Chapter 4 Baseband Data Transmission CHAPTER 4 BASEBAND

Chapter 4: Baseband Data Transmission CHAPTER 4 BASEBAND DATA TRANSMISSION Digital Communication Systems 2012 R. Sokullu 1/52

Chapter 4: Baseband Data Transmission Outline • 4. 5 Nyquist Criterion for Distortion-less Baseband Binary Channel • 4. 6 Correlative-Level Coding • 4. 7 Baseband M-ary PAM Transmission • 4. 8 Digital Subscriber Lines • 4. 9 Optimum Linear Receiver Digital Communication Systems 2012 R. Sokullu 2/52

Chapter 4: Baseband Data Transmission From the previous class. . • We explained what is a “matched filter” • The matched filter is the optimum detector of a pulse of known shape in additive white noise. • In evaluating the ability of a matched-filter receiver to overcome/remove additive white noise we see that all signals with equal energy are equally effective. • The matched filter completely removes the dependence of the peak SNR on the input waveform g(t). • But still, there is the ISI. . What do we do with it from design point of view? • This is the subject of today’s lecture… Digital Communication Systems 2012 R. Sokullu 3/52

Chapter 4: Baseband Data Transmission We defined ISI as: • Interference due to the dispersiveness – (some frequencies of the received pulse are delayed) of the communication channel which causes pulse distortion (change in shape and delay). • Also we defined the transmitted and received signal in the form of a sequence of pulses as: transmitted signal received signal Digital Communication Systems 2012 R. Sokullu 4/52

Chapter 4: Baseband Data Transmission • taking in mind that the received signal is sampled at times ti= i. Tb and using (4. 46) and (4. 47) the received signal can be expressed as: contribution of the ith pulse Noise sample at time ti residual effect due to other transmitted pulses Digital Communication Systems 2012 R. Sokullu 5/52

Chapter 4: Baseband Data Transmission 4. 5 Nyquist Criterion for Distortion. Less Baseband Binary Transmission This is the diagram of the binary transmission system From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]. Done by extracting and decoding the corresponding sequence of coefficients [ak] from the output y(t). Digital Communication Systems 2012 R. Sokullu 6/52

Chapter 4: Baseband Data Transmission Reconstructing: • extracting – sampling of the output y(t) at times t = i. Tb • decoding – the weighted pulse contribution should be free from ISI to take a proper logical decision – the weighted pulse contribution is: To be free from ISI it has to meet the condition: p(0)=1 by normalization Digital Communication Systems 2012 R. Sokullu 7/52

Chapter 4: Baseband Data Transmission • if the condition given in (4. 49) is satisfied then the pulse will be free from ISI and its form will be a perfect pulse (of course not considering the noise): • y(ti)=µai • So, the condition (4. 49), formulated in the time domain, ensures perfect symbol recovery if there is no noise. • In the next slides we will try to formulate this condition in the frequency domain. Digital Communication Systems 2012 R. Sokullu 8/52

Chapter 4: Baseband Data Transmission In the Frequency Domain: • If we consider a sequence of samples {p[n. Tb]}, where n = 0, +/-1, +/-2… in the time domain, we will have periodicity in the frequency domain (duality property) (discusses in detail in Ch. 3) which in general can be expressed as: FT of an infinite sequence of samples Rb=1/Tb bit rate (b/s) Digital Communication Systems 2012 R. Sokullu 9/52

Chapter 4: Baseband Data Transmission Pδ(f) in our case is the FT of an infinite sequence of delta functions with period Tb, weighted by the sample values of p(t). So, it can also be expressed by: where if we let m = i – k for i = k we have m = 0 and for i ≠ k we have correspondingly m ≠ 0 Digital Communication Systems 2012 R. Sokullu 10/52

Chapter 4: Baseband Data Transmission • further on if we impose the conditions (4. 49) on the sample values of p(t): . . and using the sifting property of the delta function (4. 51) can be written as: Digital Communication Systems 2012 R. Sokullu 11/52

Chapter 4: Baseband Data Transmission As we have the normalizing condition p(0) = 1 substituting (4. 52) in (4. 50) we finally get: which represents the condition for zero ISI in the frequency domain Digital Communication Systems 2012 R. Sokullu 12/52

Chapter 4: Baseband Data Transmission Conclusion: • The case when ISI is equal to zero is known as distortion-less channel. • We have derived the condition for distortionless channel both in the time (4. 49) and frequency domain (4. 53) in the absence of noise • We can formulate the Nyquist criterion for distortion-less bandpass transmission in the absence of noise as follows: Digital Communication Systems 2012 R. Sokullu 13/52

