week-5-6-recap

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

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Unformatted text preview: 21 WEEK 5 RECAP Connecting t-Domain and s-Domain Review Problems, MT Prep, and MT. 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 := 1 and j := 2 Then (15.2) and (15.3) can be simply expressed as k = , k = , k, = 1 , 2 , (15.4) = 1 , k = (15.5) or k = k, := , k = (15.6) := 1 , k = (15.7) where k, is called the Kronecker Delta. (15.4) (or (15.6) simply tells you that 1 and 2 are (mutually) orthogonal, while (15.5) (or (15.7)) means that 1 1 := || 1 || 2 = 1 = 2 2 := || 2 || 2 (15.8) || 1 || = 1 = || 2 || (15.9) where || w || denote the length (or norm) of a vector w . Hence the vectors 1 and 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 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....
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This note was uploaded on 04/01/2008 for the course EE 102 taught by Professor Levan during the Spring '08 term at UCLA.

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

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