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Introduction to Digital Communications Based on prof. Moshe Nazarathy lectures on Digital Communications 1.Overview of comm. channels and digital links 2.Optimal Detection 3.Matched Filters

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A digital Communications Link: bitstream-> TX->Analog Medium with Noise->RX > bitstream 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 Bitstream: a finite or possibly infinite sequence of random bits out of the set {0,1}, representing the information to be carried

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

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Data Source Randomly Transmit M different messages a i 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 A1 areaA2 area d

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

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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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