week-5-6-recap

week-5-6-recap - 21 WEEK 5 RECAP Connecting t-Domain and...

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21 WEEK 5 RECAP Connecting t -Domain and s -Domain Review Problems, MT Prep, and MT.
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22 NHAN LEVAN WEEK 6 RECAP DECOMPOSITION OF SIGNALS INTO ELEMENTARY SIGNALS 15. Orthogonal Representation in R 2 In R 2 any vector v can be written as — or can admit the representation, or can be represented by, a linear combination of the form (15.1) v = v x i + v y j , where the scalars v x and v y are components of v on the x -axis and y -axis, respec- tively, while the vectors i := (1 , 0) x -axis j := (0 , 1) y -axis are unit vectors 3 — i.e. of length 1 — living on the two ‘perpendicular’ (or orthog- onal — denoted by ) x -axis and y -axis, respectively. These can be expressed in terms of the “dot” product as (15.2) i · j = 0 i j, and (15.3) i · i = 1 = j · j length of i = 1 = length of j where · means the dot product (of course you remember what it is). To proceed let us rename i and j as i := φ 1 and j := φ 2 Then (15.2) and (15.3) can be simply expressed as φ k · φ = 0 , k = , k, = 1 , 2 , (15.4) = 1 , k = (15.5) or φ k · φ = δ k, := 0 , k = (15.6) := 1 , k = (15.7) where δ k, is called the Kronecker Delta. (15.4) (or (15.6) simply tells you that φ 1 and φ 2 are (mutually) orthogonal, while (15.5) (or (15.7)) means that φ 1 · φ 1 := || φ 1 || 2 = 1 = φ 2 · φ 2 := || φ 2 || 2 (15.8) || φ 1 || = 1 = || φ 2 || (15.9) where || w || denote the length (or norm) of a vector w . Hence the vectors φ 1 and φ 2 are said to have unit length, equivalently, they are unit (or normal means nor- malized) vectors. Because of (15.4) and (15.5) the vectors i and j are defined as (mutually) ortho- normal vectors . Moreover, they serve as “basis” vectors for R 2 , that is, any vector of R 2 can be uniquely written as a linear combination of the basis vectors . More- over, both i and j are “simpler” than any nonzero vector in R 2 (are they really?). This leads to the important concept of representing a signal by elementary signals , in parrticular, representation of periodic signals by complex exponentials, or — for 3 do not confuse this i with i 2 = - 1
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23 real-valued signals — by sines and cosines. These begin to sound like Fourier Series representation which we will take up in the next section. What important is the fact that the components v x and v y can be calculated from v — by means of the · product.
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