Chapter 4: Baseband Data Transmission • The frequency function P(f) eliminates ISI for samples taken at intervals Tb providing that it satisfies equation (4. 53). • It is important to note that P(f) refers to the whole system, including the transmission filter, the channel and the receiver filter in accordance with equation (4. 47): Digital Communication Systems 2012 R. Sokullu 14/52

Chapter 4: Baseband Data Transmission • As it is very unlikely in real life that a channel itself will exhibit Nyquist transfer response and the condition for distortion-less transmission is incorporated in the design of the filters used. • This is also known as Nyquist channel filtering and Nyquist channel reponse. • Often the Nyquist filtering response needed for zero ISI is split between Tx and Rx using a root raised cosine filter pair (which we will discuss later) • Next we discuss what is understood by “Ideal Nyquist channel” Digital Communication Systems 2012 R. Sokullu 15/52

Chapter 4: Baseband Data Transmission Ideal Nyquist Channel • If we have to design a filter to meet the condition in (4. 53) for recovering pulses free from ISI • one possible function that we can specify for the frequency function P(f) is the rectangular function: overall system bandwidth Digital Communication Systems 2012 R. Sokullu rect function of unit amplitude at f = 0 16/52

Chapter 4: Baseband Data Transmission Figure 4. 8 (a) Ideal magnitude response. (b) Ideal basic pulse shape. Digital Communication Systems 2012 R. Sokullu 17/52

Chapter 4: Baseband Data Transmission • in the time domain this corresponds to: • the special rate Rb is the well known Nyquist rate • the W = Rb/2 is the Nyquist bandwidth Digital Communication Systems 2012 R. Sokullu 18/52

Chapter 4: Baseband Data Transmission Conclusions: • The ideal baseband pulse transmission system (channel) which satisfies – eq. (4. 54) in the frequency domain, – eq. (4. 56) in the time domain • is known as the Ideal Nyquist Channel. • The function p(t) is regarded as the impulse response of that channel. It is in fact the impulse response of an ideal lowpass filter with pass-band magnitude response 1/2 W and bandwidth W. • It has its peak at origin, and goes through zero at integer multiples of the bit duration Tb. • If such a waveform is sampled at t = 0, +/-Tb, +/-2 Tb. . the pulses defined by µp(t-i. Tb) with amplitude µ will not interfere with each other. Digital Communication Systems 2012 R. Sokullu 19/52

Chapter 4: Baseband Data Transmission Figure 4. 9 A series of sinc pulses corresponding to the sequence 1011010. Digital Communication Systems 2012 R. Sokullu 20/52

Chapter 4: Baseband Data Transmission Unfortunately, • This is difficult to realize in practice because: – requires flat magnitude characteristic P(f) from –W to +W, and 0 elsewhere – there is no margin of error for the sampling in the receiver (p(t) decreases slowly and decays for large t) • A practical solution is the raised cosine filter mentioned before. Digital Communication Systems 2012 R. Sokullu 21/52

Chapter 4: Baseband Data Transmission Raised Cosine Spectrum • We extend the min value of W = Rb/2 to an adjustable value between W and 2 W. • We specify a condition for the overall frequency response P(f). • Specifically in the equation for the ideal frequency response (4. 53) we consider only three terms (three harmonics) and restrict the bandwidth to (-W, + W). Digital Communication Systems 2012 R. Sokullu 22/52

Chapter 4: Baseband Data Transmission • Then (4. 43) reduces to the following expression: • There are several possible band-limited functions to satisfy this equation. • Of great practical interest is the raised cosine spectrum whose frequency domain characteristic is given on the next slide: Digital Communication Systems 2012 R. Sokullu 23/52

Chapter 4: Baseband Data Transmission where f 1 and the bandwidth W are related by indicates the excess bandwidth over the ideal solution W rolloff factor Digital Communication Systems 2012 R. Sokullu BT=2 W-f 1 =W(1+α) 24/52

Chapter 4: Baseband Data Transmission Figure 4. 10 Responses for different roll-off factors. (a) Frequency response. (b) Time response. Digital Communication Systems 2012 R. Sokullu 25/52

Chapter 4: Baseband Data Transmission Remarks: • In Figure 4. 10 a we have the normalized P(f). Increasing the roll off factor we see that it gradually cuts off compared to Ideal Niquist Channel (α = 0) • In Figure 4. 10 b we have p(t) which is obtained from P(f) (4. 60) using the FT. Digital Communication Systems 2012 R. Sokullu 26/52

Chapter 4: Baseband Data Transmission factor characterizing the Ideal Nyquist Channel ensures zero crossings of p(t) at the desired time instants t = i. T factor decreasing with time, proportional to 1/ |t 2| reduces the tails of the pulse considerably below that of the INC Digital Communication Systems 2012 R. Sokullu 27/52

