Problem understanding half wave symmetry

Question

I am trying to understand half wave symmetry.

I understand the first (a) graphical image is half wave symmetry but (b) seems like even symmetry and (c) seems to be odd symmetry. I am unable to find the difference. Please guide.

Edit: I read my question and it seems to be confusing. I know the b is even symmetry and c is odd but how are they half wave symmetry?

Answer 1

Let examine the symmetry.

Above is an odd symmetry. Because half of the wave on the positive side lies in the negative frquency domain while the other half on the negative time domain lies on the positive frequency domain.

Above is an even symmetry. Since the wave on the right and left are mirror images of each other, it is a an even symmetry.

Above is a half wave symmetry.

Above is an even half wave symmetry. Since its a mirror image on both positive and negative axis of time domain and signal is divided into half.

Above is an odd half wave symmetry.

Answer 2

You seem to be assuming that it is an either/or situation. It isn't. A wave can be all three: odd (OR even), have half wave symmetry, and also have quarter wave symmetry. All your examples have half wave symmetry, (b) is even in addition to having half wave symmetry, while (c) is odd in addition to having half wave symmetry.

Answer 3

one more point to add to the above answer: what is the difference between even half wave symmetry and an even signal and similarly for odd? for a signal to be half wave symmetric the area under the curve for one period should be 0. it is not necessary for an even or odd signal