# Is it possible to consider circular convolution in case of appending the guard interval as a suffix instead of a prefix

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.

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).