In ZP-OFDM, The convolution is linear. it's different from CP-OFDM which is circular convolution leading to have one tap equalizer in frequency domain.
if I have a ZP-OFDM system, and we have a signal to transmit $X$ of length $M$ x $1$, after adding the zero padding, the resultant transmitted signal will be $x$ of length $N$ x $1$. where the received signal is:
$y = h*x + v$, where $h$,$x$,$v$,$*$ denote multi-path channel of length $L$ x $1$, transmitted signal of length $N$x$1$, AWGN, and convolution operation, respectively.
to recover the transmitted signal from the received signal $y$, let the matrix $G$ to be the MMSE equalizer built based on the toeplitz matrix $H$ equivalent to $h$. so estimated signal $x$ = $Gy$.
The question which I couldn't get a clear answer for it, Is it better to built the matrix $G$ of size $M$x$N$ and multiply it directly with $y$ which is $N$x$1$, resulting the transmitted signal of size $M$x$1$. OR, I start discard the part equivalent to the zero padding from $y$ to have the length of $M$x$1$ and then build the matrix $G$ of size $M$x$M$ to be multiplied with $y$ after removing the part equivalent into the zero padding?
In simulation, the first option gives slightly better performance and more stable. Why ? what is the explanation of each case?
EDIT
1- The length of the received signal $y$ is considered to be $N$x $1$, it means that the delay of channel is already discarded.
2- We considered the length of the guard interval to be longer than the channel delay
3- The toeplitz matrix $H$ and $G$ was built as following,
C = [h;zeros(N-L,1)];
R = [h(1), zeros(1,N-1)];
H = toeplitz(C,R);
G = inv((1/SNRv(i))*eye(N) + (H')*H) * (H');
