# Hermitian symmetry in OFDM systems

I am trying to understand the usage of Hermitian symmetry in OFDM systems and have a couple of questions regarding this.

1. What is the reason of using the Hermitian symmetry in OFDM?
2. How can we arrange the data in terms of Hermitian symmetry?
3. In OFDM system where do we arrange our data as Hermitian symmetry?
4. Is there any reference or documentation that explains Hermitian symmetry in OFDM system?

## 2 Answers

The Hermitian symmetry is used to obtain a real-valued time-domain signal. It is a special case of OFDM called discrete multitone (DMT). It exploits a property of the discrete Fourier transform (DFT), namely that the DFT of a real-valued signal has Hermitian symmetry. The motivation is usually the channel: if the signal shall be transmitted over a low-pass channel without additional radio-frequency upconversion (which requires an oscillator at the transmitter and a local oscillator at the receiver) the transmit signal must be real-valued.

Given an OFDM system with $N$ subcarriers the following conditions must hold $$c_{-k} = c^*_k\\ c_0\in \mathbb{R}$$ where $c_k$ is the complex value of subcarrier with index $k\in[-N/2\ldots N/2-1]$. This can be achieved by mapping $N/2$ complex symbols to subcarriers $0$ to $N/2-1$ and then assigning the respective complex conjugate value to the according subcarrier with negative index. This would normally take place after the mapper and before the inverse DFT (IDFT) block.

• can you please recommend a source for further details? – HappyBee Apr 18 '16 at 16:51
1. Hermitian Symmetry is used when you want the IFFT output to be real-valued, so that it could be transmitted directly over a wire, for example.

2-3. At the input to the N-point IFFT, you should have f(N-n) = conj(f(n)) where n = 0 to N/2.

• I thought n = 0 to N/2-1, isn't it? – HappyBee Apr 18 '16 at 17:18