lecture_13-15 - 58 Lecture 13-15 Subspaces We have...

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58 Lecture 13-15 Subspaces: We have introduced the notion of angle, length and distance into the space of signals. The next concept we look at is related to the notion of a plane or line in the space of signals. Note that in 3-dimensions to specify a plane (passing through the origin) we need 2 coefficients, i.e., equation of a plane in 3D is: c 1 x 1 + c 2 x 2 + x 3 = 0 To specify a line in 3-dimensions we need 4 coefficients (intersection of 2 planes): c 1 x 1 + c 2 x 2 + x 3 = 0 c 3 x 1 + c 4 x 2 + x 3 = 0 This is called algebraic characterization. It turns out that it is difficult to extend this idea to the general Hilbert space. So we look at a geometric characterization. The key observation we want to make here is that: 1. If vectors vectorx and vector y belong to a plane or a line, then the vector vectorx + vectory also belongs to that plane or that line, respectively. 2. If vector vectorx belongs to a plane or a line, then αvectorx belongs to the same plane or line, respectively. Using these ideas we can define a subspace of a Hilbert space. Definition: A set of vectors V is said to be a subspace of a Hilbert space S if the following conditions are satisfied. 1. If vectorx V , vectory V then vectorx + vectory V . 2. If vectorx V then αvectorx V for all α . Examples: 1. A line in 2D euclidean space is a subspace. 2. Let S be the set of all CT signals. Let V represents a set of CT signals that are frequency limited to the range [ 1 / 2 , 1 / 2] Hz. V is a subspace of S . 3. Let S Set of all DT signals. Let V be a set of all DT signals that are frequency limited to {− π/ 2 , π/ 2 } . V is a subspace of S . Before we go further, let us look at two concrete applications where different inner products come into play.
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59 Digital Communication: Consider a 2-input-2-output communication channel modeled as: Y = X + N where X is the input vector, N is the noise vector and Y is the output vector. All these vectors belong to R 2 . Let N be modeled as a zero mean jointly Gaussian random vector with probability density function: f N ( n ) = 1 2 πK 1 2 exp bracketleftbig 1 2 n T K 1 n bracketrightbig where K is the covariance matrix of N K = bracketleftbigg E [( N 1 E [ N 1 ]) 2 ] E [( N 1 E [ N 1 ])( N 2 E [ N 2 ])] E [( N 1 E [ N 1 ])( N 2 E [ N 2 ])] E [( N 2 E [ N 2 ]) 2 ] bracketrightbigg where N = bracketleftbigg N 1 N 2 bracketrightbigg . We want to transmit a binary digit of information over the channel. If 0 is to be transmitted, send X = s 0 and if 1 is to be transmitted, send X = s 1 where s 0 and s 1 are 2 vectors which are deterministic quantities. The decoder is given by a partition of the output space R 2 into 2 regions D 0 and D 1 . If Y D 0 then 0 is decoded and if Y D 1 then 1 is decoded. Goal: Choose D 0 and D 1 optimally to minimize probability of error ( J ) in decoding.
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