amplitude of upsampled and downsampled signal without filter Given:
$$
DTFT\{x[n]\}=X(\omega)= \begin{cases}
1 &amp; |\omega| \leq 2/\pi
\\
0 &amp; 2/ \pi &lt; |\omega| &lt; \pi 
\end{cases}\ \ \ \ \ (periodic\ 2\pi)
$$

If I downsample $X(\omega)$ by M. I get:

$$
\begin{aligned}
x_d[n]      &amp;= x[nM] \\
\\
X_d(\omega) &amp;= 1/M\ X(\omega/M)
\end{aligned}
$$

Now what happens if I up-sample $x[n]$ again by $L$?

why can't I just let $M = 1/L$ and substitute into down-sampling equation to get up-sampling equation?

$$
\begin{aligned}
x_i[n]      &amp;= x[n/L] \\
\\
X_i(\omega) &amp;= L\ X(L \omega)
\end{aligned}
$$

I'm a little bit confused because my book is telling me that upsampling equation doesn't scale the amplitude by L:

$$
X_i(\omega) = X(L\omega)
$$

Just wondering what happened to the L term when upsampling?

(page 112, Schaum's Outline, Digital Signal Processing, Second Edition)

If I look up DTFT time scaling property, I find:

$$
DTFT\{\ x[an]\ \} = 1/a X(\omega / a)
$$

Which seems to confirm that upsampler should have an L term ..
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.