[Solved] Array-sensor Homework01

1. A complex valued function f(z) ∈ C of a complex valued argument z ∈ C can always be expressed in terms of two real valued functions u(x,y),v(x,y) ∈ R of two real-valued variables x,y ∈ R:

f(z) = f(x + j · y) = u(x,y) + j · v(x,y).

In the following u(x,y),v(x,y) are to be continuously differentiable with respect to x and y in an arbitrarily small region around z. The complex derivative of f(z) with respect to z is defined as

(1)

• Write (1) in terms of ∂u/∂x and ∂v/∂x by using ∆z = ∆x, i.e. by moving parallel to the real axis to the point z.
• Repeat the exercise using ∆z = j ∆y, i.e. by moving parallel to the imaginary axis to the point z.
• In order for (1) to be uniquely defined, these two results must be the same. Whatconstraint does this impose on u(x,y) and v(x,y) ?
• Compare this result to the Cauchy-Riemann equations.

2. Let g(z,z^∗) = f(x,y) ∈ C be a function of a complex vector z = x + j y ∈ Cⁿ and its complex conjugate z^∗ = x−j·y ∈ Cⁿ with x,y ∈ Rⁿ. We have that the total differential of g and f, respectively, is

T

dg =dz^∗ (2) TT

df =dxdy. (3)

• By using the fact that dg = df, show that

(4)

. (5)

2

• From the previous result show that

(6)

. (7)

• If f(x,y) = u(x,y)+jv(x,y), where u(x,y),v(x,y) ∈ R show that the differential dg does not depend on the differential dz^∗ if g(z,z^∗) = f(x,y) is analytic, i.e. show that.

3. Consider the function

I(w,w^∗) = w^HRw ,

with w,p ∈ Cⁿ and R = R^H ∈ C^n×n.

• Is I(w,w^∗) a real valued function?
• Find a w that minimizes I(w,w^∗) by solving.
• Find a w that minimizes I(w,w^∗) by solving.
• Compare the results of 3b and 3c.

4. Solve the following constrained real-valued minimization problem

minimize (8)

subject to g(x1,x2) = 1 + x1 − 2x2 = 0 x1,x2,f,g ∈ R,(a) by solving (9) for x2 in terms of x1 and then minimizing (8).(b) by means of (real) Lagrangian multipliers.5. Solve the following constrained complex minimization problem:

(9)

minimize w (10)

 1 −j H

subject to g(w) = j 2 w, (11)  1 j 

with w ∈ C³,f ∈ R,g ∈ C² by means of complex Lagrangian multipliers.