1、1,Ch. 4 Baseband Pulse Transmission,4.1 Introduction 4.2 Matched filter 4.3 Error rate due to noise 4.4 Intersymbol interference 4.5 Nyquists criterion for distortionless baseband binary transmission 4.6 Correlative-level coding 4.7 Baseband M-ary PAM transmission 4.8 Digital subscriber lines 4.9 Op

2、timum linear receiver 4.10 Adaptive equalization 4.11 Computer experiments: Eye patterns 4.12 Summary and discussion,2,4.1 Introduction,The problem of transmission digital data over baseband channel Do not concern source of the digital data The typical channel is dispersive, which leads to intersymb

3、ol interference (ISI) Pulse shaping is adopted to correct ISI Additive channel noise Optimal detection of signals under AWGN is achieved by matched filter,3,4.2 Matched Filter,The problem: the pulse shape is known, what is the optimal receiver in AWGN channel?,Figure 4.1 Linear receiver.,The answer:

4、 maximizing the peak pulse signal-to-noise ratio,The output signal power at time T,The average output noise power,4,Matched Filter (Contd),5,Matched Filter (Contd),The Schwarzs inequality,The equality holds if, and only if,where k is an arbitrary constant.,6,Matched Filter (Contd),With the Schwarzs

5、inequality, we have,so that,when,7,Matched Filter (Contd),The impulse response of the optimum filter, the matched filter, is a time-reversed and delayed version of the input signal g(t).,8,Properties of Matched Filters,The peak pulse signal-to-noise ratio of a matched filter depends only on the sign

6、al energy-to-noise spectral density ratio.,Since,we have,The Rayleighs energy theorem,9,Properties of Matched Filters (Contd),We also have,So,The signal energy-to-noise spectral density ratio.,E: the signal energy, unit: Joules N0/2: the noise spectral density, unit: watts/Hertz is dimensionless.,10

7、,Example: Matched Filter for Rectangular Pulse,Figure 4.2(a) Rectangular pulse. (b) Matched filter output. (c) Integrator output.,11,Example: Matched Filter for Rectangular Pulse (Contd),Figure 4.3 Integrate-and-dump circuit.,For the special case of a rectangular pulse, the matched filter may be imp

8、lemented using an integrate-and-dump circuit.,12,4.3 Error Rate Due to Noise,Figure 4.4 Receiver for baseband transmission of binary-encoded PCM wave using polar NRZ signaling.,Binary PCM system Polar NRZ signaling AWGN with zero mean and PSD N0/2,13,Error Rate Due to Noise (Contd),Suppose symbol 0

9、was sent,14,Error Rate Due to Noise (Contd),the complementary error function,For large positive values of u,15,Error Rate Due to Noise (Contd),Figure 4.5 Noise analysis of PCM system. (a) Probability density function of random variable Y at matched filter output when 0 is transmitted. (b) Probabilit

10、y density function of Y when 1 is transmitted.,16,Error Rate Due to Noise (Contd),Similarly, when symbol 1 was sent,17,Error Rate Due to Noise (Contd),The average probability of symbol error,Using Leibnizs rule, differentiating Pe with respect to , and setting the result to zero, we obtain:,18,Error

11、 Rate Due to Noise (Contd),For the special case of p0 = p1 = 1/2, the optimal threshold is opt = 0, and,Figure 4.6 Probability of error in a PCM receiver.,Following an exponential law,19,4.4 Intersymbol Interference (ISI),Figure 4.7 Baseband binary data transmission system.,Using discrete PAM as an

12、example discrete pulse modulation scheme Studying binary modulation first, then generalizing to M-ary case,20,ISI (Contd),p(t) is normalized by setting p(0) = 1.,Transmission delay is set to zero in this equation.,21,ISI (Contd),Sampling y(t) at ti = iTb, we get,The desired term,The ISI,The noise te

13、rm,22,4.5 Nyquists Criterion for Distortionless Baseband Transmission,The problem: Designing the transmit and receive filters g(t) and c(t), to achieve perfect reception in the absence of noise, that is,In the above case, we have (noise is ignored):,23,Nyquists Criterion (Contd),The Nyquists criteri

