Given the standard form of MC modulation as \begin{equation}\label{xt} x(t)=\sum_{m=0}^{M-1}\sum_{n=-N/2}^{N/2-1} X[m,n]g(t-m\Delta T)e^{j2\pi n\Delta F (t-m\Delta T)}, \end{equation} the prototype pulses in the conventional OFDM are the rectangular pulses $g(t)=\Pi_{T_g}(t)$ and $r(t)=\Pi_{T_{sp}}(t)$, where $T_g\ge T_{sp}+T_{cds}$ and $T_{sp}=\frac{1}{\Delta F}$. When prototype pulses are not the default rectangular pulses, the corresponding MC waveform is called pulse-shaped OFDM (PS-OFDM) [1], as pulse shaping in MC waveforms refers to windowing or truncation via the prototype pulse $g(t)$.

    In ODDM, we have $\Delta F=\frac{1}{NT_0}$ and then $T_{sp}=NT_0$. Because the prototype pulse $g(t)$ in ODDM is the pulse train $u(t)$ not $\Pi_{NT_0}(t)$, ODDM is a kind of PS-OFDM or, more precisely, a pulse-train-shaped OFDM (PTS-OFDM).

As a popular variant of OFDM, filtered OFDM [2] [3] is closely related to the inverse discrete Fourier transform (IDFT) based implementation of OFDM and PS-OFDM^*1.

     Recall that pulse-shaping in MC waveforms is performed by the prototype pulse $g(t)$. In (\ref{xt}), the $m$th MC symbol \begin{equation}\label{xmt} x_m(t)=\sum_{n=-N/2}^{N/2-1} X[m,n]g(t-m\Delta T)e^{j2\pi n\Delta F (t-m\Delta T)}, \end{equation} can be obtained by truncating or pulse-shaping \begin{align}\label{txmt} \tilde x_m(t)= \sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi n \Delta F (t-m\Delta T)}, \end{align} using the prototype pulse $g(t)$. Clearly, $\tilde x_m(t)$ is an infinite periodic signal with period $T_{sp}=\frac{1}{\Delta F}$ and is strictly band-limited to $[-\frac{N}{2}\Delta F, (\frac{N}{2}-1)\Delta F]$. Therefore, $\tilde x_m(t)$ can be sampled at a rate of $N\Delta F$ to have $\tilde {\mathbf x}_m= [\tilde x_m[0], \cdots, \tilde x_m[N-1]]^T$, which represents $N$ samples spaced by $\frac{1}{N\Delta F}=\frac{T_{sp}}{N}$ within $T_{sp}$. Since $\tilde {\mathbf x}_m$ is equal to the IDFT of $[X[m,0],\cdots,X[m,\frac{N}{2}-1],X[m,-\frac{N}{2}],\cdots,$ $X[m,-1]]^T$ [4], we can pass cyclic extension (CE) of $\tilde {\mathbf x}_m$ denoted by \begin{equation} \tilde {\mathbf x}_m^{CE}=[\cdots, \tilde {\mathbf x}_m^T, \tilde {\mathbf x}_m^T, \tilde {\mathbf x}_m^T \cdots]^T \end{equation} through an ideal low-pass filter (LPF) with cut-off frequency $\frac{N\Delta F}{2}$ to obtain $\tilde x_m(t)$ [5]. After that, the $g(t)$-based pulse-shaping is applied to generate $x_m(t)$ in (\ref{xmt}) as \begin{align} x_m(t)=(\overbrace{\tilde {\mathbf x}_m^{CE}*\sinc(tN\Delta F)}^{\tilde x_m(t)} )\times g(t-m\Delta T) \end{align} where $*$ stands for convolution. It is important to note that, as the bandwidth of $x_m(t)$ is the same as that of $x(t)$, and is given by $B_x>N\Delta F$, $x_m(t)$ cannot be represented by $\tilde {\mathbf x}_m$ or truncated $\tilde {\mathbf x}_m^{CE}$, due to the violation of the sampling theorem.

     In practice, we can simplify the implementation by adjusting the parameters. Since $N$ is usually fixed to a power of $2$ for using the inverse fast Fourier transform (IFFT), $x_m(t)$ can become approximately band-limited to $[-\frac{N}{2}\Delta F, (\frac{N}{2}-1)\Delta F]$, if we have sufficient vacant subcarriers (VSCs) at the edges of the band. In fact, a practical OFDM system often has $\dot N < N$ subcarriers to be written as \begin{equation} x_m(t)=\sum_{\substack{n=-\dot N/2, n\ne 0}}^{\dot N/2}X[m,n]g(t-m\Delta T)e^{j2\pi n \Delta F(t-m\Delta T)}, \end{equation} whose bandwidth is $B_x=\dot N\Delta F+B_g \le N\Delta F$. Then, $\tilde {\mathbf x}_m^{CE}$ truncated (windowed) by the sampled prototype pulse $\mathbf g=[\ldots, g\left(\frac{n}{N\Delta F}\right),\ldots]^T$ represents the Nyquist-sampled or oversampled $x_m(t)$, and can thus be used to generate \begin{equation}\label{filterxmt} x_m(t)\approx \left(\tilde {\mathbf x}_m^{CE} \cdot \mathbf g \right)*\sinc\left(tN\Delta F\right), \end{equation} which is exactly the so-called filtered OFDM in [3].

