Integral of a sign function proof

Question

Can anyone please prove this expression below? I saw this in a paper and trying to see where it is coming from.

integral(sign(A*w*cos(w*t+phi))*cos(w*t), t, 0, 2*pi/w) = 4/pi*cos(phi)

Answer 1

The equality doesn't hold as you copied it, but you can compute it anyway. I will give you some tips on how to proceed:

There are two cases: (1) cos(phi) >= 0 and (2) cos(phi) < 0. Let's consider the first one, the second is similar.

Replace phi with phi - 2kpi so that -pi < phi - 2kpi <= pi. Such a k exists because cos(phi) >= 0. The result of your expression won't change (why?) and therefore we can restrict ourselves to the case -pi < phi <= pi.

Now introduce the change of variable u = wt + phi. The expression becomes:

sg(Aw)/w*integral(sg(cos(u))cos(u-phi), u, phi, phi+2pi)

So the problem reduces to compute the integral (sg(Aw)/w is constant.)

Split the integral in three:

• I1: from phi to pi/2
• I2: from pi/2 to (3/2)pi and
• I3: from (3/2)pi to phi+2pi.

In the first interval sg(cos(u)) = 1, in the second = -1, in the third =-1. So,

I1 = integral(cos(u-phi),u,phi,pi/2)
I2 = -integral(cos(u-phi),u,pi/2,(3/2)pi)
I3 = integral(cos(u-phi),u,(3/2)pi,phi+2pi)

I think you can follow from here.