The connection between compressed sensing and sparse representation

If I understand correctly, Compressed Sensing as an application of Sparse Representation is defined as:

To find linear compression schemes for huge input signals that are known to have a sparse representation, so that the input signal can be recovered efficiently from the compression (the "sketch").(ref: this question)

So based on my understanding, compressed sensing is like a precoding before analog to digital data conversion.
And my take on sparse representation is:
A signal can be represented as a linear combination of basis functions where the set of basis functions is called dictionary and data samples are much more than their features. Mathematically, in the system of linear equations $Y=DX$ where $Y \in \mathbb{R}^{n \times N} (n \ll N)$ we seek a dictionary that results in sparse representation of $Y$.

So as far as I've understood, the output of sparse representation is used in compressed sensing. In my thesis on dictionary learning, after an introduction about compressed sensing, I'd like to talk a bit about the connection between compressed sensing and sparse representation before getting into the details of sparse representation. What I fail to do is to explain the connection between the two even though I am aware of the concepts. I'd like to have a good understanding of the connection between compressed sensing and sparse representation. Any explanation would be greatly appreciated!