How to do brand perception research?

As a user researcher I know how to conduct studies around usability, design, feelings, needs, pains and other topics regarding a specific product.

What I am struggling with is transferring this knowledge/experience over to the brand perception of a company from a users perspective.

How do I measure the brand perception and how do I compare the current brand design with a redesign (validated by users)? I thought about implicit reaction time tests as one methodology but couldn't find available tools with this background.