amplitude of upsampled and downsampled signal without filter
Close, but not the right formula for the up-sampler:

$$x_i[n] = x[n/L]$$

The "up sampler" is really an expander that inserts zeros between the samples of x[n], and it has the formula:

$$
x_u[n] =\begin{cases}
x[n/L] &amp; n=0, \pm L, \pm 2L, ... \\
\\
0 &amp; otherwise
\end{cases}
$$

The DTFT of the up-sampler "expander" can be derived as follows:

using $\delta(n)$ we can rewrite $x_u[n]$ as:

$$
x_u[n] = \sum_{k=-\infty}^{\infty}x[k]\delta[n-kL]
$$

Applying definition of DTFT to $x_u[n]$:

$$
\begin{aligned}
X_u(\omega) &amp;= \sum_{n=-\infty}^{\infty} \left(x_u[n] e^{-j\omega n} \right) \\
\\
X_u(\omega) &amp;= \sum_{n=-\infty}^{\infty} \left( \sum_{k=-\infty}^{\infty}x[k]\delta[n-kL] \right) e^{-j\omega n} \\
\end{aligned}
$$

Applying sifting property of $\delta$:

$$
\begin{aligned}
X_u(\omega) &amp;= \sum_{n=-\infty}^{\infty} x[k] e^{-j\omega L k} \\
\\
X_u(\omega) &amp;= X(\omega L)
\end{aligned}
$$

Thus, the gain of up-sampler "expander" in freq domain is unity.