Chapter 4: Baseband Data Transmission Conclusions: • For α = 1 we have the most gradual cut offs and also the smallest amplitude of the tails in the time domain. • This may be interpreted as the intersymbol interference resulting from timing error decreasing as the roll off factor α is increased from 0 to 1. • The special case of α = 1 is known as the full-cosine roll off characteristic • Its frequency response is given as: Digital Communication Systems 2012 R. Sokullu 28/52

Chapter 4: Baseband Data Transmission • correspondingly in the time domain given as: The time response has two interesting properties: At t = ±Tb/2 = ± 1/4 W p(t) = 0. 5 – the pulse width measured at half amplitude is equal to the bit duration There are zero crossings at t = ± 3 Tb/2, ± 5 Tb/2, … in addition to the usual zero crossings at t = ±Tb, ± 2 Tb…. Note: These two properties are extremely important when extracting a timing signal from the received signal for synchronization. Price: Double bandwidth compared to the INC. Digital Communication Systems 2012 R. Sokullu 29/52

Chapter 4: Baseband Data Transmission Outline • 4. 5 Nyquist Criterion for Distortion-less Baseband Binary Channel • 4. 6 Correlative-Level Coding • 4. 7 Baseband M-ary PAM Transmission • 4. 8 Digital Subscriber Lines Digital Communication Systems 2012 R. Sokullu 30/52

Chapter 4: Baseband Data Transmission Introduction • Uncontrolled ISI is a problem and is undesired. • If it is added in a controlled manner it is possible to achieve a signaling rate equal to the Nyquist rate of 2 W in a channel with bandwidth W Hertz. (theoretical maximum) • Such methods are called correlative-level coding or partial-response signaling schemes. • The idea is: as the ISI included in the transmitted signal is known it can be interpreted at the receiver in a deterministic way. Digital Communication Systems 2012 R. Sokullu 31/52

Chapter 4: Baseband Data Transmission Duobinary Signaling • Duo – example of doubling the transmission capacity (class I partial response) • Assumptions: – binary input sequence [bk] of uncorrelated symbols 1 and 0, each with duration Tb – sequence is applied to a PAM producing a two-level sequence of short pulses (approximating unit impulse) – amplitude of pulses is defined as: Digital Communication Systems 2012 R. Sokullu 32/52

Chapter 4: Baseband Data Transmission – sequence is applied to a duo-binary encoder (represented by filter HI(f)) and is converted to a three-level output (in our example ( -2, 0, +2) – for every unit impulse we get two impulses spaced at Tb Figure 4. 11 Duo-binary signaling scheme (encoder, channel, receiver). Digital Communication Systems 2012 R. Sokullu 33/52

Chapter 4: Baseband Data Transmission • So, the output of the duobinary encoder ck can be expressed as follows: • [ak] is uncorrelated but [ck] is correlated sequence. • From a two-level binary uncorrelated sequence the duo-binary encoder has • The correlation between adjacent created a sequence with pulses is actually ISI introduced three levels which is in a deterministic manner correlated Digital Communication Systems 2012 R. Sokullu 34/52

Chapter 4: Baseband Data Transmission Description in the Frequency Domain • An ideal delay element, with delay Tb has a frequency response of exp(-j 2πf. Tb). • The frequency response of the simple delay-line filter in fig. 4. 11 can be given as 1 + exp(-j 2πf. Tb) • Then the overall frequency response of the filter in cascade with INC (? ? ? ) is: Digital Communication Systems 2012 R. Sokullu 35/52

Chapter 4: Baseband Data Transmission • For an INC we have: • So, the overall frequency response can be written as: • and the next slide shows the magnitude and the phase response. Digital Communication Systems 2012 R. Sokullu 36/52

Chapter 4: Baseband Data Transmission Figure 4. 12 Frequency response of the duo-binary conversion filter. (a) Magnitude response. (b) Phase response. Digital Communication Systems 2012 R. Sokullu 37/52

Chapter 4: Baseband Data Transmission Description in the Time Domain • from (4. 67) we can find the impulse response of HI(f) as: two sinc pulses displaced by Tb Digital Communication Systems 2012 R. Sokullu 38/52

Chapter 4: Baseband Data Transmission Figure 4. 13 Impulse response of the duo-binary conversion filter. Digital Communication Systems 2012 R. Sokullu 39/52

Chapter 4: Baseband Data Transmission Remarks: • consists of two Nyquist pulses (sinc pulses) which are time-displaced by Tb • the impulse response h(t) has only two distinguishable values at the sampling instants. • The response to one input pulse is spread over more than one signaling intervals • so in any one interval the response is only “partial” hence the name ‘partial-response signaling” Digital Communication Systems 2012 R. Sokullu 40/52

