This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 21 WEEK 5 RECAP Connecting tDomain and sDomain 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 xaxis and yaxis, respec tively, while the vectors i := (1 , 0) xaxis j := (0 , 1) yaxis are unit vectors 3 i.e. of length 1 living on the two perpendicular (or orthog onal denoted by ) xaxis and yaxis, 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 realvalued signals by sines and cosines. These begin to sound like Fourier Series representation which we will take up in the next section....
View
Full
Document
This note was uploaded on 04/01/2008 for the course EE 102 taught by Professor Levan during the Spring '08 term at UCLA.
 Spring '08
 Levan

Click to edit the document details