The baseband digital signals considered in this chapter are typically transmitted using RF carrier modulation. As in the case of analog modulation considered in the preceding chapter, the fundamental techniques are based on amplitude, phase, or frequency modulation. This is illustrated in Figure 5.19 for the case in which the data bits are represented by an NRZ data format. Six bits are shown corresponding to the data sequence 101001. For digital amplitude modulation, known as amplitude-shift keying (ASK), the carrier amplitude is determined by the data bit for that interval. For digital phase modulation, known as phase-shift keying (PSK), the excess phase of the carrier is established by the data bit. The phase changes can clearly be seen in Figure 5.19. For digital frequency modulation, known as frequency-shift keying
t
t
t
t 0
0 0
1 1 1
Data
ASK
PSK
FSK
Figure 5.19
Examples of digital modulation schemes.
(FSK), the carrier frequency deviation is established by the data bit. To illustrate the similarity to the material studied in Chapters 3 and 4, note that the ASK RF signal can be represented by 𝑥ASK(𝑡) = 𝐴𝑐[1 + 𝑑(𝑡)] cos(2𝜋𝑓𝑐𝑡) (5.67) where𝑑(𝑡)is the NRZ waveform. Note that this is identical to AM modulation with the only essential difference being the definition of the message signal. PSK and FSK can be similarly represented by
𝑥PSK(𝑡) = 𝐴𝑐cos[
2𝜋𝑓𝑐𝑡 + 𝜋2 𝑑(𝑡)]
(5.68) and
𝑥FSK(𝑡) = 𝐴𝑐cos [
2𝜋𝑓𝑐𝑡 + 𝑘𝑓∫
𝑡𝑑(𝛼)𝑑𝛼 ]
(5.69) respectively. We therefore see that many of the concepts introduced in Chapters 3 and 4 carry over to digital data systems. These techniques will be studied in detail in Chapters 9 and 10. However, a major concern of both analog and digital communication systems is system performance in the presence of channel noise and other random disturbances. In order to have the tools required to undertake a study of system performance, we interrupt our discussion of communication systems to study random variables and stochastic processes.
Further Reading
Further discussions on the topics of this chapter may be found in Ziemer and Peterson (2001), Couch (2013), Proakis and Salehi (2005), and Anderson (1998).
Summary
1. The block diagram of the baseband model of a digi- tal communications systems contains several components not present in the analog systems studied in the preceding chapters. The underlying message signal may be analog or digital. If the message signal is analog, an analog-to-digital converter must be used to convert the signal from analog to digital form. In such cases a digital-to-analog converter is usually used at the receiver output to convert the digi- tal data back to analog form. Three operations covered in detail in this chapter were line coding, pulse shaping, and symbol synchronization.
2. Digital data can be represented using a number of for- mats, generally referred to as line codes. The two basic classifications of line codes are those that do not have an amplitude transition within each symbol period and those that do have an amplitude transition within each symbol period. A number of possibilities exist within each of these classifications. Two of the most popular data formats are NRZ (nonreturn to zero), which does not have an ampli- tude transition within each symbol period and split phase,
which does have an amplitude transition within each sym- bol period. The power spectral density corresponding to various data formats is important because of the impact on transmission bandwidth. Data formats having an amplitude transition within each symbol period may simplify symbol synchronization at the cost of increased bandwidth. Thus, bandwidth versus ease of synchronization are among the trade-offs available in digital transmission system design.
3. A major source of performance degradation in a dig- ital system is intersymbol interference or ISI. Distortion due to ISI results when the bandwith of a channel is not suf- ficient to pass all significant spectral components of the channel input signal. Channel equalization is often used to combat the effects of ISI. Equalization, in its simplest form, can be viewed as filtering the channel output using a filter having a frequency response function that is the inverse of the frequency response function of the channel.
4. A number of pulse shapes satisfy the Nyquist pulse- shaping criterion and result in zero ISI. A simple example is the pulse defined by𝑝(𝑡) = sinc(𝑡∕𝑇 ),where𝑇 is the
sampling (symbol) period. Zero ISI results since𝑝(𝑡) = 1 for𝑡 = 0and𝑝(𝑡) = 0for𝑡 = 𝑛𝑇,𝑛≠0.
