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

2. Unipolar vs. Polar Signaling

3. ÷ 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

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

5. 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

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

7. Signal Space Analysis

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

9. 8 Level PAM
Then we can represent
(t)

10. 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)

11. (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)

12. A Complete Orthonormal Basis12

13. 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.

14. 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)

15. 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)

16. 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

17. 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

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

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

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

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

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

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

24. Implementation of correlator receiver
Bank of N correlators
Observation
vector

25. Example of matched filter receivers using basic functionsT 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)

26. Example 1. 26

27. (continued) 27

28. (continued) 28

29. Signal Constellation for Ex.129

30. 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

31. 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

32. Step 1: Construct the first basis function32

33. Step 2: Construct the second basis function33

34. Procedure for the successive signals34

35. Summary of Gram-Schmidt Prodcedure35

36. Example of Gram-Schmidt Procedure36

37. Step 1: 37

38. Step 2: 38

39. Step 3:
* No new basis function 39

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

41. Final Step: 41

42. Signal Constellation Diagram42

43. Bandpass Signals
Representation

44. 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

45. (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

46. 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

47. Constellation Diagrams47

48. 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

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