How important is video card memory bandwidth?

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For a graphics card. What will be the limitations between say a a card with 10Gb/s bandwidth versus a card with 20Gb/s bandwidth if both are using the same chip and clock speeds?
 
For a graphics card. What will be the limitations between say a a card with 10Gb/s bandwidth versus a card with 20Gb/s bandwidth if both are using the same chip and clock speeds?

It all depends on how bandwidth-limited the 10GB/s model is. If it isn't all all, the 20GB/s model won't be measurably faster. If it's very bandwidth-limited (a modern high-end GPU clearly would be with 10GB/s) then the 20GB/s model would be almost twice as fast. Of course, your mileage may vary from game to game. Also, the higher your definition and anti-aliasing setting, the higher the requirement for bandwidth.

There are reviews here and there that measure performance scaling with memory clocks, I suggest taking a look at them for more specific information. If you have a discrete graphics card yourself, then you might want to conduct your own experiments. It's not very complicated, tools like Catalyst Overdrive for AMD cards let you do it easily (I haven't overclocked an NVIDIA card in a while, so I'm not sure what tools are in favor these days). After that, all you need is a game with a benchmarking tool, or even just 3DMark.
 
Well, I got a 64-bit memory DX11 card by accident (it's for gfx programming) and I'm just wondering whether it will be really half-speed compared to the 128-bit versions. I'm guessing it depends how frequently the GPU needs to read/write to memory during an operation. I didn't know manufacturers still pulled those tricks btw.
 
Geforce GT430. My laptop has a GT425M with 128-bit so I have a few comparisons. Lots of untextured meshes in a scene seem to run about the same, but a few textured (including transparent textures) meshes indeed do run a LOT slower on the 64-bit version, so I'm guessing texture manipulation requires a lot of reading and writing to memory.
 
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Probably depends a lot on texture formats and how those textures are used.
Ordinary, mipmapped textures tend to have high texture cache hit rates and shouldn't put too much pressure on memory, at least not if they are compressed - using 4 channel float16 textures certainly will need more bandwidth.
You mentioned transparency, enabling color blend might easily hit bandwidth limitations if your shaders are simple.

btw it's often not only the half-width bus, but since those cards are saving pennies, as far as I can tell they are actually more likely to also use lower frequency memory chips too. So you might only get 1/3 of the bandwidth of a reference card. Worst offender I've seen is HD6450 with 32bit low-clocked ddr3 memory, which has only 1/6 or so of the bandwidth of the gddr5 reference card (which you can't buy but that is another story). Needless to say though people were complaining it's lagging even for basic desktop usage :).
 
The normal GT 430 is fairly well balanced, actually it has a FLOPS/bandwidth ratio almost identical to the GTX 560 Ti's. So halving (or cutting even more of, as mczak points out) the bandwidth should indeed have a pretty severe performance penalty.
 
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