 3.1
3.1 Assume that a DSB signal
𝑥𝑐(𝑡) = 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡 + 𝜙0)
is demodulated using the demodulation carrier 2 cos[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)]. Determine, in general, the demod- ulated output𝑦𝐷(𝑡). Let𝐴𝑐= 1and𝜃(𝑡) = 𝜃0, where𝜃0is a constant, and determine the mean-square error between 𝑚(𝑡)and the demodulated output as a function of𝜙0and 𝜃0. Now let𝜃0= 2𝜋𝑓0𝑡and compute the mean-square error between𝑚(𝑡)and the demodulated output.
3.2 A message signal is given by 𝑚(𝑡) =
∑5 𝑘=1
10
𝑘 sin(2𝜋𝑘𝑓𝑚𝑡)
K1
–K1
0 t
T
(a)
K2
–K2
0 t
T
(b)
K3
–K3
0 t
T
(c) Figure 3.32
and the carrier is given by
𝑐(𝑡) = 100 cos(200𝜋𝑡)
Write the transmitted signal as a Fourier series and deter- mine the transmitted power.
Section 3.2
3.3 Design an envelope detector that uses a full-wave rectifier rather than the half-wave rectifier shown in Fig- ure 3.3. Sketch the resulting waveforms, as was done in for a half-wave rectifier. What are the advantages of the full-wave rectifier?
3.4 Three message signals are periodic with period𝑇, as shown in Figure 3.32. Each of the three message signals is applied to an AM modulator. For each message signal, determine the modulation efficiency for𝑎 = 0.2,𝑎 = 0.3, 𝑎 = 0.4,𝑎 = 0.7, and𝑎 = 1.
40 25 10
0
0 T T 3T
2 2
Figure 3.33
3.5 The positive portion of the envelope of the output of an AM modulator is shown in Figure 3.33. The message signal is a waveform having zero DC value. Determine the modulation index, the carrier power, the efficiency, and the power in the sidebands.
3.6 A message signal is a square wave with maxi- mum and minimum values of 8 and−8 V, respectively.
The modulation index𝑎 = 0.7and the carrier amplitude 𝐴𝑐= 100V. Determine the power in the sidebands and the efficiency. Sketch the modulation trapezoid.
3.7 In this problem we examine the efficiency of AM for the case in which the message signal does not have symmetrical maximum and minimum values. Two mes- sage signals are shown in Figure 3.34. Each is periodic with period 𝑇, and𝜏 is chosen such that the DC value of𝑚(𝑡)is zero. Calculate the efficiency for each𝑚(𝑡)for 𝑎 = 0.7and𝑎 = 1.
3.8 An AM modulator operates with the message signal
𝑚(𝑡) = 9 cos(20𝜋𝑡) − 8 cos(60𝜋𝑡) m(t)
T t
m(t) 5
0 –1
1
–5 0
τ
T t
τ
Figure 3.34
The unmodulated carrier is given by110 cos(200𝜋𝑡), and the system operates with an index of0.8.
(a) Write the equation for𝑚𝑛(𝑡), the normalized sig- nal with a minimum value of−1.
(b) Determine⟨𝑚2𝑛(𝑡)⟩, the power in𝑚𝑛(𝑡).
(c) Determine the efficiency of the modulator.
(d) Sketch the double-sided spectrum of 𝑥𝑐(𝑡), the modulator output, giving the weights and fre- quencies of all components.
3.9 Rework Problem 3.8 for the message signal 𝑚(𝑡) = 9 cos(20𝜋𝑡) + 8 cos(60𝜋𝑡) 3.10 An AM modulator has output
𝑥𝑐(𝑡) = 40 cos[2𝜋(200)𝑡] + 5 cos[2𝜋(180)𝑡]
+5 cos[2𝜋(220)𝑡]
Determine the modulation index and the efficiency.
3.11 An AM modulator has output
𝑥𝑐(𝑡) = 𝐴 cos[2𝜋(200)𝑡] + 𝐵 cos[2𝜋(180)𝑡]
+𝐵 cos[2𝜋(220)𝑡]
The carrier power is𝑃0and the efficiency is𝐸𝑓𝑓. Derive an expression for𝐸𝑓𝑓 in terms of𝑃0,𝐴, and𝐵. Deter- mine𝐴,𝐵, and the modulation index for𝑃0= 200 𝑊 and 𝐸𝑓𝑓 = 30%.
