# Why is it that the input to an IFFT is first reordered in an OFDM Modulator

I am back with my questions! I recently came across a paper titled "Low Latency IFFT Design for OFDM Systems Supporting Full-Duplex FDD" where they claim a new method of IFFT input mapping to reduce the output latency. However, they have mentioned how in a conventional OFDM, the null subcarriers are mapped in the middle and the data subcarriers to the side. I have attached a picture of this. But my question is why is it done in the first place? Is it something to do with the efficient use of butterfly operation?

Also, please mention a reference where it is cited that conventional OFDM systems use this method of reordering before performing an IFFT operation.

Thank you

• This is the very definition of inverse dft en.wikipedia.org/wiki/…. Specifically, the $x(0)$ of iDFT is the first positive subcarrier, and go on ... – AlexTP Jun 29 at 17:13
• @AlexTP Of course I would know the definition of IFFT but you didn't specify why the subcarriers are reordered first before IFFT operation, and why not just simply feed in the data and null subcarriers without any reordering. – Amit Sravan Jun 30 at 4:12
• they're not reordered. This is just a graphical representation to match usual understanding of words like "positive". Nothing to see here, move on... – Marcus Müller Jun 30 at 7:01
• @AmitSravan maybe my comment was too short. I meant what MarcusMuller said. This is convention. If you start with time-domain signal, the first piece data of the output FFT corresponds to DC, and should be mapped to the first position of IFFT input. – AlexTP Jun 30 at 10:02

Since Discrete Fourier Basis Vectors are $$2\pi$$-Periodic, hence, negative frequencies $$-\frac{2\pi k}{N}$$ are conventionally represented as $$(2\pi - \frac{2\pi k}{N} = \frac{2\pi}{N} (N-k))$$. It has nothing to do with Butterfly Structure of FFT Algorithm.
The above statement basically means that $$-k^{th}$$ tone is represented as $$(N-k)^{th}$$ tone in DFT/FFT. That is why you are seeing that DC and positive tones are placed as it is, but negative tone $$-k$$ is placed at $$N-k$$ before taking IFFT.
It is precisely the consequence of the convention that DFT coefficients are $$X[k]$$ where $$k = 0,1,2,...,(N-1)$$. $$k$$ goes from $$0$$ to $$(N-1)$$ and not from $$(-\frac{N}{2})$$ to $$(\frac{N}{2} - 1)$$.
So, DC + Positive $$\frac{N_d}{2}$$ tones are placed as first $$\frac{N_d}{2}+1$$ values, and then Negative $$\frac{N_d}{2}$$ tones are placed in reverse order from the end, i.e. $$-1^{th}$$ tone becomes $$(N-1)^{th}$$ value, $$-2^{nd}$$ tone becomes $$(N-2)^{nd}$$ value and so on to $$-\frac{N_d}{2}^{th}$$ tone becomes $$(N-\frac{N_d}{2})^{th}$$ value. The rest of the $$(N - (N_d +1))$$ tones are zero padded. This array of values becomes the IFFT input.