FFT output scaling

I am not familiar with your fft scaling but here is explanation of your statements.

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full_range_real_out[26:25] <= {real_in[15],real_in[15]};

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this is sign bit repeated twice to sign extend the value

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full_range_real_out[24:0] <= {real_in[15:0],9'b0};

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this is left shift by inserting 9 zeros on new LSBs. the original bits 15:0 are in effect multiplied by 2^9

since 9 zero LSBs, then 15:0 make 25 bits while full range is 27 bits then sign bit is repeated twice.