3.12 An AM modulator has output
𝑥𝑐(𝑡) = 25 cos[2𝜋(150)𝑡] + 5 cos[2𝜋(160)𝑡]
+5 cos[2𝜋(140)𝑡]
Determine the modulation index and the efficiency.
Σ x(t) y(t) g(t) m(t)
cosωct
Square-law
device Filter
+ +
Figure 3.35
3.13 An AM modulator is operating with an index of 0.8.
The modulating signal is
𝑚(𝑡) = 2 cos(2𝜋𝑓𝑚𝑡) + cos(4𝜋𝑓𝑚𝑡) +2 cos(10𝜋𝑓𝑚𝑡)
(a) Sketch the spectrum of the modulator output showing the weights of all impulse functions.
(b) What is the efficiency of the modulation process?
3.14 Consider the system shown in Figure 3.35. Assume that the average value of𝑚(𝑡)is zero and that the maxi- mum value of|𝑚(𝑡)|is𝑀. Also assume that the square-law device is defined by𝑦(𝑡) = 4𝑥(𝑡) + 2𝑥2(𝑡).
(a) Write the equation for𝑦(𝑡).
(b) Describe the filter that yields an AM signal for 𝑔(𝑡). Give the necessary filter type and the fre- quencies of interest.
(c) What value of𝑀 yields a modulation index of 0.1?
(d) What is an advantage of this method of modula- tion?
Section 3.3
3.15 Assume that a message signal is given by 𝑚(𝑡) = 4 cos(2𝜋𝑓𝑚𝑡) + cos(4𝜋𝑓𝑚𝑡) Calculate an expression for
𝑥𝑐(𝑡) = 1
2 𝐴𝑐𝑚(𝑡) cos(2𝜋𝑓𝑐𝑡) ±1
2 𝐴𝑐𝑚(𝑡) sin(2𝜋𝑓̂ 𝑐𝑡) for𝐴𝑐= 10. Show, by sketching the spectra, that the result is upper-sideband or lower-sideband SSB depending upon the choice of the algebraic sign.
3.16 Redraw Figure 3.10 to illustrate the generation of upper-sideband SSB. Give the equation defining the upper- sideband filter. Complete the analysis by deriving the ex- pression for the output of an upper-sideband SSB modu- lator.
fc f
*fc 0
Figure 3.36
3.17 Squaring a DSB or AM signal generates a frequency component at twice the carrier frequency. Is this also true for SSB signals? Show that it is or is not.
Section 3.4
3.18 Prove analytically that carrier reinsertion with en- velope detection can be used for demodulation of VSB.
3.19 Figure 3.36 shows the spectrum of a VSB signal.
The amplitude and phase characteristics are the same as described in Example 3.3. Show that upon coherent de- modulation, the output of the demodulator is real.
Section 3.5
3.20 Sketch Figure 3.20 for the case where 𝑓LO= 𝑓𝑐− 𝑓IF.
3.21 A mixer is used in a short-wave superheterodyne receiver. The receiver is designed to receive transmitted signals between 10 and 30 MHz. High-side tuning is to be used. Determine an acceptable IF frequency and the tuning range of the local oscillator. Strive to generate a design that yields the minimum tuning range.
3.22 A superheterodyne receiver uses an IF frequency of 455 kHz. The receiver is tuned to a transmitter hav- ing a carrier frequency of 1100 kHz. Give two permissi- ble frequencies of the local oscillator and the image fre- quency for each. Repeat assuming that the IF frequency is 2500 kHz.
Section 3.6
3.23 A DSB signal is squared to generate a carrier com- ponent that may then be used for demodulation. (A tech- nique for doing this, namely the phase-locked loop, will be studied in the next chapter.) Derive an expression that illustrates the impact of interference on this technique.
Section 3.7
3.24 A continuous-time signal is sampled and input to a holding circuit. The product of the holding time and the sampling frequency is𝜏𝑓𝑠. Plot the amplitude response of the required equalizer as a function of𝜏𝑓𝑠. What problem, or problems, arise if a large value of𝜏is used while the sampling frequency is held constant?
