Differential Encoding
3.7 Pulse Shaping
The shape of a pulse at the output or receiving end of a transmission channel is determined by the spectrum of the input pulse, the frequency response of the transmitter, the frequency response of the channel, and the frequency response of the receiver. This is shown diagrammatically in Fig. 3.7.1.
Input pulse
Transmission channel
Output pulse
t Figure 3.6.1 Linear distortion of a pulse that produces “ringing.”
Denoting the input pulse spectrum by Vi(f), the transmit filter by HT(f), the channel frequency response by HC(f), and the receive filter by HR(f), the spectrum of the output pulse is given by
(3.7.1) In general there will be a transmission delay, and for present purposes this is included in the channel response HC(f). To correctly shape the output pulse, its spectrum is shaped by adjustment of HT(f) or HR(f), or both. One major advantage of digital systems is that shaping the spectrum in this way has no effect on the information content of the signal, in contrast to analog systems, in which the message spectrum, and hence the analog waveshape, is directly affected by the frequency response of the transmission system. The designer also has control over the input pulse, although this is usually chosen to be rectangular. The frequency response of the channel is seldom within the control of the designer.
Figure 3.7.2 shows in a general way how pulse shaping can be used to avoid ISI. Three rectangular pulses representing 010are shown at the input to the system. The output pulses have tails that overlap, but the zeros in the tails are arranged to occur at the sampling instants. Pulses that are shaped to eliminate ISI are known as Nyquist pulses.One particular shape that is widely used is the raised-cosine response,named
Vo(f)Vi(f)HT(f)HC(f)HR(f)
Vi(f) Vo(f)
Vo(f) = Vi(f)HT(f)HC(f)HR(f) Vo(t) = F–1
Vo(f)HT(f) HC(f) HR(f)
Figure 3.7.1 Factors affecting the spectrum of an output pulse.
HT(f)HC(f)HR(f)
Sampling clock
Samples and regenerated pulses
t t t t
t t t
Figure 3.7.2 Pulse shaping to avoid ISI.
after the shape of its spectrum. The spectrum is sketched in Fig. 3.7.3, where Ais the peak value of the pulse in the time domain and Tsymis the symbol period, not the pulse width. As will be shown shortly, the pulse is spread out in time, and therefore a precise pulse width cannot be defined. The frequency spectrum of the raised cosine pulse can be described mathematically by
(3.7.2) The frequenciesf1and Bare determined by a design parameter known as the roll-off factor, denoted here by the symbol and by the symbol period. The design equations are
(3.7.3) (3.7.4)
The roll-off factor is a specified parameter that lies between the limits 0 1. Strictly, the raised cosine response is a theoretical model, but it is one that can be closely approximated in practice for moderate to high values of . It is left as an exercise for the student to show that when 0 the raised cosine spectrum becomes rectangular in shape, and this is usually referred to as the ideal low-pass response. Recalling that the symbol rate is given by Rsym1/Tsymthe equations relating to the raised cosine pulse can be rewritten in terms of the symbol rate if desired.
The time waveform corresponding to the raised-cosine spectrum is obtained by taking the inverse Fourier transform of Vo(f). The mathematical details will be omitted here, but the result is
(3.7.5) It will be seen that the sinc function, previously encountered in the frequency domain, now enters into the time domain also. The shape of the pulse is shown in Fig. 3.7.4 for a roll-off factor of 0.25. The pulse
(t)A sinc t
Tsym cos tTsym 1(2tTsym)2 f1 1
2Tsym B1
2Tsym 0 for f B
ATsym cos2 (ff1)
2(Bf1) for f1 f B Vo(f) ATsym, for f f1
ATb
0 f1 B
Vo(f)
f Figure 3.7.3 Raised-cosine spectrum.
has its maximum at t0, and it has periodic zeros at integer multiples of the symbol period,tkTsym. This can be seen by rewriting the pulse equation as
(3.7.6) The sinc function is equal to zero for integer values of kand is maximum at unity for k0. The cosine term is also unity for k0. Therefore, sampling the pulse at k0 yields (0) A, while sampled at any other integer value,kyields zero output. This means that a waveform consisting of a sequence of such pulses will have no ISI. Denoting the basic pulse shape by
(3.7.7) allows the waveform to be written in the form given by Eq. (3.4.1):
(3.7.8) For example, the resultant binary waveform for the sequence 010is shown in Fig. 3.7.5. Although the pulses interfere with one another, by sampling at integer multiples of the bit period, ISI is avoided. In prac- tice, some mistiming is likely to occur (referred to as timing jitter), which results in sampling at points where the ISI is not zero. However, the timing jitter is kept to a minimum by careful design. With the raised cosine pulse, the denominator term [1 (2k)2] increases rapidly for large values of k, so the tails of the pulse decrease rapidly, which helps to minimize the ISI.
