Today I learned how to make nice html presentations from markdown with reveal.js.
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How a Half Band Interpolator works.
We start with 16 samples: $ x[nT_s] \text{ for } n \in [0,15]$.
As we have 16 samples, $\frac{1}{f_0} = T_0 = 16 T_s$.
When we perform the IDFT and DFT on our signal, we get:
$$ IDFT : x[nT_s] = \frac{1}{16} \sum_{k=0}^{15} X(k) e^{2 \pi i f_0 k n T_s} \\
= \frac{1}{16} \sum_{k=0}^{15} X(k) e^{\frac{2 \pi i k n}{16}} .$$
We have expressed our signal in terms of a basis that consists of the 16th roots of unity.
$$ DFT : X(k) = \sum_{n = 0}^{15} x[nT_s] e^{\frac{-2 \pi i k n f_0}{f_s} } \\
= \sum_{n = 0}^{15} x[nT_s] e^{\frac{-2 \pi i k n}{16} }.$$
The DFT at $X(k)$ gives the component in our signal along the $k$-th 16th root of unity.
We’ve expressed our sample in terms of the basis $\{e^{\frac{2 \pi i k}{16}} | k \in [-7,8] \} $ with 16 elements that is periodic modulo $2 \pi$.
Our discrete-time frequency is:
$$\hat{\omega} = \frac{2 \pi k }{16} = \frac{2 \pi k T_s}{T_s} = \frac{2 \pi k f_0}{f_s}. $$
Follow the sampling theorem and limiting $f_s \geq 2 f_0$, constrains
$k$ to lie between -7 and 8.
We’d like to double our sample rate using only the data points that we already have. Mathematically this is,
$$x[nT_s] = x\left [(2n)\frac{T_s}{2}\right ] = \frac{1}{16}\sum_{k=-7}^{8}X(k) e^{\frac{2\pi k }{16 T_s}\frac{T_s}{2} 2n } \\
= \frac{1}{16}\sum_{k=-7}^{8}X(k) e^{\frac{2\pi k 2n }{32} } .$$
We’ve changed the basis to $\{e^{\frac{2 \pi i k}{32}} | k \in [-7,8] \} $,
changed the time variable to $2n$, and kept the original coefficients $X(k)$.
We have a new discrete time frequency:
$$ 2 \hat{\omega}' = \hat{\omega} .$$
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Since the data actually is sampled at $\omega$, we now have spectral images
at multiples of $\hat {\omega} = 2 \pi$ and thus at multiples of $\hat{\omega}' = \pi$.
Now the spectral images lie in the domain -\pi < $\hat{\omega}' \leq \pi$!
There is aliasing! We need to apply a low pass filter to remove the unwanted images.