Section 3.8
3.25 A continuous data signal is quantized and transmit- ted using a PCM system. If each data sample at the receiv- ing end of the system must be known to within±0.25% of the peak-to-peak full-scale value, how many binary sym- bols must each transmitted digital word contain? Assume that the message signal is speech and has a bandwidth of 4 kHz. Estimate the bandwidth of the resulting PCM signal (choose𝑘).
3.26 A delta modulator has the message signal 𝑚 (𝑡) = 3 sin 2𝜋(10)𝑡 + 4 sin 2𝜋(20)𝑡
Determine the minimum sampling frequency required to prevent slope overload, assuming that the impulse weights 𝛿0are0.05𝜋.
3.27 Five messages bandlimited to 𝑊 , 𝑊 , 2𝑊 , 4𝑊 , and4𝑊 Hz, respectively, are to be time-division multi- plexed. Devise a commutator configuration such that each signal is periodically sampled at its own minimum rate and the samples are properly interlaced. What is the minimum transmission bandwidth required for this TDM signal?
3.28 Repeat the preceding problem assuming that the commutator is run at twice the minimum rate. What are the advantages and disadvantages of doing this?
3.29 Five messages bandlimited to 𝑊 , 𝑊 , 2𝑊 , 5𝑊 , and7𝑊 Hz, respectively, are to be time-division multi- plexed. Devise a sampling scheme requiring the minimum sampling frequency.
3.30 In an FDM communication system, the transmitted baseband signal is
𝑥(𝑡) = 𝑚1(𝑡) cos(2𝜋𝑓1𝑡) + 𝑚2(𝑡) cos(2𝜋𝑓2𝑡) This system has a second-order nonlinearity between trans- mitter output and receiver input. Thus, the received base- band signal𝑦(𝑡)can be expressed as
𝑦(𝑡) = 𝑎1𝑥(𝑡) + 𝑎2𝑥2(𝑡)
Assuming that the two message signals,𝑚1(𝑡)and𝑚2(𝑡), have the spectra
𝑀1(𝑓) = 𝑀2(𝑓) = Π (𝑓
𝑊 )
sketch the spectrum of𝑦(𝑡). Discuss the difficulties en- countered in demodulating the received baseband signal.
In many FDM systems, the subcarrier frequencies𝑓1 and 𝑓2are harmonically related. Describe any additional prob- lems this presents.
Computer Exercises
3.1 In Example 3.1 we determined the minimum value of𝑚(𝑡)using MATLAB. Write a MATLAB program that provides a complete solution for Example 3.1. Use the FFT for finding the amplitude and phase spectra of the transmitted signal𝑥𝑐(𝑡).
3.2 The purpose of this exercise is to demonstrate the properties of SSB modulation. Develop a computer pro- gram to generate both upper-sideband and lower-sideband SSB signals and display both the time-domain signals and the amplitude spectra of these signals. Assume the mes- sage signal
𝑚(𝑡) = 2 cos(2𝜋𝑓𝑚𝑡) + cos(4𝜋𝑓𝑚𝑡)
Select both𝑓𝑚and𝑓𝑐so that both the time and frequency axes can be easily calibrated. Plot the envelope of the SSB signals, and show that both the upper-sideband and the lower-sideband SSB signals have the same envelope.
Use the FFT algorithm to generate the amplitude spectrum for both the upper-sideband and the lower-sideband SSB signal.
3.3 Using the same message signal and value for𝑓𝑚used in the preceding computer exercise, show that carrier rein-
sertion can be used to demodulate an SSB signal. Illustrate the effect of using a demodulation carrier with insufficient amplitude when using the carrier reinsertion technique.
3.4 In this computer exercise we investigate the prop- erties of VSB modulation. Develop a computer program (using MATLAB) to generate and plot a VSB signal and the corresponding amplitude spectrum. Using the program, show that VSB can be demodulated using carrier reinser- tion.
3.5 Using MATLAB simulate delta modulation.
Generate a signal, using a sum of sinusoids, so that the bandwidth is known. Sample at an appropriate sampling frequency (no slope overload). Show the stairstep approx- imation. Now reduce the sampling frequency so that slope overload occurs. Once again, show the stairstep approxi- mation.
3.6 Using a sum of sinusoids as the sampling frequency, sample and generate a PAM signal. Experiment with var- ious values of𝜏𝑓𝑠. Show that the message signal is recov- ered by lowpass filtering. A third-order Butterworth filter is suggested.
CHAPTER4