Noise Peaks Near Nyquist Frequency
You're doing interpolation "wrong", at least in the sense I assume you want it to happen.

So what you'd normally do is just take every input sample, and insert 15 zeros in between each of them. Spectrally, the resulting signal would be your input signal, plus 14 images of it, spaced 1/15 of the output bandwidth.

Now, your approach of sample and hold interpolation is the same signal, i.e. the zero padded one, unshifted, plus shifted by one sample, plus shifted by two..., plus shifted by 14 samples. So let $x$ be the input $y$ be the zero-padded and $w$ be your repetition-interpolated signal:

$$\begin{align*}
x[n]&amp;:=\text{input}\\
y[n] &amp;= \begin{cases}
x[\frac{n}{15}]&amp; \text{for}\, n\, \mathrm{mod}\, 15 = 0\\
0&amp; \text{else}
\end{cases}\\
w[n] &amp;= x\left[\left\lfloor \frac{n}{15}\right\rfloor\right]\\
&amp;= x[n] + x[n-1] + \dots + x[n-14]\\
&amp;= \sum_{l=0}^{14}x[n-l]
\end{align*}$$

Hence, the spectrum of $w$ is just that of $y$, superimposed with 14 phase-rotated versions of itself. 

$$\begin{align*}
W[z] 
&amp;= \sum_{l=0}^{14}\mathcal{Z}\left\{x[n-l]\right\}
\end{align*}$$

In fact, you've built a signal whose 15 polyphase decomposition components are identical!

Now, that will lead to interesting effects. In contrast, if you did the usual interpolation, i.e. took my $y[n]$ and low-pass filtering it to its original Nyquist bandwidth (i.e. $\frac{1}{15}$ of the output rate) it, none of the  polyphase components of the resulting signal wouldn't be identical to the previous one -- they'd incorporate a phase shift that ensures continuity.