Digital Signaling Digital Signaling Vector Representation Bandwidth Estimation Binary Signaling Multilevel Signaling Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

Digital Signaling Mathematical Representation of the waveform: Voltage (or current) waveform for digital signals: Example: Message ‘X’ from a digital source - code word “0001101”

Digital Signaling Baud (Symbol Rate) : D = N/T0 symbols/sec ; N- number of dimensions used in T0 sec. Bit Rate : R = n/T0 bits/sec ; n- number of data bits sent in T0 sec. Binary (2) Values More than 2 Values Binary signal Multilevel signal How to detect the data at the receiver?

Vector Representation Orthogonal function space corresponds to orthogonal vector space :

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

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

wk takes only BINARY values Bandwidth Estimation The lower bound for the bandwidth of the waveform w(t) is given by the Dimensionality Theorem Binary Signaling: wk takes only BINARY values Waveform: Example: Binary signaling from a digital source: M=256 distinct messages M = 2n = 28 = 256  Each message ~ 8-bit binary words T0=8 ms – Time taken to transmit one message; Code word: 01001110 w1= 0, w2= 1, w3= 0, w4= 0, w5= 1, w6= 1, w7= 1, w8= 0 Case 1: Rectangular Pulse Orthogonal Functions: : unity-amplitude rectangular pulses;

Bandwidth Estimation (Binary Signaling) 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: Bandwidth:  Null BW > lower bound BW

Which wave shape gives lower bound BW? Binary Signaling Which wave shape gives lower bound BW?

Binary Signaling Case 2: sin(x)/x Pulse Orthogonal Functions Minimum Bandwidth Where Ts=Tb for the case of Binary signaling. Receiver end: How are we going to detect data? Orthogonal series coefficients wk are needed. Sample at MIDPOINT of each interval

Multilevel Signaling Lower bound BW: For N=8 pulses, T0=8 ms => B=500Hz. B Reduces, if N Reduces: So wk should take more than 2 values ( 2- binary signaling) If wk’s have L>2 values  Resultant waveform – Multilevel signal Multilevel data : Encoding l-bit binary data  into L-level : DAC

Multilevel Signaling (Example) M=256-message source ; L=4; T0=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 w1= -3, w2= -1, w3= +3, w4= +1 Bit rate : k bits/second Different Baud ( symbol rate): k baud Relation :

Multilevel Signaling - Example B=1/Ts=D=500 Hz B=N/2T0=250Hz How can the data be detected at the receiver? Sampling at midpoint of Ts=2 ms interval for either case (T=1, 3, 5, 7 ms)

Binary-to-multilevel polar NRZ Signal Conversion 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=lTb where the Symbol Rate or the BAUD Rate D=1/Ts=1/lTb. The symbols are converted to a L level (L=2l ) 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=2l D=1/Ts=1/lTb=R/l

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