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You will not receive the same signal.

Let's try another analogy.

Imagine a perfect ball of dough. That perfect ball of dough is called an Isotropic Radiator, essentially, a perfect sphere of signal. This type of antenna doesn't actually exist because it would be an antenna with no length.

Now, every time you increase the gain from the Isotropic Radiator, you push down on the dough and flatten it a bit.

When you increase the gain, you flatten the dough a bit more. So, you'll see that a 6dBi antenna has a flatter, wider pattern than the 3dBi antenna.

Yes, the EIRP is the same, but with the higher gain the power is more concentrated. It's like my analogy in my last post. A magnifying glass doesn't increase the power of the sun, it just focuses it. Just like an antenna.

Gene