In my experience, clients often want their website to be the first step towards a new brand presence. But even if we just do a relatively simple redesign; we are changing the way the brand is perceived.

Do you measure the impact of a (re)design on the way a brand is perceived?

I currently don't. And I think I might be missing out. Possible metrics could be: Awareness, Brand recall, Traffic, Reach, Impressions, Differentiation, Perceived quality, Customer satisfaction, Repeat visits/purchases, Customer lifetime value, etc..

Is brand perception a part of UX design for you? And if so; how does it play a role in your process?