Baseband Pulse Transmission Introduction Matched Filter Error Rate

Baseband Pulse Transmission • • • Introduction Matched Filter Error Rate Due to Noise Inter-Symbol Interference Nyquist’s Criterion of Distortionless Baseband Binary Transmission Correlative-Level Coding Baseband M-ary PAM Transmission Digital Subscriber Lines Optimum Linear Receiver Adaptive Equalization 2

Introduction • Digital data: broad spectrum with a significant low-frequency content. • Baseband transmission of digital data requires low- pass channel with a large enough bandwidth • Typically the channel is despersive. Its frequency response deviates from that of an ideal low-pass filter. • Inter-Symbol Interference: effect of adjacent pulses on each other • ISI >> major source of bit errors • Pulse shaping >> reducing ISI • Channel noise >> another source of bit errors • Detection of a pulse signal of known waveform immersed in additive white noise >> matched filter 3

Matched Filter (1) Linear receiver: Filter input: Filter output: Peak pulse signal to noise ratio: 4

Matched Filter (2) Filter output Sampled output PSD of noise Average power of output noise Peak pulse signal to noise ratio 5

Matched Filter (3) Problem: for a given G(f), which H(f) makes the η maximum? Schwarz’s inequality : for two complex functions φ1(x) and φ2(x) in the real x, satisfying the conditions and we may write equality holds if, and only if 6

Matched Filter (3) for our problem: 7

Matched Filter (4) Property: The peak pulse signal-to-noise ratio of a matched filter depends only on the ratio of the signal energy to the power spectral density of the white noise at the filter input. Independent of g(t) waveform 8

Matched Filter (5) 9

Error Due to Noise (1) Two possible kinds of error: Error of the first kind: Symbol 1 is chosen when a 0 was actually transmitted Error of the second kind: Symbol 0 is chosen when a 1 was actually transmitted. 10

Error Due to Noise (2) 11

Error Due to Noise (3) Complementary Error Function: Upper band: 12

Error Due to Noise (4) Transmitted signal energy per bit: 13

Inter-symbol Interference (1) {bk}: Incoming binary sequence consists of symbols 1 and 0, each with duration Tb {ak}: Sequence of PAM short pulses 14

Inter-symbol Interference (2) ISI Noise 15

Nyquist’s Criterion for Distortionless Baseband Binary Transmission (1) Normalization Ignoring noise 16

Nyquist’s Criterion for Distortionless Baseband Binary Transmission (2) The frequency function P(f) eliminates intersymbol interference for samples taken at intervals Tb provided that it satisfies: Note: P(f) refers to the overall system, incorporating the transmit filter, the channel, and the receive filter. Ideal Nyquist Channel: 17

Nyquist’s Criterion for Distortionless Baseband Binary Transmission (3) 1. P(f) has abrupt transitions which is physically unrealizable. 2. p(t) has low rate of decay and there is practically no margin for timing error. 18

Nyquist’s Criterion for Distortionless Baseband Binary Transmission (4) 19

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Nyquist’s Criterion for Distortionless Baseband Binary Transmission (5) Raised Cosine Spectrum: 22

Nyquist’s Criterion for Distortionless Baseband Binary Transmission (6) Full-Cosine Rolloff (α=1): • • Useful in timing extraction form the received signal Required double the ideal Nyquist channel α=0 α=1 (minimum) 23

Correlative-Level Coding (1) Duobinary Signaling Scheme Decision Feedback: 24

Correlative-Level Coding (2) Precoding (for Avoiding Error Propagation): 25

Correlative-Level Coding (3) Example: Binary Sequence {bk} Precoded sequence {dk} Two-level Sequence {ak} Duobinary coder output {ck} Binary sequence obtained by decision rule 1 +1 0 1 +1 +2 1 0 -1 0 0 1 -1 +1 -2 0 1 0 -1 0 0 0 -1 -2 0 0 1 0 1 26

