Introduction to Analog And Digital Communications

1. Introduction to Analog And Digital Communications Second Edition Simon Haykin, Michael Moher

2. Chapter 6 Baseband Delta Transmission 6.1 Baseband Transmission of Digital Data 6.2 The Intersymbol Interference Problem 6.3 The Nyquist Channel 6.4 Raised-Cosine Pulse Spectrum 6.5 Baseband Transmission of M-ary Data 6.6 The Eye Pattern 6.7 Computer Experiment : Eye Diagrams for Binary and Quaternary Systems 6.8 Theme Examples : Equalization 6.9 Summary and Discussion

3. The transmission of digital data over a physical communication channel is limited by two unavoidable factors • Intersymbol interference • Channel noise • Lesson 1 : Understanding of the intersymbol interference problem and how to cure it is of fundamental importance to the design of digital communication systems • Lesson 2 : The raised cosine spectrum provides a powerful mathematical tool for baseband pulse-shaping designed to mitigate the intersymbol interference problem • Lesson 3 : The eye pattern is a visual indicator of performance, displaying the physical limitations of a digital data transmission system in an insightful manner

4. 6.1 Baseband Transmission of Digital Data • In this chapter, we emphasize the use of discrete pulse-amplitude modulation • Discrete pulse-amplitude modulation is simple to analyze • It is the most efficient form of discrete pulse modulation in terms of both power and bandwidth use • The analytic techniques developed for handling discrete pulse-amplitude modulation may be extended to other discrete-pulse modulation techniques using phase or frequency • In discrete PAM (Pulse-Amplitude Modulation) • The amplitude of transmitted pulses is varied in a discrete manner in accordance with an input stream of digital data Fig. 6.1

5. Back Next Fig.6.1

6. The level-encoded signal and the discrete PAM signal are • The channel output is

7. 6.2 The Intersymbol Interference Problem • We may express the receive-filter output as the modified PAM signal

8. Pulse-shaping problem • Given the channel transfer function, determine the transmit-pulse spectrum and receive-filter transfer function so as to satisfy two basic requirements • Intersymbol interference is reduced to zero • Transminssion bandwidth is conserved Residual phenomenon, intersymbol interference (ISI) Without ISI

9. 6.3 The Nyquist Channel • Nyquist Channel • The optimum solution for zero intersymbol interference at the minimum transmission bandwidth possible in a noise-free environment • the condition for zero ISI, it is necessary for the overall pulse shape p(t), the inverse Fourier transform of the pulse spectrum P(f), to satisfy the condition

10. The overall pulse spectrum is defined by the optimum brick-wall function • The brick-wall spectrum defines B0 as the minimum transmission bandwidth for zero intersymbol interference • The optimum pulse shape is the impulse response of an ideal low-pass channel with an amplitude response in the passband and a bandwidth B0 Fig. 6.2

11. Back Next Fig.6.2

12. Two difficulties that make its use for a PAM system impractical • The system requires that the spectrum P(f) be flat from –B0 to B0, and zero else-where • The time function p(t) decreases as 1/|t| for large |t|, resulting in a slow rate of decay • To pursue the timing error problem under point 2, consider Eq. (6.5) and sample the y(t) at t=∆t

13. 6.4 Raised-Cosine Pulse Spectrum • To ensure physical realizability of the overall pulse spectrum P(f), the modified P(f) decreases toward zero gradually rather than abruptly • Flat portion, which occupies the frequency band 0≤|f| ≤f1 for some parameter f1 to be defined • Roll-off portion, which occupies the frequency band f1≤|f| ≤2B0-f1 • One full cycle of the cosine function defined in the frequency domain, which is raised up by an amount equal to its amplitude • The raised-cosine pulse spectrum

14. The roll-off factor • The amount of intersymbol interference resulting from a timing error ∆t decreases as the roll-off factor is increased form zero to unity. • For special case of α=1 Fig. 6.3

15. Back Next Fig.6.3

16. Transmission-Bandwidth Requirement • The transmission bandwidth required by using the raised-cosine pulse spectrum is • Excess channel • The transmission bandwidth requirement of the raised-cosine spectrum exceeds that of the optimum Nyquist channel • When the roll-off factor is zero, the excess bandwidth is reduced to zero • When the roll-off factor is unity, the excess bandwidth is increased to B0.

