10-401-03-RandomSignalsProcesses

10-401-03-RandomSign - EE 401 Digital Communications Random Variables and Stochastic(Random Processes  Random variables  Stochastic(Random

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Unformatted text preview: EE 401 Digital Communications Random Variables and Stochastic (Random) Processes  Random variables  Stochastic(Random) Processes, Gaussian process  Noise in communication systems 2 EE 401 Digital Communications Deterministic or random signals Deterministic signal  no uncertainty on its value at any time  explicit mathematical expressions  sin(x), exp(x) Random signal  some degree of uncertainty, => no explicit expression  value of the signal cannot be known in advance But it could be described by some statistical parameters;  Mean  Variance  Average power  Spectral distribution of power etc. 3 EE 401 Digital Communications Experiment, event and outcomes An experiment with k-th possible outcomes  Set of all possible outcomes: sample space, S  Single or multiple sample points in S: event  Single sample point: elementary event  Occurence of an event makes occurence of another event impossible: mutually exclusive events s 1 s k event S 4 EE 401 Digital Communications Examples: Discrete or Continuous Experiment-1: A coin is tossed (discrete)  Outcomes: heads (H) or tails(T)  Sample space: S={H,T} Experiment-2: A die is tossed(discrete)  Outcomes: 1,2,3,4,5,6  Sample space: S={12,3,4,5,6} Set of all real numbers betweem 0 to 100 (continuous)  Sample space: S={x: x is a real number between 0 and 100} 5 EE 401 Digital Communications Set notation<=>space Sets A, B and C  Union  Intersection  Mutually exclusive sets S A ∩ B A C B A ∩ B ∩ C C ∩ A B ∩ C 6 EE 401 Digital Communications Probability (Pr) axioms  Sample space might be  discrete (tossing a die) or  continuous (measument of voltage at the output of a noise source)  Probability measure P is a function: assigning a non- negative number to an event A in the sample space S Three properties of probability 1.0  P [ A ] 1 2. P [ S ]= 1 3. A, and B are mutually exclusive events P [ A ∪ B ]= P [ A ] P [ B ] Tossing a coin three times  Sample space? (TTT,..,HHH)  Probability of two heads?  P {2 heads}  Probability of one head and two tails ?  P {1 head and 2 tails}? s 1 s k S A B C 1 Pr events 7 EE 401 Digital Communications .. more on Pr axioms Additional axioms derived: 1.P [  A ]= 1 − P [ A ] where  A is the complement of the event 2.When A and B are NOT mutually exclusive , P [ A ∪ B ]= P [ A ] P [ B ]− P [ A ∩ B ] P [ A ∩ B ] is the joint probability 3.When A 1, A 2, ... , A m are mutually exclusive and all possible outcomes of an experiment , P [ A 1 ] P [ A 2 ] ..  P [ A m ]= 1 8 EE 401 Digital Communications Conditional Pr : Bayes’ theorem Conditional probability: an experiment involving the events A and B P [ B ∣ A ] :probability of B given that A is occured or conditional probability of B given A P [ B ∣ A ]= P [ A ∩ B ] P [ B ] resulting P [ A ∩ B ]= P [ A ∣ B ] P [ B ] P [ A ∩ B ]= P [ B ∣ A ] P [ A ] P [ B ∩ A ]= P [ A ∩ B ] P [ B...
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This note was uploaded on 12/15/2010 for the course EE EE 401 taught by Professor Alikara during the Spring '10 term at Ankara Üniversitesi.

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10-401-03-RandomSign - EE 401 Digital Communications Random Variables and Stochastic(Random Processes  Random variables  Stochastic(Random

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