Chapter 4: Baseband Data Transmission Detection of Duobinary Signals • Based on equation (4. 66): • Let denote the estimate of the original pulse ak as conceived by the receiver at time t = kt • Then Digital Communication Systems 2012 R. Sokullu 41/52

Chapter 4: Baseband Data Transmission • So, if ck is received without an error, and the previous estimate was right, then the current estimate of ak will also be correct. • this technique is using stored information from the previous symbol – known as decision feedback. • But, once error is made it propagates forward, because each new decision depends on the previous. • To avoid errors: using precoding (EXCLUSIVE OR) Digital Communication Systems 2012 R. Sokullu 42/52

Chapter 4: Baseband Data Transmission Figure 4. 14 A precoded duobinary scheme; details of the duobinary coder are given in Figure 4. 11. Digital Communication Systems 2012 R. Sokullu 43/52

Chapter 4: Baseband Data Transmission • The pre-coded binary sequence [dk] is applied to a PAM producing the corresponding two level sequence of short pulses [ak]. (-1 or +1) • Duo-binary encoding is linear, but pre-coding is not. • When we put (4. 72) and (4. 74) together we get: Digital Communication Systems 2012 R. Sokullu 44/52

Chapter 4: Baseband Data Transmission • so the decision is taken as: Figure 4. 15 Detector for recovering original binary sequence from the pre-coded duo-binary coder output. Digital Communication Systems 2012 R. Sokullu 45/52

Chapter 4: Baseband Data Transmission Outline • 4. 5 Nyquist Criterion for Distortion-less Baseband Binary Channel • 4. 6 Correlative-Level Coding • 4. 7 Baseband M-ary PAM Transmission • 4. 8 Digital Subscriber Lines Digital Communication Systems 2012 R. Sokullu 46/52

Chapter 4: Baseband Data Transmission • In this model the PAM block produces binary pulses (two possible amplitude levels) • A PAM in a baseband M-ary PAM system the pulse amplitude modulator produces one of M possible amplitude levels (M>2). • Next slide – example for quaternary (M=4) – each level represents a dibit (pairs of bits) Digital Communication Systems 2012 R. Sokullu 47/52

Chapter 4: Baseband Data Transmission Figure 4. 20 Output of a quaternary system. (a) Waveform. (b) Representation of the 4 possible dibits, based on Gray encoding. Digital Communication Systems 2012 R. Sokullu 48/52

Chapter 4: Baseband Data Transmission • In an M-ary system we have the following relationships: – – more than one bit is encoded in a symbol forming an alphabet if symbols differ with only 1 bit we have Gray Coding each symbol is represented by one amplitude level each symbol duration Tb; all symbols equally likely and statistically independent • Then: – symbol rate is 1/Tb – rate of emitting symbols per second (baud rate) – for binary - bit rate is 1/Tb – rate of sending bits – for quaternary – 4 possible bits; sent in combinations of two – so: 1 baud = 2 bits per second – Generalized: 1 baud = log 2 M bits per second or Digital Communication Systems 2012 R. Sokullu 49/52

Remarks: Chapter 4: Baseband Data Transmission • In a given channel bandwidth an M-ary system can transmit bits at a rate log 2 M faster than the corresponding binary system. • To realize this the M-ary system requires more transmission power – average of M 2/log 2 M compared to binary. Algorithm of an M-ary system • • • Source symbols converted into an M-ary PAM pulse train (PAM block) Pulse train is shaped by a transmit filter Transmit over a channel (AWGN) – noise and distortion. Passed trough a received filter. Sampled at appropriate rate in sync with the transmitter Each sample compared to a threshold – decision is taken which symbol was transmitted • Same as binary PAM – ISI, noise, timing errors; same procedures for designing transmit and receive filter but more complex. Digital Communication Systems 2012 R. Sokullu 50/52

Chapter 4: Baseband Data Transmission Outline • 4. 5 Nyquist Criterion for Distortion-less Baseband Binary Channel • 4. 6 Correlative-Level Coding • 4. 7 Baseband M-ary PAM Transmission • 4. 8 Digital Subscriber Lines Digital Communication Systems 2012 R. Sokullu 51/52

Chapter 4: Baseband Data Transmission Reading Assignments • Reading assignment 2: 4. 8 Digital Subscriber Lines – pp. 277 – 282 • Reading Assignment 3: 4. 9 Optimum Linear Receiver – pp. 282 – 287. Digital Communication Systems 2012 R. Sokullu 52/52