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Unformatted text preview: Midterm 2 - Solutions ECEIOI Fall 2008 1. For convenience, defme the rectangular pulse rect(t):= 1 if OStSl o otherwise The two pulse shapesare then A+------. 51 (t) 2A A o T o T (a.. I. The simplest of the manypossible non-orthogonal bases of the spaceisjust S1(t) and s2(t) themselves. (b, : I. From observation, the two terms comprising S1 (t) are orthogonal and one of them is sit). A directrouteto an orthonormal basisis to normalize the components. This gives smce (sketchit and see) A more difficult way to get an orthonormal basisis to applythe Gram-Schmidt procedure. Here it'sbest to startwith S2(t). Starting with Sl (t) is too horrible to contemplate, although it eventually givesa different, and equally correct, result So,normalizing sit) gives Project Sl (t) onto the s2(t) subspaceto getthe approximation S Ihat(t)='V 2(t)J: S1 (th 2(t)dt .= ~orect(i)-(Ao~=Aorect(i) This leaves the error e 1(t)=s 1(t) - s Ihat(t)=AOSin(hoi) Normalization gives the nextorthonormal basisfunction which are the sameoneswe obtained by inspection. for OStST OStST (c, : The signal constellation is obtained by calculating the coefficients of the signals with respectto the orthonomal basis. The various possible orthormal bases are all relatedbyrespectto the orthonomal basis....
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- Spring '08
- Linear Algebra, Orthonormal basis, ZL, Spectral efficiency, basis function