Move API docs generation to docs/api

docs/api/index.rst

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+Pipecat API Reference Docs
+==========================
+
+Welcome to Pipecat's API reference documentation!
+
+Pipecat is an open source framework for building voice and multimodal assistants.
+It provides a flexible pipeline architecture for connecting various AI services,
+audio processing, and transport layers.
+
+Quick Links
+-----------
+
+* `GitHub Repository <https://github.com/pipecat-ai/pipecat>`_
+* `Website <https://pipecat.ai>`_
+
+
+API Reference
+-------------
+
+Core Components
+~~~~~~~~~~~~~~~
+
+* :mod:`pipecat.frames`
+* :mod:`pipecat.processors`
+* :mod:`pipecat.pipeline`
+
+Audio Processing
+~~~~~~~~~~~~~~~~
+
+* :mod:`pipecat.audio`
+* :mod:`pipecat.vad`
+
+Services
+~~~~~~~~
+
+* :mod:`pipecat.services`
+
+Transport & Serialization
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+* :mod:`pipecat.transports`
+* :mod:`pipecat.serializers`
+
+Utilities
+~~~~~~~~~
+
+* :mod:`pipecat.clocks`
+* :mod:`pipecat.metrics`
+* :mod:`pipecat.sync`
+* :mod:`pipecat.transcriptions`
+* :mod:`pipecat.utils`
+
+.. toctree::
+   :maxdepth: 2
+   :caption: API Reference
+   :hidden:
+
+   api/pipecat.audio
+   api/pipecat.clocks
+   api/pipecat.frames
+   api/pipecat.metrics
+   api/pipecat.pipeline
+   api/pipecat.processors
+   api/pipecat.serializers
+   api/pipecat.services
+   api/pipecat.sync
+   api/pipecat.transcriptions
+   api/pipecat.transports
+   api/pipecat.utils
+   api/pipecat.vad
+
+Indices and tables
+==================
+
+* :ref:`genindex`
+* :ref:`modindex`
+* :ref:`search`