ECE 4710: Lecture #10 1 Digital Signaling  What is appropriate way to mathematically represent the waveform of a digital signal?  What is the bandwidth of the digital signal?  BW depends on pulse shape used to represent digital data  Only indirectly related to bandwidth of analog signal bandwidth via sampling frequency f s  Digital waveform can be represented by series summation of N orthogonal functions  N is dimension of orthogonal function set = # of  ( t ) functions required to represent all possible waveforms for digital signal  w k represents the digital data (e.g. 101  w 1 = 1, w 2 = 0, w 3 = 1)

ECE 4710: Lecture #10 2 Orthogonal Functions  What is orthogonal?  satisfies mathematical condition  Example: sin( t ) and cos( t )

ECE 4710: Lecture #10 3 Orthogonal Functions  Orthogonal  Another word is “perpindicular” »Sine and cosine are 90° out of phase  In complex domain »Orthogonal characteristic of sine/cosine »Cosine  Real axis »Sine  Imaginary axis »Used to represent vector R  Uniqueness of orthogonal characteristic enables the vector representation  Many other types of orthogonal function sets  |R|cos  |R|sin  Real Im

ECE 4710: Lecture #10 4 Symbol & Bit Rate  For N dimension waveform set transmitted over T 0 seconds:  Symbol Rate = D = N / T 0 (symbols/sec or sps) »Also called baud rate  outdated »Please use symbol rate (sps) in this class  Bit Rate or Data Rate = R = n / T 0 (bits/s or bps)  If w k ’s have binary values then n = N  and D = R »2 states only per symbol  binary signaling  If w k ’s have more than 2 possible states  and D  R

ECE 4710: Lecture #10 5 Vector Representation  Orthogonal function space can be represented in vector space by where w is an N dimensional vector and the set {  j } is orthogonal set of N directional vectors  Shorthand notation for w is row vector  Note book uses bold w for vector representation

ECE 4710: Lecture #10 6 Vector Representation  Three-bit binary signal s(t) represented by 3-bit waveform Let p(t)  So t 5 V T T 0 = 3T t 5 V T  Bit-Shape Waveform Functional Space

ECE 4710: Lecture #10 7 Vector Representation  Orthogonal Function Set t 5 V T p(t)p(t) t T 2T 3T p1(t)p1(t) t 5 V T 2T 3T p2(t)p2(t) t 5 V T 2T 3T p3(t)p3(t)

ECE 4710: Lecture #10 8 Vector Representation  Orthogonal Vector Space N = 3 dimensions  2 N = 8 possible messages for each symbol

ECE 4710: Lecture #10 9 Digital Signal BW  Lower bound for digital signal BW  Lower bound only achieved for sin( x )/ x pulse shape  Other real pulse shapes will have larger BW  Binary Signal Example  M = 256 possible message & n = 8-bit binary words  T = 1 msec so T 0 = 8 msec  Example message = so

ECE 4710: Lecture #10 10 Binary Signal Example Bit Rate = R = n / T 0 = 8 / 8 ms = 1 kbps Symbol Rate = D = R = 1 ksps since it is a binary signal Rectangular Pulse Shape, T b = 1 ms sin(x)/x Pulse Shape, T b = 1 ms

ECE 4710: Lecture #10 11 Binary Signal BW  Rectangular Pulse  FNBW = 1 / T = 1 / 1 msec = 1 kHz  Digital source info transmitted with digital waveform  Sin( x ) / x Pulse  Smooth rounded corners have much less frequency content  Digital source information transmitted with analog waveform  Pulse shape has no ISI if sampled exactly at midpoint of bit period  see sampling points in previous figure  Absolute BW = minimum BW = 0.5 D = 500 Hz  Other Pulse Shapes  Filter rectangular pulses to reduce BW  Studied next

ECE 4710: Lecture #10 12 Multi-Level Signaling  Multi-level signaling  Binary signals have L = 2 states/symbol »“0” = 0 V and “1” = +5V  Multi-level signaling has L > 2 states/symbol »# bits / symbol = log 2 ( L )  Two possible benefits: 1)For same symbol period, if L  then # bits per unit time   data rate is increased OR 2) If L  we can increase the symbol period to maintain the same data rate  BW  1 / T s so BW will be reduced

ECE 4710: Lecture #10 13 Multi-Level Signaling  Binary to Multi-Level Conversion for L = 4 Binary InputOutput Level  Example message =  -3, -1, +3, +1  Same message as previous binary signal example

ECE 4710: Lecture #10 14 Multi-Level Signaling Bit Rate = R = n / T 0 = 8 / 8 ms = 1 kbps Symbol Rate = D = N / T 0 = 4 / 8 ms = 500 sps N = 4 Dimensions { w 1, w 2, w 3, w 4 } = {-3, -1, +3, +1} Multi-Level Rectangular Pulse Shape Multi-Level sin(x)/x Pulse Shape T s = 2 msec

ECE 4710: Lecture #10 15 Multi-Level Signal BW  Rectangular Pulse  FNBW = 1 / T s = 1 / 2 msec = 500 Hz  Sin( x ) / x Pulse  Absolute BW = minimum BW = 0.5 D = 250 Hz  BW’s are 2  smaller than same message for binary signal  Data rate kept the same  Symbol period increased by factor of 2  BW   Alternate approach would be to keep same BW and increase data rate by factor of 2