Signal Processing and Linear Systems-B.P.Lathi copy

In convolution also we follow the same procedure e

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Unformatted text preview: , C2, a nd longer a n a pproximation b ut a n equality C3 3.3 Signal representation by Orthogonal Signal Set exists, for which (3.33) is no N e(t) = f (t) - L cnxn(t) a re t he projections (components) of f on C 3X3 f· Xi X l, X 2, We show in Appendix 3A t hat E e, t he energy of t he e rror signal e(t), is minimized if we choose (3.35a) Ci=-- Xi ' Xi = 1 - - 2 f 'Xi I Xil (3.39a) i = 1, 2, 3 (3.35b) Note t hat t he e rror in t he a pproximation is zero when f is a pproximated in terms of three m utually o rthogonal vectors: X l,X2, a nd X 3. T he reason is t hat f is a t hree-dimensional vector, a nd t he vectors X l,X2, a nd X 3 r epresent a c omplete s et of orthogonal vectors in three-dimensional space. Completeness here means t hat i t is impossible t o find another vector X 4 in this space, which is o rthogonal to all t he t hree vectors X l,X2, a nd X 3. Any vector in this space can t hen be represented (with zero e rror) in terms of these t hree vectors. Such vectors are known as b asis vectors. I f a s et of vectors { xd is n ot complete, t he error in t he a pproximation will generally n ot b e zero. Thus, in t he three-dimensional case discussed above, i t is generally not possible to represent a vector f in terms of only two basis vectors without a n e rror. T he choice o f basis vectors is n ot unique. In fact, a set of basis vectors corresponds t o a p articular choice of coordinate system. Thus, a 3-dimensional vector f may be represented in many different ways, depending on t he c oordinate system used. 3.3-2 Orthogonal Signal Space m #n (3.36) m =n I f t he energies E n = 1 for all n, t hen t he s et is normalized a nd is called a n o rthonormal s et. An orthogonal set can always be normalized by dividing x n(t) by f fn for all n . Now, consider approximating a signal f (t) over t he interval [ tl, t 2J by a set of N real, m utually o rthogonal signals Xl(t), X2(t), . .. , XN(t) as q Xl(t) + C2X2(t) + ... + CNXN(t) (3.37a) N = L CnXn(t) n =l T he e rror e(t) i n t he a pproximation (3.37) 1 =En 1t2 f(t)xn(t) dt n =I,2, . .. , N (3.39b) t, For this choice of t he coefficients Cn, i t is shown in Appendix 3A t hat t he e rror signal energy E e is given by (3.40) Observe t hat t he e rror energy E e generally decreases as N , t he n umber of terms, is increased because t he t erm Ck 2 E k is nonnegative. Hence, i t is possible t hat t he error energy - > 0 as N - > 0 0. W hen t his happens, t he o rthogonal signal set is s aid to be c omplete. I n this case, Eq. (3.37b) is no more a n a pproximation b ut a n equality (3.41) We s tart w ith real signals first, a nd t hen e xtend the discussion t o complex signals. We proceed with our signal approximation problem using clues a nd insights developed for vector approximation. As before, we define orthogonality of a real signal s et Xl(t), X2(t), . .. , XN(t) over interval [ tl, t 2J as f (t) ~ (3.38) n =l (3.34) In this case, q Xl, C 2X2, a nd a nd X 3, respectively; t hat is, 185 (3.37b) where t he coefficients C n a re given by Eq. (3.39). Because t he e rror signal energy approaches zero, it follows t hat t he energy of f (t) is now equal t o t he...
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