Correlative-Level Coding (4) Modified Duobinary Signaling 27

Correlative-Level Coding (5) Generalized form of Correlative-level Coding (Partial Response Signaling) Type of class N w 0 w 1 w 2 I 2 1 1 II 3 1 2 1 III 4 2 1 -1 IV 3 1 0 -1 V 5 -1 0 2 w 3 w 4 Duobinary Modified Duobinary 0 -1 28

Correlative-Level Coding (6) Useful Characteristics: • Binary data transmission over a physical baseband channel can be accomplished at a rate close to Nyquist rate, using realizable filters with gradual cutoff characteristics. • Different spectral shapes can be produced, appropriate for the application at hand. Price: A larger signal-to-noise ratio is required to yield the same average probability of symbol error, because of an increase in the number of signal levels used. 29

Baseband M-ary PAM Transmission 30

Digital Subscriber Lines (1) Twisted pairs for High data rate, Full duplex, Digital transmission capability Duplex Operation Time Compression Multiplexing Echo Cancellation 31

Digital Subscriber Lines (2) Simplified hybrid transformer Squared magnitude response of twisted pairs: 32

Digital Subscriber Lines (3) 33

Digital Subscriber Lines (4) Line Codes for Digital Subscriber Lines 1. The power spectral density of the transmitted signal should be zero at zero frequency, since no DC transmission through a hybrid transformer is possible. 2. The power spectral density of the transmitted signal should be low at high frequencies for the following reasons: • Transmission attenuation in a twisted pair is most severe at high frequencies. • Crosstalk between adjacent twisted pair increases dramatically at high frequencies because of increased capacitive coupling. v Manchester (large bandwidth NEXT & ISI) v Modified Duobinary (moderate bandwidth, min ISI) v Bipolar(AMI) (slightly inferior to M. D. ) ü 2 B 1 Q(4 PAM) (greatest baud reduction, best NEXT & ISI) 34

Digital Subscriber Lines (5) Line Codes for Digital Subscriber Lines • Using the 2 B 1 Q as the line code and VLSI implementation of a transceiver that incorporates adaptive equalizers and echo cancellers, it is possible to achieve a bit error rate of 10 -7 with 12 d. B noise margin, when 1 percent worst-case NEXT is present, operating full duplex at 160 Kbps in the vast majority of twisted-pair is an accepted performance criterion for digital subscriber lines. • Noise margin is the amount of receiver noise (including uncancelled echo) that can be tolerated without exceeding the 10 -7 error rate. 35

Digital Subscriber Lines (6) Asymmetric Digital Subscriber Lines 1. Data transmission downstream (toward the subscriber) at bit rates of up to 9 Mbps 2. Data transmission upstream (away form the subscriber) at bit rates of up to 1 Mbps 3. Plain Old Telephone Service (POTS) 36

Optimum Linear Receiver (1) • Channel noise acting alone Matched filter • Intersymbol interference acting alone Pulse-shaping filter • In real life, channel noise and ISI act together: 1. Zero forcing equalizer ISI forced to zero, ignoring the effect of the channel noise enhancement 2. Mean-square error criterion reducing the effects of both channel noise and ISI Performs as well as, or often better than zero-forcing. 37

Optimum Linear Receiver (2) Mean-square error criterion 1 2 3 4 5 6 38

Optimum Linear Receiver (3) ( 1) 39

Optimum Linear Receiver (4) ( 2) ( 3) (4) (5) 40

Optimum Linear Receiver (5) ( 6) 1… 6 Minimum J 41

Optimum Linear Receiver (6) Power spectral density of the sequence [q(k. Tb)]: C(f) of the optimum linear receiver is periodic with period 1/Tb. Optimum linear receiver is cascade of two basic components: Ø A matched filter whose impulse response is q(-t), where q(t) = g(t)*h(t). Ø A Transversal (tapped-delay-line) equalizer whose frequency response is the inverse of the periodic function Sq(f)+ (N 0/2). 42

Optimum Linear Receiver (7) Cascade connection of matched filter and transversal equalizer 43

Adaptive Equalization 44

Eye Diagram (1) 45

Eye Diagram (2) 46