1. Introduction to Digital Communications
Based on prof. Moshe Nazarathy lectures on Digital Communications
Overview of comm. channels and digital links
Optimal Detection
Matched Filters

2. A digital Communications Link: bitstream-> TX->Analog Medium with Noise->RX > bitstream
Bitstream: a finite or possibly infinite sequence of random bits
out of the set {0,1}, representing the information to be carried
All media in Nature are analog –
A purely digital medium exists only in math.
“Underneath every digital communications link there resides an analog medium”
The TX: Digital->Analog
The RX: Analog->Digital
The objective of a communication link:
Receiving a bitstream at the TX and faithfully reproducing it at the RX at maximum rate and with minimum power

3. Complete digital communication link
Redundant
check-bits
insertion
Data
compression
A/D
QUANTIZATION

4. Data Source Randomly Transmit M different messages ai every T sec.
The amount of information is measured using entropy :
Maximum information is achieved when :

5. Data Encoder Source Encoder: Data Compression (Zip ..)
Channel Encoder: Redundant check-bits insertion (CRC, Turbo, etc)

6. Modulator Converts M digital messages to M analog signals :
Limitations for choosing :
Energy
Amplitude
Bandwidth

7. P-MOD example: QPSK transmitter mapping pairs of bits to one of four signals

8. 4-level PAM transmission

9. Communication Media - Fiber
In our course the channel can be described by:
LTI transfer function of analog medium
Additive Noise
Mathematical model for this channel is :

10. Communication Media - Fiber
b(t) – Fiber Impulse response
Optical mode propagation constants
Disspersion
n(t) – system noise
Laser noise
Modulator
Amplifiers noise
Photo-detector noise

11. Receiver
Receives random analog signal R(t) and matches it to one of M possibilities
Optimal decision is required. We choose Pr(Error) as our optimization parameter.

12. Scalar Detection Problem
We look at special case when M=2 and we transmit scalar amplitudes s1 and s2 with probability 1/2 :

13. Detection Princple The detector defines 2 areas A1 and A2 S1 S2 d
A1 area
A2 area

14. Detection Princple Optimum performance is achieved for :
If we choose s1=-A and s2=A, then d=0 and Error probability of the detector is:

15. The gaussian Q-function^
Gaussian integral function
or Q-function
=Prob. of “upper tail” of normalized gaussian r.v.

16. Time dependent Detection
Problem formulation:
If Pr(s1)=Pr(S2) optimal detection rule is

17. Time dependent Detection
Detection rule can be written as :
If we assume equal power symbols :

18. Error Probability Calculation
We assume 2 signals s1(t) and s(2) with correlation ρ:
We define a new random processes X, n1 and n2 such as:

19. Error Probability Calculation
Z=n1-n2 is combination of Gaussian processes and therefore also Gaussian

20. Error Probability Calculation
Special cases:
ρ=0 : Orthogonal signals
ρ=-1 : Antipodal signals

21. Antipodal transmission operational point: For 10^-5 Error Probability, SNR must be 9.6 dB
Figure 1.41:

22. Matched Filters We already saw that our decision algorithm is :
It is more convenient to write it in form:
is called matched filter for signal

23. Simple Detector Block Diagram
R(t)
H1(t)
H2(t)
HM(t)
Choose the biggest Ak
Estimated Data
Matched filter is chosen according to following parameters:
Transmitter modulation format
Channel transfer function b(t)

24. Matched Filter and SNR
Lets assume general MF with following characteristics:
In this case after MF the system SNR is:
It can be noticed that when
We achieve optimal performance with