5. A popular technique for implementing zero-ISI con- ditions is to use identical filters in both the transmitter and receiver. If the frequency resonse function of the channel is known and the underlying pulse shape is defined, the fre- quency response function of the transmitter/receiver filters can easily be found so that the Nyquist zero-ISI condition is satisfied. This technique is typically used with pulses having raised cosine spectra.
6. A zero-forcing equalizer is a digital filter that operates upon a channel output to produce a sequence of samples satisfying the Nyquist zero-ISI condition. The implemen- tation takes the form of a tapped delay line, or transver- sal, filter. The tap weights are determined by the inverse of the matrix defining the pulse response of the channel.
Attributes of the zero-forcing equalizer include ease of implementation and ease of analysis.
7. Eye diagrams are formed by overlaying segments of signals representing𝑘data symbols. The eye diagrams, while not a quantitative measure of system performance, provide a qualitative measure of system performance. Sig- nals with large vertical eye openings display lower levels of intersymbol interference than those with smaller ver- tical openings. Eyes with small horizontal openings have high levels of timing jitter, which makes symbol synchro- nization more difficult.
8. Many levels of synchronization are required in digital communication systems, including carrier, symbol, word, and frame synchronization. In this chapter we considered only symbol synchronization. Symbol synchronization is typically accomplished by using a PLL to track a compo- nent in the data signal at the symbol frequency. If the data format does not have discrete spectral lines at the symbol rate or multiples thereof, a nonlinear operation must be applied to the data signal in order to generate a spectral component at the symbol rate.
Drill Problems
5.1 Which data formats, for a random (coin toss) data stream, have (a) zero dc level; (b) built in redundancy that could be used for error checking; (c) discrete spectral lines present in their power spectra; (d) nulls in their spectra at zero frequency; (e) the most compact power spectra (measured to first null of their power spectra)?
(i) NRZ change;
(ii) NRZ mark;
(iii) Unipolar RZ;
(iv) Polar RZ;
(v) Bipolar RZ;
(vi) Split phase.
5.2 Tell which binary data format(s) shown in Fig- ure 5.2 satisfy the following properties, assuming random (fair coin toss) data:
(a) Zero DC level;
(b) A zero crossing for each data bit;
(c) Binary 0 data bits represented by 0 voltage level for transmission and the waveform has nonzero DC level;
(d) Binary 0 data bits represented by 0 voltage level for transmission and the waveform has zero DC level;
(e) The spectrum is zero at frequency zero (𝑓 = 0 Hz);
(f) The spectrum has a discrete spectral line at fre- quency zero (𝑓 = 0Hz).
5.3 Explain what happens to a line-coded data se- quence when passed through a severely bandlimited channel.
5.4 What is meant by a pulse having the zero-ISI prop- erty? What must be true of the pulse spectrum in order that it have this property?
5.5 Which of the following pulse spectra have inverse Fourier transforms with the zero-ISI property?
(a) 𝑃1(𝑓) = Π (𝑇𝑓)where𝑇 is the pulse duration;
(b) 𝑃2(𝑓) = Λ (𝑇𝑓∕2);
(c) 𝑃3(𝑓) = Π (2𝑇𝑓);
(d) 𝑃4(𝑓) = Π (𝑇𝑓) + Π (2𝑇𝑓).
5.6 True or false: The zero-ISI property exists only for pulses with raised cosine spectra.
5.7 How many total samples of the incoming pulse are required to force the following number of zeros on either side of the middle sample for a zero-forcing equalizer?
(a) 1; (b) 3; (c) 4; (d) 7; (e) 8; (f) 10.
5.8 Choose the correct adjective: A wider bandwidth channel implies (more) (less) timing jitter.
5.9 Choose the correct adjective: A narrower band- width channel implies (more) (less) amplitude jitter.
5.10 Judging from the results of Figures 5.16.and 5.18, which method for generating a spectral component at the
data clock frequency generates a higher-power one: the squarer or the delay-and-multiply circuit?
5.11 Give advantages and disadvantages of the carrier modulation methods illustrated in Figure 5.19.
Problems
Section 5.1
5.1 Given the channel features or objectives below.
For each part, tell which line code(s) is (are) the best choice(s).
(a) The channel frequency response has a null at 𝑓 = 0hertz.
(b) The channel has a passband from 0 to 10 kHz and it is desired to transmit data through it at 10,000 bits/s.
(c) At least one zero crossing per bit is desired for synchronization purposes.