The raised-cosine response when 0 has special significance. For this condition it is seen from Eqs. (3.7.3) and (3.7.4) that Bf1l/2Tsym. The spectrum shown in Fig. 3.7.6(a) becomes rectangular, and it is the narrowest bandwidth spectrum that still avoids ISI. It is referred to as the ideallow-pass spectrum because the rectangular shape cannot be realized in practice. It does, however, provide a reference against which system performance can be assessed.
(t)k ∞
∞ ak p(tkTsym) p(t)sinc tTsym cos tTsym 1(2tTsym)2 (kTsym) A sinc k cos k
1 (2k)2
1
0
–0.2
–5 5
v(t)
t Figure 3.7.4 Raised-cosine pulse with 0.25.
The corresponding pulse in the time domain is obtained from Eq. (3.7.5) as
(3.7.9) This is shown in Fig. 3.7.6. The pulse samples taken at intervals kTsymk/Rsymare given by Asine k, which is equal to Afor k0 and is zero for all other values of k(recalling that kis integer). The problem with the ideal response, apart from the fact that it cannot be realized in practice, is that the pulse tails do not decrease nearly as rapidly as when 0, and so the ISI that might occur as a result of timing jitter could be severe.
For the ideal response
(3.7.10) Rsym
2 B 1
2Tsym A sinc Rsymt (t) A sinc t
Tsym
1
0
–1.2 –5
0 1 0 t
v(t)
5
Figure 3.7.5 Raised-cosine pulses with 0.25 for the binary sequence 010.
1
–0.4–5 5
p = 1 p = 0
t v(t)
Figure 3.7.6 Raised-cosine response for 0 and 1.
This shows that the signaling rate in symbols per second is equal to twice the ideal bandwidth. An important parameter in digital communications is the ratio of bit rate to the spectrum bandwidth. Denoting this ratio by , then for the raised-cosine response
(3.7.11) The units for are bps/Hz, which in fact is a dimensionless quantity, since hertz are measured in cycles per second, and both cycles and bits are dimensionless. However, using bps/Hz helps to keep track of the meaning of the parameter.
For the ideal bandwidth system ( 0), 2mand thus increasing mmakes more efficient use of available bandwidth in terms of bps/Hz. However, more complex circuitry is needed to implement M-ary coding. Also, if it is desired to maintain the same separation between levels as in the binary system (a con- dition required to maintain the same performance against noise), then higher power levels are required for the M-ary system.
It is seen that the pulses at the receiver do not have well-defined shapes (such as rectangular) and, in addition, noise is an inescapable part of any communication system, which will further distort the wave- shape. The result is that the waveform at the receiver will bear very little resemblance to the well-defined line waveforms described in Section 3.4. The function of the pulse regenerator in the receiver is to regener- ate a “clean” waveform from the distorted and noisy signal, from which the data can be recovered with as few errors as possible. This topic is taken up again in Chapter 12.
PROBLEMS
3.1. Briefly discuss the difference between analog and digital signals. A thermostat is set so that it gener- ates a voltage of 5 V at its low setting and a voltage of 10 V at its high setting. Are these signal lev- els analog or digital or is the situation indeterminate?
3.2. The thermostat in Problem 3.1 generates a continuous voltage proportional to temperature between its low and high limits. Is this an analog or digital signal?
3.3. Define and explain the terms binit, bit,and baud.Which of these are symbols?
3.4. A discrete source contains 32 symbols, each of which has equal probability of being selected for transmission. Calculate the information in each symbol. What is the information content in the trans- mission of two successive symbols?
3.5. In a certain binary transmission system, the probability of a binary 1being transmitted is 0.6. What is the probability of a binary 0being transmitted? What is the information content in each binit?
3.6. The transmission rate in a binary system is 3000 binits per second. What is the rate in bauds? Given that the rate is also equal to 3000 bps, what condition is implied by this statement?
3.7. For the source alphabet of Problem 3.4, determine the number of binits required to encode each sym- bol into a binary code. Given that the binary code is transmitted at the rate of 1000 bps, what is the source-symbol transmission rate?
