7.5 Consider the complex regression model (7.29) in the form Y = XB+V , where Y = (Y1, Y2, . . . YL) denotes the observed DFTs after they have been re-indexed and X = (X1,X2, . . . ,XL) is a matrix containing the reindexed input vectors. The model is a complex regression model with Y = Y c−iY s,X = Xc−iXs,B = Bc−iBs, and V = V c−iV s denoting the representation in terms of the usual cosine and sine transforms. Show the partitioned real regression model involving the 2L×1 vector of cosine and sine transforms, say

is isomorphic to the complex regression regression model in the sense that the real and imaginary parts of the complex model appear as components of the vectors in the real regression model. Use the usual regression theory to verify (7.28) holds. For example, writing the real regression model as y = xb + v, the isomorphism would imply

Transcribed Image Text:

(Y) = (X; X) (B.) *-*) (B) + (v.),