18. Two additional Properties of the Raised-Cosine Pulse Spectrum • Property 1 • The roll-off protion of the spectrum P(f) exhibits odd symmetry about the midpoints f=±B0 • A unique characterization of the roll-off portion of the raised-cosine spectrum

19. Fig. 3.24(a) and 6.4(a), they are basically of an identical mathematical form, except for two minor differences : • The baseband raised-cosine pulse spectrum P(f) of Fig. 6.4(a) is centered on the origin at f=0, whereas the vestigial sideband spectrum of Fig. 3.24(a) is centered on the sinusoidal carrier frequency fc • The parameter fv in Fig. 6.4(a) refers to the excess bandwidth measured with respect to the ideal brick-wall solution for zero intersymbol interference, whereas the parameter fv in Fig. 3.24(a) refers to the excess bandwidth measured with respect to the optimum bandwidth attainable with single sideband modulation.. Fig. 3.24 Fig. 6.4

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21. Back Next Fig.6.4

22. Property 2 • The finite summation of replicas of the raised-cosine pulse spectrum, spaced by 2B0 hertz, equals a constant • Sampling the modified pulse response p(t) at the rate 1/2B0,

23. Finally, nothing that the Fourier transform of the delta function is unity, Eq. (6.29) is merely another way of describing the desired form shown in Eq. (6.27) • Given the modified pulse shape p(t) for transmitting data over an imperfect channel using discrete pulse-amplitude modulation at the data rate 1/T, the pulse shape p(t) eliminates intersymbol interference if, and only if, its spectrum P(f) satisfies the condition

24. Root Raised-Cosine Pulse Spectrum • A more sophisticated form of pulse shaping for baseband digital data transmission is to use the root raised-cosine pulse spectrum • The pulse shaping is partitioned equally between two entities • The combination of transmit-filter and channel constitutes one entity. With H(f) known and P(f) defined by Eq. (6.17) for a prescribed roll-off factor, we may use Eq. (6.31) to determine the frequency response of the transmit filter. • The receive filter constitutes the other entity. Hence, for the same roll-off factor we may use Eqs. (6.17) and (6.32) to determine the frequency response of the receive-filter.

25. 6.5 Baseband Transmission of M-ary Data • The output of the line encoder takes on one of M possible amplitude levels with M>2. • Signaling rate (symbol rate) • 1/T • Symbols per second, bauds • The symbol duration T of the M-ary PAM system is related to the bit duration Tb of a binary PAM system

26. 6.6 The Eye Pattern • Eye Pattern • Be produced by the synchronized superposition of successive symbol intervals of the distorted waveform appearing at the output of the receive-filter prior to thresholding • From an experimental perspective, the eye pattern offers two compelling virtues • The simplicity of generation • The provision of a great deal of insightful information about the characteristics of the data transmission system, hence its wide use as a visual indicator of how well or poorly a data transmission system performs the task of transporting a data sequence across a physical channel.

27. Timing Features • Three timing features pertaining to binary data transmission system, • Optimum sampling time : The width of the eye opening defines the time interval over the distorted binary waveform appearing at the output of the receive-filter • Zero-crossing jitter : in the receive-filter output, there will always be irregularities in the zero-crossings, which, give rise to jitter and therefore non-optimum sampling times • Timing sensitivity : This sensitivity is determined by the rate at which the eye pattern is closed as the sampling time is varied. Fig. 6.5

28. Back Next Fig.6.5

29. The Peak Distortion for Intersymbol Interference • In the absence of channel noise, the eye opening assumes two extreme values • An eye opening of unity, which corresponds to zero intersymbol interference • An eye opening of zero, which corresponds to a completely closed eye pattern; this second extreme case occurs when the effect of intersymbol interference is severe enough for some upper traces in the eye pattern to cross with its lower traces. Fig. 6.6