2m 1 Rb
(1 )Rsym2 Rb
B
3.8. The binary code in Problem 3.7 is recoded into a quaternary code. Determine the code-symbol transmission rate.
3.9. A rectangular pulse has an amplitude of 5 V and a width of 3 ms. Express this in the notation of Section 3.3 for (a) a pulse centered on the zero time reference, (b) a pulse delayed by 7 ms from the zero time reference, and (c) a pulse advanced by 2 ms from the zero time reference. Sketch all three pulses.
3.10. A rectangular pulse of width and amplitude Astarts at t0. Write the expression for the pulse in the notation of Section 3.3 and sketch the pulse.
3.11. A pulse shape can be described by p(t) sin tfor 0 t 1 s and zero for all other values of t.
(a) Sketch the pulse. Write the expressions for similar pulses and sketch these for (b) a time delay of 0.5 s, and (c) a time advance of 0.5 s.
3.12. Explain what is meant by (a) return-to-zero (RZ) and (b) not-return-to-zero (NRZ) pulses. Discuss briefly the reasons why both types are used in practice. Information bits of period Tbare encoded in rectangular pulses of both types. Denoting the pulse width by , express the bits in the notation of Section 3.3 when the bit period is centered about t0 and the pulse width for the RZ pulse is one- half the bit period.
3.13. For a unipolar binary signal, the akterms of Eq. (3.4.1) for k 3,2,1, 0, 1 are 0, 1, 1, 0, 1.
Write out the corresponding terms of Eq. (3.4.1) for a bit period of 1 s, showing the time range for each term.
3.14. A finite binary message 11100is transmitted as a unipolar NRZ-L waveform using a rectangular pulse of height 5 V for binary l’s. Calculate the energy dissipated in a l-Ω, load resistor when the waveform is developed across this, the pulse width being 2 ms.
3.15. Explain why sinc x1 for x0, and sinc x0 for x 1,2,...
3.16. Given that sinc x0.2, determine the value of x.
3.17. Plot the function sinc xfor xin steps of 0.1 over the range 3 x3.
3.18. For a unipolar NRZ-L waveform using rectangular pulses of height 2Avolts, the dc component of power is A2for a load resistance of 1 Ω. State the conditions that apply to the waveform for this rela- tionship to hold. The power density spectrum for such an NRZ-L waveform is given by Eq. (3.4.4).
Derive the expression for power spectrum density at zero frequency. What is the distinction between this and the dc component of power?
3.19. A unipolar binary waveform has the spectrum density function given by Eq. (3.4.4). Given that the basic pulse for the waveform has a magnitude of 5 V and a width of 2 s, calculate the double-sided spectrum density at a frequency of 425 kHz, stating clearly the units for this. What would be the one- sided spectrum density at this frequency?
3.20. Assuming that the curve given by Eq. (3.4.3) can be approximated as flat at its peak value for a fre- quency range of 5%/Tbabout zero, calculate the power (for a 1 Ωload) in this range centered at zero frequency. The pulse amplitude is 1 V and width is 5 s.
3.21. What is the main advantage of the polar NRZ-L waveform compared to the unipolar version? A dig- ital signal consisting of an infinitely long stream of random and equiprobable binary symbols is encoded as a polar NRZ-L waveform. The pulses are rectangular of amplitude 1 V and width 3 ms.
Assuming the waveform is developed across a 1-Ωresistor, calculate (a) the dc power, (b) the aver- age signal power, and (c) the power spectrum density at zero frequency.
3.22. A finite binary sequence 10101110is encoded as a polar NRZ-L using rectangular pulses of height 3 V and width 5 s. Sketch the waveform and determine the waveform energy, assuming a l-Ωload resistance.
3.23. Given that the average signal power in an infinitely long, random, polar NRZ-L waveform is A2, where Ais the pulse height, and using Eq. (3.4.4), deduce the value of the area under the (sinc f Tb)2 curve from f to f .
3.24. Explain what is meant by dc wander, and why this is to be avoided in a digital transmission system.
3.25. State the main advantages and main disadvantages of the Manchester code. A finite binary sequence 111000 is to be transmitted using the Manchester code. Sketch the waveform, and calculate the energy in the waveform assuming a 1-Ωload resistance and rectangular pulses of 1 V.
3.26. Calculate the power spectrum density for an infinitely long random binary sequence encoded as a Manchester waveform at the following frequencies: (a) zero, and (b) 0.5/Tb, 1/Tb, and 2/Tb. The pulse height is 5 V and the bit period is 1 ms.
3.27. Explain what is meant by an alternate mark inversion (AMI) code and why this is also referred to as a pseudoternary code. Sketch the AMI waveform for the binary sequence 11010011.
