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Unformatted text preview: EE263 S. Lall 2009.10.21.02 Homework 5 Due Thursday 10/29 1. Projection matrices. 44020 A matrix P ∈ R n × n is called a projection matrix if P = P T and P 2 = P . (a) Show that if P is a projection matrix then so is I − P . (b) Suppose that the columns of U ∈ R n × k are orthonormal. Show that UU T is a projection matrix. (Later we will show that the converse is true: every projection matrix can be expressed as UU T for some U with orthonormal columns.) (c) Suppose A ∈ R n × k is full rank, with k ≤ n . Show that A ( A T A ) − 1 A T is a projection matrix. (d) If S ⊆ R n and x ∈ R n , the point y in S closest to x is called the projection of x on S . Show that if P is a projection matrix, then y = Px is the projection of x on range( P ). (Which is why such matrices are called projection matrices . . . ) 2. Gradient of some common functions. 42090 Recall that the gradient of a differentiable function f : R n → R , at a point x ∈ R n , is defined as the vector ∇ f ( x ) = ∂f ∂x 1 . . . ∂f ∂x n , where the partial derivatives are evaluated at the point x . The first order Taylor approx- imation of f , near x , is given by ˆ f tay ( z ) = f ( x ) + ∇ f ( x ) T ( z − x ) . This function is affine, i.e. , a linear function plus a constant. For z near x , the Taylor approximation ˆ f tay is very near f . Find the gradient of the following functions. Express the gradients using matrix notation. (a) f ( x ) = a T x + b , where a ∈ R n , b ∈ R . (b) f ( x ) = x T Ax , for A ∈ R n × n . (c) f ( x ) = x T Ax , where A = A T ∈ R n × n . (Yes, this is a special case of the previous one.) 3. Least-squares deconvolution. 46050 A communications channel is modeled by a finite-impulse-response (FIR) filter: y ( t ) = n − 1 summationdisplay τ =0 u ( t − τ ) h ( τ ) , where u : Z → R is the channel input sequence, y : Z → R is the channel output, and h (0) , . . ., h ( n − 1) is the impulse response of the channel. In terms of discrete-time convolution we write this as y = h ∗ u . You will design a deconvolution filter or equalizer which also has FIR form: z ( t ) = m − 1 summationdisplay τ =0 y ( t − τ ) g ( τ ) , where z : Z → R is the filter output, y is the channel output, and g (0) , . . . , g ( m − 1) is the impulse response of the filter, which we are to design. This is shown in the block diagram below....
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- Fall '08
- Pixel, Impulse response, projection matrix, S. Lall, line integral measurements