30. Back Next Fig.6.6

31. Noise margin • In a noisy environment, • The extent of eye opening at the optimum sampling time provides a measure of the operating margin over additive channel noise • Eye opening • Plays an important role in assessing system performance • Specifies the smallest possible noise margin • Zero peak distortion , which occurs when the eye opening is unity • Unity peak distortion, which occurs when the eye pattern is completely closed. • The idealized signal component of the receive-filter output is defined by the first term in Eq. (6.10) • The intersymbol interference is defined by the second term Fig. 6.7

32. Back Next Fig.6.7

33. Eye pattern for M-ary Transmission • M-ary data transmission system uses M encoded symbols • The eye pattern for an M-ary data transmission system contains (M-1) eye openings stacked vertically one on top of the other. • It is often possible to find asymmetries in the eye pattern of an M-ary data-transmission system, which are caused by nonlinearities in the communication channel or other parts of the system.

34. 6.7 Computer Experiment : Eye Diagrams for Binary and Quanternary Systems • Fig. 6.8(a) and 6.8(b) show the eye diagrams for a baseband PAM transmission system using M=2 and M=4. • Fig. 6.9(a) and 6.9(b) show the eye diagrams for these two baseband-pulse transmission systems using the same system parameters as before, but this time under a bandwidth-limited condition. Fig. 6.8 Fig. 6.9

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36. Back Next Fig.6.9

37. 6.8 Theme Example : Equalization • An efficient approach to high-speed transmission of digital data • Discrete pulse-amplitude modulation (PAM), • Linear modulation scheme • Transversal filter • Delay line, whose taps are uniformly spaced T second apart; T is the symbol duration • Adjustable weights, which are connected to the taps of the delay line • Summer, which adds successively delayed versions of the input signal, after they have been individually weighted. • Adjustable transversal equalizer (transversal equalizer) • With channel equalization as the function of interest and the transversal filter with adjustable coefficients as the structure to perform. Fig. 6.10

38. Back Next Fig.6.10

39. Zero-Forcing Equalization • To proceed with the solution to the equalization problem, consider then the composite system depicted in Fig. 6.11 • The first subsystem characterized by the impulse response c(t) represents the combined action of the transmit-filter and communication channel • The second subsystem characterized by the impulse response heq(t) accounts for pulse shaping combined with residual-distortion equalization in the receiver. Fig. 6.11

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41. discrete convolution sum,

42. Equivalently, in matrix form we may write • Since the zero-forcing equalizer ignores the effect of additive channel noise, the equalized system does not always offer the best solution to the intersymbol interference problem

43. How Could the Receiver Determine the {ck}? • A pilot-assisted training session • For the binary data sequence applied to the transmitter input, use a deterministic sequence of 1s and 0s that is noise-like in character, hence the reference to this sequence as a pseudo-noise (PN) sequence. • The PN sequence is known a priori to the receiver. Accordingly, with the receiver synchronized to the transmitter, the receiver is enabled to know when to initiate the training session • Finally, knowing the transmitted PN sequence and measuring the corresponding channel output, it is straight-forward matter for the receiver to estimate the sequence {ck} representing the sampled impulse response of the transmit-filter and channel combined.

44. 6.9 Summary and Discussion • Baseband data transmission, for which the channel is of a low-pass type • Band-pass data transmission, for which the channel is of a band-pass type • The intersymbol interference problem, which arises due to imperfections in the frequency response of the channel • ISI refers to the effect on that pulse due to cross-talk or spillover from all other signal pulses in the data stream applied to the channel input • A corrective measure widely used in practice is to shape the overall pulse spectrum of the baseband system, starting from the source of the message signal all the way to the receiver. • The eye pattern portrays the degrading effects of timing jitter, ISI, channel noise • ISI is a signal-dependent phenomenon, it therefore disappears when the information-bearing signal is switched off. • Noise is always there, regardless of whether there is data transmission or not. • Another corrective measure for dealing with the ISI; channel equalization

45. Back Next Fig.6.12 Fig. 6.12

46. Back Next Fig.6.13 Fig. 6.13

47. Back Next Fig.6.14 Fig. 6.14

48. Back Next Fig.6.15 Fig. 6.15