Digital Signaling Vector Representation Bandwidth Estimation Binary Signaling

Digital Signaling Ø Vector Representation Ø Bandwidth Estimation Ø Binary Signaling Ø Multilevel Signaling Huseyin Bilgekul Eeng 360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University Eeng 360 1

Digital Signaling Ø How do we mathematical represent the waveform of a digital signal? Ø How do we estimate the bandwidth of the waveform? Ø Example: Message ‘X’ for ASCII computer keyboard - code word “ 0001101” Ø What is the data rate? Eeng 360 2

Digital Signaling Ø Baud (Symbol Rate) : D = N/T 0 symbols/sec ; N- number of dimensions used in T 0 sec. Ø Bit Rate : R = n/T 0 bits/sec ; n- number of data bits sent in T 0 sec. Binary (2) Values Binary signal More than 2 Values Multilevel signal Eeng 360 3

Ø How to detect the data at the receiver (after transmission over a channel)? Formal way is to evaluate the orthogonal series coefficient. It is also true that eq. (3 -30) is the optimal way of detecting the received signal that is corrupted by AWGN noise. This is known as Match Filtering or Matched Filter Detection. Eeng 360 4

Vector Representation Ø Orthogonal function space corresponds to orthogonal vector space : Eeng 360 5

Vector Representation of a Binary Signal Ø Examine the representation in next slide for the waveform of a 3 -bit (binary) signal. This signal can be directly represented by, . Ø Orthogonal function approach Eeng 360 6

Vector Representation of a Binary Signal A 3 bit Signal waveform Bit shape pulse Orthogonal Function Set Vector Representation of the 3 bit signal Eeng 360 7

Bandwidth Estimation Ø The lower bound for the bandwidth of the waveform w(t) is given by the Dimensionality Theorem Ø Binary Signaling: Waveform: wk takes only BINARY values Example: Binary signaling from a digital source: M=256 distinct messages M = 2 n = 28 = 256 Each message ~ 8 -bit binary words T 0=8 ms – Time taken to transmit one message; Code word: 01001110 w 1= 0, w 2= 1, w 3= 0, w 4= 0, w 5= 1, w 6= 1, w 7= 1, w 8= 0 Ø Case 1: Rectangular Pulse Orthogonal Functions: : unity-amplitude rectangular pulses; Eeng 360 8

Bandwidth Estimation (Binary Signaling) Ø Receiver end: How are we going to detect data? Orthogonal series coefficients wk are needed. Sample anywhere in the bit interval The Lower Bound : The actual Null Bandwidth: Null BW > lower bound BW Eeng 360 9

Binary Signaling Ø Case 2: sin(x)/x Pulse Orthogonal Functions Minimum Bandwidth Where Ts=Tb for the case of Binary signaling. Ø Receiver end: How are we going to detect data? Orthogonal series coefficients wk are needed. Sample at MIDPOINT of each interval Lower bound BW: For N=8 pulses, T 0=8 ms => B=500 Hz. Eeng 360 10

Binary Signaling 0 1 0 0 1 1 1 To recover the digital data at the receiver, we sample received wavform at the right time instants (SYNCHRONIZATION) and from the sample values a decision is made about the value of the transmitted bit at that time instant. Eeng 360 11

Binary Signaling Which wave shape gives lower bound BW? 0 1 0 0 1 1 1 Individual Pulses Total Waveform Eeng 360 12

Binary Signaling Using Sa Shape 1 0 1 0 Eeng 360 13

Binary Signaling Using Raised Cosine Shape Eeng 360 14

Multilevel Signaling Ø B Reduces, if N Reduces: So wk should take more than 2 values ( 2 - binary signaling) Ø If wk’s have L>2 values Resultant waveform – Multilevel signal Ø Multilevel data : Encoding l-bit binary data into L-level : DAC Eeng 360 15

Multilevel Signaling (Example) M=256 -message source ; L=4; T 0=8 ms Encoding Scheme: A 2 -Bit Digital-to-Analog Converter Binary Input Output Level (l=2 bits) (V) 11 +3 10 +1 00 -1 01 -3 Binary code word - 01001110 w 1= -3, w 2= -1, w 3= +3, w 4= +1 Bit rate : k bits/second Different Baud ( symbol rate): k baud Relation : Eeng 360 16

Multilevel Signaling - Example B=1/Ts=D=500 Hz B=N/2 T 0=250 Hz Ø How can the data be detected at the receiver? Ø Sampling at midpoint of Ts=2 ms interval for either case (T=1, 3, 5, 7 ms) Eeng 360 17

Multilevel Signaling - Example 0 1 1 0 -3 +1 +3 +1 Individual Pulses Total Waveform Eeng 360 18

Binary-to-multilevel polar NRZ Signal Conversion Ø Binary to multilevel conversion is used to reduce the bandwidth required by the binary signaling. • Multiple bits (l number of bits) are converted into words having SYMBOL durations Ts=l. Tb where the Symbol Rate or the BAUD Rate D=1/Ts=1/l. Tb. • The symbols are converted to a L level (L=2 l ) multilevel signal using a l-bit DAC. • Note that now the Baud rate is reduced by l times the Bit rate R (D=R/l). • Thus the bandwidth required is reduced by l times. Ts: Symbol Duration L: Number of M ary levels Tb: Bit Duration l: Bits per Symbol L=2 l D=1/Ts=1/l. Tb=R/l Eeng 360 19

Binary-to-multilevel Polar NRZ Signal Conversion (c) L = 8 = 23 Level Polar NRZ Waveform Out Eeng 360 20