You must have a few things mixed up.
1) First, M-theory is not a theory about the beginning of the universe, it's a theory about what the basic entities in nature are (namely, 2- and 5-dimensional branes, which are formed from the 0-branes of "Matrix theory", and which become 1-dimensional fundamental strings when some of M-theory's 11 dimensions are compactified). Ekpyrotic cosmology - a cyclic universe where two branes clash and rebound repeatedly, like cymbals - is inspired by the string-theory idea of braneworlds, but it's not the only or even the standard cosmological picture in string theory.
2) Gravitational waves are just a standard phenomenon in any theory that contains Einstein's gravity (general relativity), including string theory. They have not been observed directly, but the 1993 physics Nobel was awarded for measurements of a binary pulsar system which is spiraling inwards because it is losing energy at a rate consistent with the production of gravitational waves.
You must be talking about observations from the early universe, e.g. features of the cosmic microwave background. That data *is* used to distinguish between different theories of how it all began: big bang vs ekpyrotic, or one theory of inflation versus another. Presumably the brane collision in the ekpyrotic cosmology does produce distinctive effects, but I don't know what they are. If you search recent papers on arxiv.org you might find something.