In OFDM system, the cyclic prefix is added into the head of time-domain signal to enable the circular convolution with the channel, and then perform one-tap frequency-domain equalization at the receiver. However, what's about if we added the cyclic prefix into the end of the time-domain signal, is there a way to consider it as a circular convolution and then perform one-tap equalizer?

For example, let’s have the frequency-domain signal $s \in \mathbb{C}^{N \times 1}$, $N$ is the number of subcarriers. It’s multiplied with $N \times N$ DFT matrix to have the time-domain signal $S$; $S = F^H \times s$. Let’s add the cyclic prefix as a suffix instead of prefix. So, we will have $S_{cp} = \begin{bmatrix} S & S_{1+N_{cp}} \end{bmatrix}$ , where $S_{1+N_{cp}}$ is the first $N_{cp}$ samples of time-domain signal $S$. When transmitting that signal through a multipath channel, and have the received signal $r = h * S_{cp}$; can we still perform Fourier transform to $S_r$ and one-tap equalizer in that case?, $s_r \in \mathbb{C}^{N \times 1}$ is the received signal $r$ after removing the added cyclic prefix.


1 Answer 1


You raised an interesting question. The purpose of padding prefix in each time-domain OFDM symbol is in order to do frequency-domain equalization (EQ). The reason why EQ is needed is to remove (or reduce) the inter-symbol-interference (ISI) in the received signal. For example, a received signal with ISI may be expressed as $r(t)=x(t)+αx(t-τ)$, where $x(t)$ is the desired signal, $τ>0$ is a time delay, and $α$ is a (complex) attenuation. In other words, the desired signal $x(t)$ is interfered by adding a delayed (and attenuated) copy of itself.

For instance, suppose $x(t)$ consists of 1024 time bins (before padding prefix), and $τ$ is equivalent to 100 time bins. So, the attenuated copies of the first 924 bins of $x(t)$ are added to the 101-th up to the 1024 bins. Thus, $x(101)$~$x(1024)$ are interfered by the attenuated copies of $x(1)$~$x(924)$. Note that $x(1)$~$x(100)$ are not interfered by any copy of $x(t)$. This makes the frequency EQ impossible, because frequency EQ requires that the ISI exits in a circular way. That is, in this example, $x(1)$~$x(100)$ need to be interfered by $x(925)$~$x(1024)$. The solution is to pad more than 100 last bins (as prefix) to the front of $x(t)$, so that after the same delay, attenuated copies of $x(925)$~$x(1024)$ are added to $x(1)$~$x(100)$ while the attenuated copies of $x(1)$~$x(924)$ are added to $x(101)$~$x(1024)$.

In short, providing a guard period between adjacent symbols is not the only reason for prefix. More importantly, because the interfering signal is a delayed copy of the desired signal with a delay $τ>0$, we need a copy of the ending part of the signal to be padded in front of the symbol.

Can the prefix be replaced by a suffix padded at the end of the symbol ? Obviously, the answer is No, unless the delay $τ$ were negative, meaning a non-casual system (that is, the desired signal were interfered by a future copy of the signal).


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