(d) Built-in redundancy is desired for error-checking purposes.
(e) For simplicity of detection, distinct positive pulses are desired for ones and distinct negative pulses are desired for zeros.
(f) A discrete spectral line at the bit rate is desired from which to derive a clock at the bit rate.
5.2 For the±1-amplitude waveforms of Figure 5.2, show that the average powers are:
(a) NRZ change---𝑃ave= 1W;
(b) NRZ mark---𝑃ave= 1W;
(c) Unipolar RZ---𝑃ave= 14W;
(d) Polar RZ---𝑃ave=12 W;
(e) Bipolar RZ---𝑃ave=14 W;
(f) Split phase---𝑃ave= 1W;
5.3
(a) Given the random binary data sequence 0 1 1 0 0 0 1 0 1 1. Provide waveform sketches for:
(i) NRZ change;
(ii) Split phase.
(b) Demonstrate satisfactorily that the split-phase waveform can be obtained from the NRZ wave- form by multiplying the NRZ waveform by a
±1-valued clock signal of period𝑇.
5.4 For the data sequence of Problem 5.3 provide a waveform sketch for NRZ mark.
5.5 For the data sequence of Problem 5.3 provide wave- form sketches for:
(a) Unipolar RZ;
(b) Polar RZ;
(c) Bipolar RZ.
5.6 A channel of bandwidth 4 kHz is available. Deter- mine the data rate that can be accommodated for the fol- lowing line codes (assume a bandwidth to the first spectral null):
(a) NRZ change;
(b) Split phase;
(c) Unipolar RZ and polar RZ (d) Bipolar RZ.
Section 5.2
5.7 Given the step response for a second-order But- terworth filter as in Problem 2.65c, use the superposition and time-invariance properties of a linear time-invariant system to write down the filter’s response to the input
𝑥 (𝑡) = 𝑢 (𝑡) − 2𝑢 (𝑡 − 𝑇 ) + 𝑢 (𝑡 − 2𝑇 )
where𝑢 (𝑡)is the unit step. Plot as a function of𝑡∕𝑇 for (a)𝑓3𝑇 = 20and (b)𝑓3𝑇 = 2.
5.8 Using the superposition and time-invariance prop- erties of an RC filter, show that (5.27) is the response of a lowpass RC filter to (5.26) given that the filter’s response to a unit step is[
1 − exp (−𝑡∕𝑅𝐶)] 𝑢 (𝑡) . Section 5.3
5.9 Show that (5.32) is an ideal rectangular spectrum for𝛽 = 0. What is the corresponding pulse-shape func- tion?
5.10 Show that (5.31) and (5.32) are Fourier-transform pairs.
5.11 Sketch the following spectra and tell which ones satisfy Nyquist’s pulse-shape criterion. For those that do,
find the appropriate sample interval,𝑇, in terms of𝑊. Find the corresponding pulse-shape function𝑝 (𝑡) .(Recall thatΠ(
𝑓 𝐴
)
is a unit-high rectangular pulse from−𝐴2 to 𝐴
2; Λ(
𝑓 𝐵
)
is a unit-high triangle from−𝐵to𝐵.) (a) 𝑃1(𝑓) = Π(
𝑓 2𝑊
)+ Π(
𝑓 𝑊
) (b) 𝑃2(𝑓) = Λ(
𝑓 2𝑊
)+ Π(
𝑓 𝑊
) (c) 𝑃3(𝑓) = Π(
𝑓 4𝑊
)− Λ(
𝑓 𝑊
) (d) 𝑃4(𝑓) = Π(
𝑓−𝑊 𝑊
)+ Π(
𝑓+𝑊 𝑊
) (e) 𝑃5(𝑓) = Λ(
𝑓 2𝑊
)− Λ(
𝑓 𝑊
) 5.12 If||𝐻𝐶(𝑓)||=[
1 + (𝑓∕5000)2]−1∕2
, provide a plot for ||𝐻𝑇(𝑓)||=||𝐻𝑅(𝑓)|| assuming the pulse spectrum 𝑃RC(𝑓)with 1
𝑇 = 5000Hz for (a)𝛽 = 1; (b)𝛽 = 12. 5.13 It is desired to transmit data at 9 kbps over a chan- nel of bandwidth 7 kHz using raised-cosine pulses. What is the maximum value of the roll-off factor,𝛽, that can be used?