14、on,24,Ideal Nyquist Channel,The Nyquists criterion,The overall system bandwidth, also called the Nyquist bandwidth,The ideal Nyquist channel,The Nyquist rate,25,Ideal Nyquist Channel (Contd),Figure 4.8 (a) Ideal magnitude response. (b) Ideal basic pulse shape.,Having zeros at t = Tb, 2Tb, ,26,Ideal

15、Nyquist Channel (Contd),Figure 4.9 A series of sinc pulses corresponding to the sequence 1011010.,27,Ideal Nyquist Channel (Contd),Practical difficulties of the ideal Nyquist channel Unrealizable magnitude characteristic of P(f) Has a slow rate of decay of p(t) (p(t) decreases as 1/|t| for large |t|

16、),28,Ideal Nyquist Channel (Contd),The effect of timing error,The ISI caused by timing error t, possible to diverge,29,Raised Cosine Spectrum,The rolloff factor,The transmission bandwidth,30,Raised Cosine Spectrum (Contd),Figure 4.10 Responses for different rolloff factors. (a) Frequency response. (

17、b) Time response.,31,Raised Cosine Spectrum (Contd),Ensures zero crossings,1/|t|2 decreasing from large |t|,ISI from timing error decreases as is increased from zero to unity.,32,Raised Cosine Spectrum (Contd), = 1, the full-cosine rolloff characteristic,At t = Tb/2 = 1/4W, p(t) = 0.5, that is, the

18、pulse width measured at half amplitude is Tb; There are zero crossing at t = nTb/2, n = 2, 3, Channel bandwidth is 2W,33,Example 4.2: Bandwidth Requirement of the T1 System,34,4.6 Correlative-Level Coding,Also called partial-response signaling Introducing ISI to achieve the maximum signaling rate of

19、 2W symbols/s in a bandwidth of W Hz The implementation is realizable and perturbation-tolerant,35,Duobinary Signaling(Class I Partial Response),Figure 4.11 Duobinary signaling scheme.,ak = 1,ck = ak + ak-1 = 0 or 2,36,Duobinary Signaling (Contd),Decaying with 1/|t|2,37,Duobinary Signaling (Contd),F

20、igure 4.12 Frequency response of the duobinary conversion filter. (a) Magnitude response. (b) Phase response.,38,Duobinary Signaling (Contd),Figure 4.13 Impulse response of the duobinary conversion filter.,hI(t) has two distinguishable values at the sampling instants,39,Duobinary Signaling (Contd),D

21、etection:,Decision feedback,Drawbacks: error propagation, which can be avoided by precoding.,Figure 4.14 A precoded duobinary scheme.,40,Duobinary Signaling (Contd),Figure 4.15 Detector for recovering original binary sequence from the precoded duobinary coder output.,41,Example 4.3 Duobinary Coding

22、with Precoding,Table 4.1 Illustrating Example 4.3 on duobinary coding,42,Modified Duobinary Signaling (Class IV Partial Response),Figure 4.16 Modified duobinary signaling scheme.,43,Modified Duobinary Signaling (Contd),Decaying with 1/|t|2,44,Modified Duobinary Signaling (Contd),Figure 4.17 Frequenc

23、y response of the modified duobinary conversion filter. (a) Magnitude response. (b) Phase response.,No DC component,45,Modified Duobinary Signaling (Contd),Figure 4.18 Impulse response of the modified duobinary conversion filter.,hIV(t) has three distinguishable values at the sampling instants,46,Ge

24、neralized Form of Correlative-Level Coding (Partial-Response Signaling),Figure 4.19 Generalized correlative coding scheme.,Partial-response signaling requires a larger SNR.,47,4.7 Baseband M-ary PAM Transmission,The symbol duration,The bit duration,The number of symbols in the constellation,bauds: s

