this sum is not a function of time. \[CorrAvval\]

Since $B{T_\mathrm{s}}\leq 1$, the sum in (\[eq:bandlimited\]) has only one nonzero term (i.e., $Q(0)$ can be nonzero whereas $Q(2\pi n /{T_\mathrm{s}})=0$ for all $n\neq 0$ due to (\[eq:bandwidthLimitation\])).

As a result of Corollary \[CorrAvval\], (\[eq:DC\]) for the regular Nyquist pulses and root-Nyquist pulses considered in Sec. \[sec:Pulse-shaping\] and \[sec:RootPulse-shaping\] (but not SRC and SDJ) can be written as $$\mu=\left(\hat{a}-L\right)\max_{0\leq t< {T_\mathrm{s}}}\sum_{k=-\infty}^{\infty}\left|q(t-k{T_\mathrm{s}})\right|-L\frac{Q(0)}{{T_\mathrm{s}}},\label{eq:DC_New}$$ where $Q(0)=\overline{q}{T_\mathrm{s}}$ for all pulses, see (\[eq:Overbar\_q\]). It appears that solving the summation in (\[eq:DC\_New\]) is impossible analytically even for simple pulses.

For bandlimited pulses where $B{T_\mathrm{s}}\leq 1$, the transmitted signal (\[eq:TxSignal\]) is unchanged if all constellation points in $\mathcal{C}$ are shifted by a constant offset. \[CorrSevom\]

Since the chosen pulse has limited bandwidth given by (\[eq:bandwidthLimitation\]), using (\[eq:coroll1\]) given in Corollary \[CorrAvval\], the transmitted signal (\[eq:TxSignal\]) can be written as $$\begin{aligned}
x(t)=A\left(\mu+\sum_{k=-\infty}^{\infty}\left(a_{k}-L+L\right)q\left(t-k{T_\mathrm{s}}\right)\right)\nonumber\\ = A\left(\mu+\sum_{k=-\infty}^{\infty}\left(a_{k}-L\right)q\left(t-k{T_\mathrm{s}}\right)+L\frac{Q(0)}{{T_\mathrm{s}}}\right)\label{eq:alaki}.\end{aligned}$$ Substituting the required bias given by (\[eq:DC\_New\]), (\[eq:alaki\]) can be written as $$x(t)=A\Bigg(\left(\hat{a}-L\right)\max_{0\leq t< {T_\mathrm{s}}}\left[\sum_{i=-\infty}^{\infty}\left|q(t-i{T_\mathrm{s}})\right|\right]+\sum_{k=-\infty}^{\infty}\left(a_{k}-L\right)q\left(t-k{T_\mathrm{s}}\right)\Bigg).\label{eq:symmetric}$$ It can be seen that (\[eq:symmetric\]) only depends on symbols through $\hat{a} - L$ and $a_{k}-L$. Both terms are independent of the constellation offset.

Theorem \[CorrSevom\] shows that for narrow-band pulses defined in (\[eq:bandwidthLimitation\]), the constellation offset does not have 