5.14
(a) Show by a suitable sketch that the trapezoidal spectrum given below satisfies Nyquist’s pulse- shaping criterion:
𝑃 (𝑓) = 2Λ (𝑓∕2𝑊 ) − Λ (𝑓∕𝑊 )
(b) Find the pulse-shape function corresponding to this spectrum.
∑
∑
Delay Delay
Delay τm
Gainβ
Delay
βN
y(t)
x(t)
z(t) y(t)
β3
β1 β2
(b) (a)
+ + x(t– m)
β τ
∆
∆
∆
Figure 5.20
Section 5.4
5.15 Given the following channel pulse response sam- ples:
𝑝𝑐(−3𝑇 ) = 0.001 𝑝𝑐(−2𝑇 ) = −0.01 𝑝𝑐(−𝑇 ) = 0.1 𝑝𝑐(0) = 1.0 𝑝𝑐(𝑇 ) = 0.2 𝑝𝑐(2𝑇 ) = −0.02 𝑝𝑐(3𝑇 ) = 0.005
(a) Find the tap coefficients for a three-tap zero- forcing equalizer.
(b) Find the output samples for𝑚𝑇 = −2𝑇 , −𝑇 , 0, 𝑇 , and2𝑇.
5.16 Repeat Problem 5.15 for a five-tap zero-forcing equalizer.
5.17 A simple model for a multipath communications channel is shown in Figure 5.20(a).
(a) Find𝐻𝑐(𝑓) = 𝑌 (𝑓)∕𝑋(𝑓)for this channel and plot||𝐻𝑐(𝑓)||for𝛽 = 1and 0.5.
(b) In order to equalize, or undo, the channel-induced distortion, an equalization filter is used. Ideally, its frequency response function should be
𝐻eq(𝑓) = 1 𝐻𝑐(𝑓)
if the effects of noise are ignored and only dis- tortion caused by the channel is considered. A tapped-delay-line or transversal filter, as shown in Figure 5.20(b), is commonly used to approxi- mate𝐻eq(𝑓). Write down a series expression for 𝐻eq′(𝑓) = 𝑍(𝑓)∕𝑌 (𝑓).
(c) Using (1 + 𝑥)−1= 1 − 𝑥 + 𝑥2− 𝑥3+ … ,|𝑥|
< 1,find a series expression for1∕𝐻𝑐(𝑓). Equat- ing this with𝐻eq(𝑓)found in part (b), find the values for𝛽1, 𝛽2, … , 𝛽𝑁, assuming𝜏𝑚= Δ.
5.18 Given the following channel pulse response:
𝑝𝑐(−4𝑇 ) = −0.01; 𝑝𝑐(−3𝑇 ) = 0.02; 𝑝𝑐(−2𝑇 )
= −0.05; 𝑝𝑐(−𝑇 ) = 0.07; 𝑝𝑐(0) = 1;
𝑝𝑐(𝑇 ) = −0.1; 𝑝𝑐(2𝑇 ) = 0.07; 𝑝𝑐(3𝑇 )
= −0.05; 𝑝𝑐(4𝑇 ) = 0.03;
(a) Find the tap weights for a three-tap zero-forcing equalizer.
(b) Find the output samples for𝑚𝑇 = −2𝑇 , − 𝑇 , 0, 𝑇 , 2𝑇 .
5.19 Repeat Problem 5.18 for a five-tap zero-forcing equalizer.
Section 5.5
5.20 In a certain digital data transmission system the probability of a bit error as a function of timing jitter is given by
𝑃𝐸 = 1
4exp (−𝑧) +1 4exp
[
−𝑧 (
1 − 2|Δ𝑇| 𝑇
)]
where𝑧is the signal-to-noise ratio,|Δ𝑇|, is the timing jit- ter, and𝑇is the bit period. From observations of an eye di- agram for the system, it is determined that|Δ𝑇|∕𝑇 = 0.05 (5%).
(a) Find the value of signal-to-noise ratio,𝑧0,that gives a probability of error of10−6for a timing jitter of 0.
(b) With the jitter of 5%, tell what value of signal-to- noise ratio,𝑧1,is necessary to maintain the prob- ability of error at10−6.Express the ratio𝑧1∕𝑧0
in dB, where[ 𝑧1∕𝑧0
]
dB= 10 log10( 𝑧1∕𝑧0
).Call this the degradation due to jitter.