    Due to its low implementation complexity and high bandwidth flexibility via adjusting $\dot N$, filtered OFDM, as shown in Fig. 1 where the LPF is the interpolation filter in the DAC, has been widely adopted in practical OFDM systems [6] [7].      For ODDM signal, we have the $m$th symbol \begin{align} \check{x}_m(t)=\sum_{n=-N/2}^{N/2-1}X[m,n]u(t-m\frac{T_0}{M})e^{j2\pi\frac{n}{NT_0} (t-m\frac{T_0}{M})}, \nonumber \end{align} which is $ \dot {x}_m(t)=\sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi\frac{n}{NT_0} (t-m\frac{T_0}{M})} $ pulse-shaped by $u(t)=\sum_{\dot n=0}^{N-1}a(t-\dot nT_0)$. Let $\check{x}_m^{\dot n}(t)$ denote $\check{x}_m(t)$ within the duration of $\dot n$th elementary pulse in $u(t-m\frac{T_0}{M})$, we have \begin{align} \check {x}_m^{\dot n}(t)&=\sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi\frac{n}{NT_0} (t-m\frac{T_0}{M})} a(t-\dot n T_0-m\frac{T_0}{M}), \nonumber \\ & \approx \sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi\frac{n\dot n}{N}} a(t-\dot n T_0-m\frac{T_0}{M}), \nonumber \end{align} for $\dot n T_0+(m-Q)\frac{T_0}{M}\le t \le \dot n T_0+(m+Q)\frac{T_0}{M}$ when $2Q\ll M$ [8]. Then, we can obtain \begin{align} \check {x}_m(t) \approx \sum_{\dot n=0}^{N-1}\sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi\frac{n\dot n}{N}} a(t-\dot n T_0-m\frac{T_0}{M}), \end{align} where $\sum_{n=-N/2}^{N/2-1}X[m,n]e^{j2\pi\frac{n\dot n}{N}}, 0\le \dot n \le N-1$ are exactly the IDFT of $[X[m,0],\cdots,X[m,\frac{N}{2}-1],X[m,-\frac{N}{2}],\cdots,$ $X[m,-1]]^T$ denoted by $\tilde {\mathbf x}_m$ in (\ref{filterxmt}). As a result, the $m$th ODDM symbol can be approximately generated as [5] [8]. \begin{equation}\label{filtercheckxmt} \check{x}_m(t)\approx \tilde {\mathbf x}_m*a(t). \end{equation} Recall that the fundamental frequency of the ODDM is $\Delta F=\frac{1}{NT_0}$. For a conventional filtered OFDM with the same fundamental frequency, the cut-off frequency of the LPF or interpolation filter is $\frac{1}{2T_0}$. Considering $a(t)$ has a cut-off frequency greater than $\frac{M}{2T_0}$, the approximate ODDM in (\ref{filtercheckxmt}) is a wideband filtered OFDM.

For the time being, other pulses or pulse-trains that can form (bi)orthogonal WH subsets same as the DDOP are still unknown. Nevertheless, because the DDOP is basically a time-limited and band-limited periodic impulse train, it has another representation.

     As shown in [9] [10], the DDOP can be obtained by applying a rectangular window $\Pi_{NT_0}\left(t+\frac{T_0}{2}\right)$ to a periodic impulse train \begin{align} \ddot u(t)=\sum_{n=-\infty}^{\infty} \delta (t-nT_0), \end{align} and then filtering the truncated impluse train using a filter with the impulse response $a(t)$, where the order of windowing and filtering can be exchanged.

     Because the frequency domain representation of $\ddot u(t)$ is a Fourier series, it can also be written as a frequency domain periodic impulse train having infinite number of frequency bins as \begin{align} \ddot U(f)=\frac{1}{T_0}\sum_{m=-\infty}^{\infty} \delta (f-\frac{m}{T_0}), \end{align} which indicates that $\ddot u(t)$ can be rewritten as \begin{align} \ddot u(t)=\sum_{n=-\infty}^{\infty} \delta (t-nT_0)=\frac{1}{T_0}\sum_{m=-\infty}^{\infty} e^{j2\pi \frac{m}{T_0}t}. \end{align} By passing $\ddot u(t)$ through $a(t)$ and then applying the rectangular window, we have another representation of the DDOP given by \begin{align}\label{utf} u(t)=\frac{1}{T_0}\left(\sum_{m=-\frac{\check M}{2}}^{\frac{\check M}{2}} A\left(\frac{m}{T_0}\right)e^{j2\pi \frac{m}{T_0}t}\right)\times \Pi_{NT_0}\left(t+\frac{T_0}{2}\right), \end{align} where $(\check M+1)$ frequency bins or complex sinusoids are spaced by $1/T_0$ under the envelope of $A(f)$.

     It is interesting to observe that by letting $a(t)=\sinc(\frac{tM}{T_0})$, we have $\check M=M$ and $A(\frac{m}{T_0})=A(0), \forall m$ in (\ref{utf}). Then, the summation over $m$ represents a Dirichlet kernel, which converges to a periodic extension of $\sinc(\frac{tM}{T_0})$ with period $T_0$, as $M\gg 2Q$ [11].

    Meanwhile, it should be noted that the elementary pulse $a(t)$ in the DDOP $u(t)$ can be a Nyquist pulse [8] [10], such as the $\sinc(\frac{tM}{T_0})$ pulse. As mentioned in [8], the purpose of using the root Nyquist pulse is to easily achieve a matched filtering at the receiver. When we use a Nyquist pulse as $a(t)$, the matched filtering may be simplified to the sampling.