3.28. A digital message consists of an infinitely long random sequence of equiprobable binits, which is encoded as an AMI waveform. What is the probability of a mark being encoded as a negative pulse?
What is the average signal power in such a waveform? What is the power spectrum density at zero frequency? A pulse height of Avolts and a load resistor of 1 Ωmay be assumed.
3.29. A binary sequence 111000111is to be transmitted on a digital link. Compare the average energies for the following waveforms: (a) polar NRZ-L, (b) Manchester, and (c) AMI. The same pulse width and peak value are used in all cases.
3.30. Give the reason for the use of high-density bipolar codes, and show that these are a development of the AMI code. A binary sequence 100000011is encoded as an HDB3 code. Sketch the resulting waveform.
3.31. The binary sequence 101000000001110010is to be encoded in the HDB3 code. Sketch the resulting waveform.
3.32. The binary sequence 10000011111000011001is to be encoded in the HDB3 code. Sketch the result- ing waveform.
3.33. Draw and compare the waveform for the binary sequence 10100010000110000encoded in (a) AMI and (b) HDB3 line codes.
3.34. The binary sequence shown in Problem 3.33 is differentially encoded. Given that a reference 1bit is inserted ahead of the sequence, write out the differentially encoded sequence.
3.35. A binary stream is being generated at the rate of 64 kbps and is to be encoded into a quaternary (four- level) code in real time. Calculate the symbol rate for the quaternary code.
3.36. A polar M-ary code has eight levels, the spacing between levels being 1 V. Write down the permissi- ble voltage levels for the code.
3.37. A polar M-ary code has five levels, the spacing between levels being 1 V. Write down the permissi- ble voltage levels for the code.
3.38. A binary stream 1110110001010001is partitioned in groups of two and encoded in quaternary code.
Sketch the resulting waveform.
3.39. Explain what is meant by intersymbol interference. A rectangular pulse of width 1 ms is transmitted through a channel that can be modeled as an RClow-pass filter with a time constant of 1 s. At the receiver the pulses are sampled at their midpoint. Is ISI likely to be significant?
3.40. A rectangular pulse of width 1 s is transmitted through a channel that can be modeled as an RClow- pass filter with a time constant of 1 s. At the receiver the pulses are sampled at their midpoint. Is ISI likely to be significant?
3.41. Discuss briefly the factors that affect the shape of the output pulse in a digital transmission system.
3.42. Explain what is meant by the raised-cosine response.A transmission system has rectangular input pulses of width 5 ms, and the output response is shaped to be a raised-cosine curve with a roll-off factor of 0.5. Determine the cutoff bandwidth of the raised-cosine response. Determine also the frequency at which the raised-cosine section of the curve starts.
3.43. A transmission system has rectangular input pulses, and the output response is shaped to be a raised- cosine curve with a roll-off factor of 0.5. The transmission rate is 3000 bps. Determine the cutoff bandwidth of the raised-cosine response. Determine also the frequency at which the raised-cosine section of the curve starts.
3.44. A transmission system has a raised-cosine output for a rectangular input pulse of width 2 ms and a pulse amplitude of 1 V. Plot the raised-cosine response for roll-off factors of (a) 1, (b) 0.5, and (c) 0.
3.45. For the raised-cosine responses specified in Problem 3.44, plot the corresponding pulses in the time domain.
3.46. Show that for the raised-cosine response Tb1/(Bf1) and (B f1)/(Bf1).
3.47. A rectangular pulse is transmitted through a channel that has a raised-cosine output, the frequency parameters for which are f1350 kHz and B650 kHz. The peak value of the output pulse in the time domain is 1 V. Plot the output pulse as a function of time up to the third null.
3.48. The binary sequence 111is coded as a rectangular pulse sequence A,A,Aand transmitted through a channel having a raised-cosine response. Plot the output waveform as a function of t/Tb given that the roll-off factor is 0.6.
3.49. Develop a MATLAB program to generate a random binary sequence.
3.50. Generate and plot the sinc(x) function using MATLAB.
3.51. Generate and plot a train of ten sinc(x) pulses.
3.52. Using dibits, encode and plot the waveform for the binary sequence “000110111000111101”.
3.53. Plot the binary sequence “000110110111” when applied to a QPSK modulator.
3.54. Plot the binary sequence “01010101001000” when applied to an AMI encoder.
3.55. Plot the binary sequence “001100011111110” when applied to a 8-ary QAM.
3.56. Explore the following MATLAB functions to generate waveforms: (a) rectpuls (b) saw-tooth (c)tripulsand (d) pulstran.
3.57. Generate qam using the modulate(.)command in MATLAB.