Unipolar vs. Polar Signaling Signal Space Representation Baseband Receiver Unipolar vs. Polar Signaling Signal Space Representation

Unipolar vs. Polar Signaling

÷ ø ö ç è æ = N T A Q E P 2N Error Probability of Binary Signals Unipolar Signaling ÷ ø ö ç è æ = 2 N T A Q E P d b 2N

Polar Signaling (antipodal) ÷ ø ö ç è æ = 2 4 N E Q T A P b d

Orthogonal Bipolar ( Antipodal ) æ ö E P = ç ÷ Q ç ÷ N è ø Bipolar signals require a factor of 2 increase in energy compared to Orthagonal Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than Orthagonal

Error Performance Degradation Two Types of Error-Performance Degradation Power degradation Signal distortion (a) Loss in Eb/No. (b) Irreducible PB caused by distortion.

Signal Space Analysis

8 Level PAM S0(t) S1(t) S2(t) S3(t) S4(t) S5(t) S6(t) S7(t)

8 Level PAM Then we can represent (t)

8 Level PAM We can have a single unit height window (t) as the receive filter And do the decisions based on the value of z(T) We can have 7 different threshold values for our decision (we have one threshold value for PCM detection) In this way we can cluster more & more bits together and transmit them as single pulse But if we want to maintain the error rate then the transmitted power = f (clustered no of bitsn)

(t) as a unit vector One dimensional vector space (a Line) If we assume (t) as a unit vector i, then we can represent signals s0(t) to s7(t) as points on a line (one dimensional vector space) S1(t) S2(t) S3(t) S4(t) S5(t) S6(t) S7(t) S0(t) (t)

A Complete Orthonormal Basis 12

Signal space What is a signal space? Why do we need a signal space? Vector representations of signals in an N-dimensional orthogonal space Why do we need a signal space? It is a means to convert signals to vectors and vice versa. It is a means to calculate signals energy and Euclidean distances between signals. Why are we interested in Euclidean distances between signals? For detection purposes: The received signal is transformed to a received vectors. The signal which has the minimum distance to the received signal is estimated as the transmitted signal.

Signal space To form a signal space, first we need to know the inner product between two signals (functions): Inner (scalar) product: Properties of inner product: = cross-correlation between x(t) and y(t)

Signal space – cont’d The distance in signal space is measure by calculating the norm. What is norm? Norm of a signal: Norm between two signals: We refer to the norm between two signals as the Euclidean distance between two signals. = “length” of x(t)

Signal space - cont’d N-dimensional orthogonal signal space is characterized by N linearly independent functions called basis functions. The basis functions must satisfy the orthogonality condition where If all , the signal space is orthonormal. Orthonormal basis Gram-Schmidt procedure

Signal space – cont’d Any arbitrary finite set of waveforms where each member of the set is of duration T, can be expressed as a linear combination of N orthonogal waveforms where . where Vector representation of waveform Waveform energy

Signal space - cont’d Waveform to vector conversion Vector to waveform conversion

Schematic example of a signal space Transmitted signal alternatives Received signal at matched filter output

Example of distances in signal space The Euclidean distance between signals z(t) and s(t):

Example of an ortho-normal basis functions Example: 2-dimensional orthonormal signal space Example: 1-dimensional orthonornal signal space T t

Example of projecting signals to an orthonormal signal space Transmitted signal alternatives

Implementation of matched filter receiver Bank of N matched filters Observation vector

Implementation of correlator receiver Bank of N correlators Observation vector

Example of matched filter receivers using basic functions T t T t 1 matched filter T t Number of matched filters (or correlators) is reduced by 1 compared to using matched filters (correlators) to the transmitted signal. Reduced number of filters (or correlators)

Example 1. 26

(continued) 27

(continued) 28

Signal Constellation for Ex.1 29

Notes on Signal Space Two entirely different signal sets can have the same geometric representation. The underlying geometry will determine the performance and the receiver structure for a signal set. In both of these cases we were fortunate enough to guess the correct basis functions. Is there a general method to find a complete orthonormal basis for an arbitrary signal set? Gram-Schmidt Procedure 30

Gram-Schmidt Procedure Suppose we are given a signal set: We would like to find a complete orthonormal basis for this signal set. The Gram-Schmidt procedure is an iterative procedure for creating an orthonormal basis. 31

Step 1: Construct the first basis function 32

Step 2: Construct the second basis function 33

Procedure for the successive signals 34

Summary of Gram-Schmidt Prodcedure 35

Example of Gram-Schmidt Procedure 36

Step 1: 37

Step 2: 38

Step 3: * No new basis function 39

Step 4: * No new basis function. Procedure is complete 40

Final Step: 41

Signal Constellation Diagram 42

Bandpass Signals Representation

Representation of Bandpass Signals Bandpass signals (signals with small bandwidth compared to carrier frequency) can be represented in any of three standard formats: 1. Quadrature Notation s(t) = x(t) cos(2πfct) − y(t) sin(2πfct) where x(t) and y(t) are real-valued baseband signals called the in-phase and quadrature components of s(t) 44

(continued) Complex Envelope Notation where is the complex baseband or envelope of . Magnitude and Phase where is the magnitude of , and is the phase of . 45

Key Ideas from I/Q Representation of Signals We can represent bandpass signals independent of carrier frequency. The idea of quadrature sets up a coordinate system for looking at common modulation types. The coordinate system is sometimes called a signal constellation diagram. Real part of complex baseband maps to x-axis and imaginary part of complex baseband maps to the y-axis 46

Constellation Diagrams 47

Interpretation of Signal Constellation Diagram Axis are labeled with x(t) and y(t) In-phase/quadrature or real/imaginary Possible signals are plotted as points Symbol amplitude is proportional to distance from origin Probability of mistaking one signal for another is related to the distance between signal points Decisions are made on the received signal based on the distance of the received signal (in the I/Q plane) to the signal points in the constellation

For PAM signals How??? Binary PAM 4-ary PAM T t