1. Eeng 360 1 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

2. Eeng 360 2 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 “0001101”  What is the data rate?

3. Eeng 360 3  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?

4. Eeng 360 4  Orthogonal function space corresponds to orthogonal vector space : Vector Representation

5. Eeng 360 5 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

6. Eeng 360 6 Vector Representation of a Binary Signal Bit shape pulse A 3 bit Signal waveform Orthogonal Function Set Vector Representation of the 3 bit signal

7. Eeng 360 7 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: 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  Case 1: Rectangular Pulse Orthogonal Functions: : unity-amplitude rectangular pulses; w k takes only BINARY values Waveform:  Binary Signaling:

8. Eeng 360 8 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)

9. Eeng 360 9 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. 0 1 0 0 1 1 1

10. Eeng 360 10 Binary Signaling 0 1 0 0 1 1 1 Individual Pulses Total Waveform Which wave shape gives lower bound BW?

11. Eeng 360 11 Binary Signaling Using Sa Shape 1 0 0 1 0

12. Eeng 360 12 Binary Signaling Using Raised Cosine Shape

13. Eeng 360 13  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.

14. Eeng 360 14 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

15. Eeng 360 15 Multilevel Signaling (Example) Encoding Scheme: A 2-Bit Digital-to-Analog Converter Binary InputOutput Level (l=2 bits) (V) 11 +3 10 +1 00 -1 01 -3 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 - 01001110 Bit rate : k bits/second Baud ( symbol rate): k baud Different Relation :

16. Eeng 360 16  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

17. Eeng 360 17 Multilevel Signaling - Example 0 1 1 0 1 1 1 0 -3 +1 +3 +1 Individual Pulses Total Waveform

18. Eeng 360 18 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.

19. Eeng 360 19 Binary-to-multilevel Polar NRZ Signal Conversion (c) L = 8 = 2 3 Level Polar NRZ Waveform Out