Eeng 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

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

Eeng  Baud (Symbol Rate) : D = N/T 0 symbols/sec ; N- number of dimensions used in T 0 sec.  Bit Rate : R = n/T 0 bits/sec ; n- number of data bits sent in T 0 sec. Binary (2) Values More than 2 Values Binary signal Multilevel signal Digital Signaling  How to detect the data at the receiver?

Eeng  Orthogonal function space corresponds to orthogonal vector space : Vector Representation

Eeng 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

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

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

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

Eeng Binary Signaling 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 Binary Signaling Individual Pulses Total Waveform Which wave shape gives lower bound BW?

Eeng Binary Signaling Using Sa Shape

Eeng Binary Signaling Using Raised Cosine Shape

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

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

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

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

Eeng Multilevel Signaling - Example Individual Pulses Total Waveform

Eeng Binary-to-multilevel polar NRZ Signal Conversion T s : Symbol Duration L: Number of M ary levels T b : Bit Duration l: Bits per Symbol L=2l D=1/Ts=1/lT b =R/l  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 T s =lT b where the Symbol Rate or the BAUD Rate D=1/T s =1/lT b. 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.

Eeng Binary-to-multilevel Polar NRZ Signal Conversion (c) L = 8 = 2 3 Level Polar NRZ Waveform Out