t< {T_\mathrm{s}}$ is that $\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})$ and $\sum_{k=-\infty}^{\infty}|q(t-k{T_\mathrm{s}})|$ are periodic functions with period equal to ${T_\mathrm{s}}$. Since for all pulses defined in Sec. \[sec:Pulse-shaping\] and \[sec:RootPulse-shaping\], $q(t)$ rescales with ${T_\mathrm{s}}$ as $q(t)=v(t/{T_\mathrm{s}})$ for some function $v(t)$, then $\mu$ is independent of ${T_\mathrm{s}}$.

To simplify (\[eq:DC\]), Lemma \[LemFS\] and Corollary \[CorrAvval\] will be helpful, since they prove that the second term in (\[eq:DC\]) does not change over time.

\[LemFS\] For an arbitrary pulse $q(t)$, $$\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})=\frac{1}{{T_\mathrm{s}}}\sum_{n=-\infty}^{\infty}Q\left(\frac{2\pi n}{{T_\mathrm{s}}}\right)e^{\frac{j2\pi nt}{{T_\mathrm{s}}}}.$$

Since $f(t)=\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})$ is a periodic function with period ${T_\mathrm{s}}$, it can be expanded as a Fourier series. Its Fourier series coefficients are $$\begin{aligned}
C_{n} & =\frac{1}{{T_\mathrm{s}}}\intop_{-{T_\mathrm{s}}/2}^{{T_\mathrm{s}}/2}f(t)e^{-\frac{j2\pi nt}{{T_\mathrm{s}}}}dt\nonumber \\
&=\frac{1}{{T_\mathrm{s}}}\intop_{-{T_\mathrm{s}}/2}^{{T_\mathrm{s}}/2}\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})e^{-\frac{j2\pi nt}{{T_\mathrm{s}}}}dt.\label{eq:Fourier Coeffs}\end{aligned}$$ Since both $n$ and $k$ are integers, $e^{j2\pi nk}=1$. As a result, (\[eq:Fourier Coeffs\]) can be written as $$\begin{aligned}
C_{n} & =\frac{1}{{T_\mathrm{s}}}\intop_{-{T_\mathrm{s}}/2}^{{T_\mathrm{s}}/2}\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})e^{-\frac{j2\pi n}{{T_\mathrm{s}}}(t-k{T_\mathrm{s}})}dt\nonumber \\
& =\frac{1}{{T_\mathrm{s}}}\intop_{-\infty}^{\infty}q(t)e^{-\frac{j2\pi nt}{{T_\mathrm{s}}}}dt=\frac{1}{{T_\mathrm{s}}}Q\left(\frac{2\pi n}{{T_\mathrm{s}}}\right).\nonumber \end{aligned}$$ Hence, $$f(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{\frac{j2\pi nt}{{T_\mathrm{s}}}}=\frac{1}{{T_\mathrm{s}}}\sum_{n=-\infty}^{\infty}Q\left(\frac{2\pi n}{{T_\mathrm{s}}}\right)e^{\frac{j2\pi nt}{{T_\mathrm{s}}}},\label{eq:bandlimited}$$ which proves the lemma.

The usefulness of this lemma follows from the fact that for bandlimited pulses $q(t)$, (\[eq:bandlimited\]) is reduced to a finite number of terms. As a special case, we have the following corollary.

If $q(t)$ is a bandlimited pulse defined in (\[eq:bandwidthLimitation\]), where $B{T_\mathrm{s}}\leq 1$, then (\[eq:bandlimited\]) can be written as $$f(t)=\sum_{k=-\infty}^{\infty}q(t-k{T_\mathrm{s}})=\frac{1}{{T_\mathrm{s}}}Q(0).\label{eq:coroll1}$$In other words, for such $q(t)$, 