(c) Recalculate parts (a) and (b) for a probability of error of 10−4. Is the degradation due to jitter better or worse than for a probability of error of 10−6?
5.21
(a) Using the superposition and time-invariance properties of a linear time-invariant system find the response of a lowpass RC filter to the input 𝑥 (𝑡) = 𝑢 (𝑡) − 2𝑢 (𝑡 − 𝑇 ) + 2𝑢 (𝑡 − 2𝑇 ) − 𝑢 (𝑡 − 3𝑇 )
Plot for 𝑇 ∕𝑅𝐶 = 0.4, 0.6, 1, 2 on separate axes. Use MATLAB to do so.
(b) Repeat for−𝑥 (𝑡). Plot on the same set of axes as in part a.
(c) Repeat for𝑥 (𝑡) = 𝑢 (𝑡). (d) Repeat for𝑥 (𝑡) = −𝑢 (𝑡).
Note that you have just constructed a rudimentary eye diagram.
5.22 It is desired to transmit data ISI free at 10 kbps for which pulses with a raised-cosine spectrum are used. If the channel bandwidth is limited to 5 kHz, ideal lowpass, what is the allowed roll-off factor,𝛽?
5.23
(a) For ISI-free signaling using pulses with raised- cosine spectra, give the relation of the roll-off factor, 𝛽, to data rate, 𝑅 = 1∕𝑇, and channel bandwidth,𝑓max(assumed to be ideal lowpass).
(b) What must be the relationship between 𝑅 and 𝑓maxfor realizable raised-cosine spectra pulses?
Section 5.6
5.24 Rewrite the MATLAB simulation of Example 5.8 for the case of an absolute-value type of nonlinearity. Is the spectral line at the bit rate stronger or weaker than for the square-law type of nonlinearity?
5.25 Assume that the bit period of Example 5.8 is𝑇 = 1 second. That means that the sampling rate is𝑓𝑠= 10sps becausensamp = 10in the program. Assuming that a 𝑁FFT= 5000point FFT was used to produce Figure 5.16 and that the 5000th point corresponds to𝑓𝑠 justify that the FFT output at bin 1000 corresponds to the bit rate of 1∕𝑇 = 1bit per second in this case.
Section 5.7
5.26 Referring to (5.68), it is sometimes desirable to leave a residual carrier component in a PSK-modulated waveform for carrier synchronization purposes at the receiver. Thus, instead of(5.68), we would have
𝑥PSK(𝑡) = 𝐴𝑐cos[
2𝜋𝑓𝑐𝑡 + 𝛼 𝜋2 𝑑(𝑡)]
, 0 < 𝛼 < 1
Find𝛼so that 10% of the power of𝑥PSK(𝑡)is in the carrier (unmodulated) component.
(Hint: Usecos (𝑢 + 𝑣)to write𝑥PSK(𝑡)as two terms, one dependent on𝑑 (𝑡)and the other independent of𝑑 (𝑡). Make
use of the facts that𝑑 (𝑡) = ±1and cosine is even and sine is odd.)
5.27 Referring to(5.69)and using the fact that𝑑 (𝑡) = ±1 in𝑇-second intervals, find the value of𝑘𝑓 such that the
peak frequency deviation of𝑥FSK(𝑡)is 10,000 Hz if the bit rate is 1000 bits per second.
Computer Exercises
5.1 Write a MATLAB program that will produce plots like those shown in Figure 5.2 assuming a random binary data sequence. Include as an option a Butterworth channel filter whose number of poles and bandwidth (in terms of bit rate) are inputs.
5.2 Write a MATLAB program that will produce plots like those shown in Figure 5.10. The Butterworth channel filter poles and 3-dB frequency should be inputs as well as the roll-off factor,𝛽.
5.3 Write a MATLAB program that will com- pute the weights of a transversal-filter zero-
forcing equalizer for a given input pulse sample sequence.
5.4 A symbol synchronizer uses a fourth-power device instead of a squarer. Modify the MATLAB program of Computer Example 5.3 accordingly and show that a use- ful spectral component is generated at the output of the fourth-power device. Rewrite the program to be able to select between square-law, fourth-power law, and delay- and-multiply with delay of one-half bit period. Compare the relative strengths of the spectral line at the bit rate to the line at DC. Which is the best bit sync on this basis?
CHAPTER6