participles instead of sub.clauses of reason and time

[Elvis]

Hi,
I need someone to explain how do we replace the mentioned grammatical point above.
Thanks a lot in advance for any help.
The many examples you will write down the better it will be to understand the case.

Many thanks from the depth of my heart.

[Buddhaheart]

Look at this complex sentence with an adverb clause of time: I woke up after I heard the noise.
Hearing the noise, I woke up. Hearing is a participle qualifying the pronoun ‘I’. It’s a verbal adjective.

Here’s an example with a sub-clause of reason: I did it because I thought all was safe.
Thinking all was safe, I did it. Thinking is a participle qualifying the pronoun ‘I’. It’s a verbal adjective.

[fluffyhamster]

I wonder if it's possible to properly hear something whilst still asleep. You could just say that something (e.g. the loud music and/or dog next door) woke you.

'Hearing a sound, I got up' however would be fine with this fussy huffy fluffyhamster.