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Early WRXes were notorious for having really laggy turbo systems, due to restrictive exhaust with multiple cats, and even a cat before the turbo (which is a really bad idea as far as response goes). I think that got at least a bit better in the later ones, but I'm not sure by how much.

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Most OEM turbos are on the small side, designed to produce lots of torque at low RPMs and with little lag. My old B5 S4 (2.7L, twin KKK K03s, stock), for example, made peak torque at like 2000 RPM, and it fell off a lot above 5000 (redline of 7000). Except for the slight whooshing noise, you almost couldn't tell that that car had turbos at all, and it made noticably better low-end torque than the 4.2L naturally aspirated V8 in the B6 S4 that I replaced it with.

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One other important difference in forced induction systems is how they respond to altitude. Positive displacement superchargers (and I think most traditional centrifugal ones, too) behave like naturally aspirated cars -- as the altitude rides and ambient pressure drops, the torque curve drops proportionally but stays the same shape. Turbo cars, OTOH, will spin the turbos faster to make up for the altitude losses to some degree. Raising altitude on a turbo will raise the boost threshold (minimum RPM to make full boost) and increase lag, but the peak power will not drop anywhere close to as much as a supercharged or NA car. If it's got MAP-referenced electric boost control and enough headroom in the turbo, it may not even drop the peak power at all.

The downside that I encounter for a turbo at autox is that the torque at a given RPM/TPS point can change as the boost rises. This makes powering out of corners a bit harder to get right than with a naturally aspirated or supercharged car, but it's far from an insurmountable problem.

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