notes4 - ECE 562 Fall 2011 September 7, 2011 Signal Space...

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ECE 562 Fall 2011 September 7, 2011 Signal Space Concepts In order to proceed with the design and analysis of digital communication systems (in complex baseband) it is important for us to understand some properties of the space in which the complex message bearing signal s ( t ) lies. Inner Product and Norm Let x ( t ) and y ( t ) be complex valued signals with t [ a, b ]. If a and b are not specified, it is assumed that t ( -∞ , ). Definition 1. (Inner Product) h x, y i Δ = Z b a x ( t ) y * ( t ) du . (1) The inner product satisfies the necessary axioms: ± h x, y i = h y, x i * ² h x + y, z i = h x, z i + h y, z i ³ h αx, y i = α h x, y i , for any complex number α . ´ h x, x i ≥ 0, and h x, x i = 0 iff x ( t ) = 0 for all t . Signals x ( t ) and y ( t ) are said to be orthogonal if h x, y i = 0. The orthogonality of x ( t ) and y ( t ) is sometimes denoted by x y . Definition 2. (Norm) The inner product defined above induces the following norm: k x k = p h x, x i . (2) It is easy to show that the above quantity is a valid norm in that it satisfies the required axioms. (Based on
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notes4 - ECE 562 Fall 2011 September 7, 2011 Signal Space...

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