Find the frequency response of a system
I'll go through it in the $z$-domain. First, we find the transfer function $H_1(z) = \frac{V(z)}{X(z)}$. As you noted, in the time domain, $x[n]$ and $v[n]$ are related as follows:

$$
v[n] = x[n] + g * v[n - M]
$$

Take the $z$-transform of the above and you get:

$$
V(z) = X(z) + z^{-M}G(z) V(z)
$$

taking advantage of the convolution property, which states that $x_1[n] * x_2[n] \Leftrightarrow X_1(z)X_2(z)$, as well as the time-delay property, which states that $x[n-M] \Leftrightarrow z^{-M}X(z)$. Now, we can rearrange terms in the above to get:

$$
H_1(z) = \frac{V(z)}{X(z)} = \frac{1}{1 - z^{-M}G(z)}
$$

Now, find the transfer function $H_2(z) = \frac{Y(z)}{V(z)}$. They are related in the time domain as follows:

$$
y[n] = v[n-M] - g * v[n]
$$

Take the $z$-transform of the above to yield:

$$
Y(z) = z^{-M} V(z) - G(z)V(z)
$$

Rearrange terms again to get:

$$
H_2(z) = \frac{Y(z)}{V(z)} = z^{-M} - G(z)
$$

Now, you can get the overall transfer function $H(z)$ by multiplying the two:

$$
\begin{align}
H(z) &amp;= H_1(z) H_2(z) = \frac{V(z)}{X(z)}\frac{Y(z)}{V(z)} = \frac{Y(z)}{X(z)} \\
&amp;= \frac{1}{1-z^{-M}G(z)} \left(z^{-M} - G(z)\right) \\
&amp;= \frac{-G(z) + z^{-M}}{1-z^{-M}G(z)}
\end{align}
$$

which has some nice symmetry to it. The overall response will be dependent upon the system $g$ in the feedback path. If you want to evaluate the frequency response, just let $z = e^{j\omega}$:

$$
H(\omega) = \frac{-G(\omega) + e^{-j\omega M}}{1-e^{-j\omega M}G(\omega)}
$$