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**Introduction to Digital Communications**

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

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**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

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**Complete digital communication link**

Redundant check-bits insertion Data compression A/D QUANTIZATION

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**Data Source Randomly Transmit M different messages ai every T sec.**

The amount of information is measured using entropy : Maximum information is achieved when :

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**Data Encoder Source Encoder: Data Compression (Zip ..)**

Channel Encoder: Redundant check-bits insertion (CRC, Turbo, etc)

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**Modulator Converts M digital messages to M analog signals :**

Limitations for choosing : Energy Amplitude Bandwidth

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

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**4-level PAM transmission**

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**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 :

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**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

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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.

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**Scalar Detection Problem**

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

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**Detection Princple The detector defines 2 areas A1 and A2 S1 S2 d**

A1 area A2 area

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**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:

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**The gaussian Q-function**

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

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**Time dependent Detection**

Problem formulation: If Pr(s1)=Pr(S2) optimal detection rule is

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**Time dependent Detection**

Detection rule can be written as : If we assume equal power symbols :

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**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:

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**Error Probability Calculation**

Z=n1-n2 is combination of Gaussian processes and therefore also Gaussian

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**Error Probability Calculation**

Special cases: ρ=0 : Orthogonal signals ρ=-1 : Antipodal signals

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**Antipodal transmission operational point: For 10^-5 Error Probability, SNR must be 9.6 dB**

Figure 1.41:

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**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

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**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)

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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

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