[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 = xjy 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 Cⁿⁿ.

• 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.