I am trying to understand the whole structure and concepts about caching. As we use TLB for fast mapping virtual addresses to physical addresses, in case if we use virtually-indexed, physically-tagged L1 cache, can one overlap the virtual address translation with the L1 cache access?
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Yes, that's the whole point of a VIPT cache.
Since the virtual addresses and physical one match over the lower bits (the page offset is the same), you don't need to translate them. Most VIPT caches are built around this (note that this limits the number of sets you can use, but you can grow their associativity instead), so you can use the lower bits to do a lookup in that cache even before you found the translation in the TLB.
This is critical because the TLB lookup itself takes time, and the L1 caches are usually designed to provide as much BW and low latency as possible to avoid stalling the often much-faster execution.
If you miss the TLB and suffer an even greater latency (either some level2 TLB or, god forbid, a page walk), it's less critical since you can't really do anything with the cache lookup until you compare the tag, but the few cycles you did save in the TLB hit + cache hit case should be the common case on many applications, so that's usually considered worthy to optimize and align the pipelines for.