25、ymbols per second 1 baud = log2M bits per second,48,M-ary PAM (Contd),Figure 4.20 Output of a quaternary system. (a) Waveform. (b) Representation of the 4 possible dibits, based on Gray encoding.,Gray encoding,49,4.8 Digital Subscriber Lines,A digital subscriber line (DSL) provides connection betwee

26、n end user and central office (that is, the local loop),Figure 4.21 Block diagram depicting the operational environment of digital subscriber lines.,50,DSL (Contd),DSL transmits digital data over twisted pairs DSL provides full-duplex transmission Time compression multiplexing Simple to implement Re

27、quiring a higher data rate Echo-cancellation mode Need hybrid and echo canceller Do not require a higher data rate Is commonly used,51,DSL (Contd),Figure 4.22 Full-duplex operation using (a) time compression multiplexing, and (b) echo-cancellation.,52,DSL (Contd),Impairments during transmission (the

28、 echo-cancellation mode) Echo: leakage from transmitter to receiver at the same end Intersymbol interference (ISI) Crosstalk: interference from adjacent twisted pairs in a cable Near-end crosstalk (NEXT) Far-end crosstalk (FEXT),53,DSL (Contd),Figure 4.24 (a) Near-end crosstalk (NEXT). (b) Far-end c

29、rosstalk (FEXT).,54,DSL (Contd),Figure 4.25 Model of twisted-pair channel.,55,Line Code for DSLs,Desirable features of signal transmitting over DSLs No DC component PSD should be low at high frequencies, since at high frequencies: Transmission attenuation becomes severe Crosstalk increases dramatica

30、lly,56,Line Code for DSLs (Contd),Potential candidates Manchester code Modified duobinary code Bipolar code 2B1Q code Two bits is encoded into one 4-level symbol Having the best performance,57,Asymmetric Digital Subscriber Lines (ADSLs),ADSL supports three services in a frequency division way Downst

31、ream data transmission Up to 9 Mb/s Upstream data transmission Up to 1 Mb/s Plain old telephone service (POTS) A splitter and bidirectional filter are used for the FDM Supports high data rate applications, such as video-on-demand (VOD),58,ADSLs (Contd),Figure 4.26 (a) Illustrating the different band

32、 allocations for an FDM-based ADSL system. (b) Block diagram of splitter performing the function of multiplexer or demultiplexer. Note: both filters in the splitter are bidirectional filters.,59,4.9 Optimum Linear Receiver,The problem is designing of optimum receiver for a linear channel with both I

33、SI and additive noise Two main solutions Zero-forcing equalizer Simple to implement Poor performance with low SNR Minimum-mean square error (MMSE) equalizer,60,MMSE Equalizer,61,MMSE Equalizer (Contd),The mean-square error is defined as:,Differentiating J with respect to c(t), and setting the result

34、 to zero, we have:,62,MMSE Equalizer (Contd),The MMSE equalizer,63,MMSE Equalizer (Contd),A MMSE equalizer is the cascade connection of two components A matched filter q(-t) A transversal (tapped-delay-line) equalizer Requiring infinite number of taps In practice, using sufficient number of taps Whe

35、n the tap delay is Tb, the equalizer is named “synchronous”,64,MMSE Equalizer (Contd),Figure 4.27 Optimum linear receiver consisting of the cascade connection of matched filter and transversal equalizer.,65,Practical Considerations,In practice, the channel is time varying Using adaptive receiver to

36、realize the full transmission capability Adaptive implementation of both the matched filter and the equalizer in a combined manner Synchronous equalizer and fractionally spaced equalizer (FSE),66,4.10 Adaptive Equalizer,Figure 4.28 Block diagram of adaptive equalizer.,67,Adaptive Equalizer (Contd),T

37、wo operation modes of the adaptive equalizer Training mode A known training sequence is used to adjust equalizer taps The pseudo noise (PN) sequence is usually used as training sequence Deterministic periodic sequence Having noise-like characteristics Decision-directed mode (tracking mode) The mode for normal data transmission Equalizer taps are adjusted by previous dec