diff --git a/_notes/fourier_series.md b/_notes/fourier_series.md
new file mode 100644
index 0000000000000000000000000000000000000000..9e6a35ecd9ebc681c37d566c1e49f6b9346c9186
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+---
+title: Fourier Series
+---
+
+## Definition
+
+Let $$f: \mathbb{R} \rightarrow \mathbb{C}$$ be a reasonably well behaved $$L$$-periodic function
+(i.e. $$f(x) = f(x + L)$$ for all $$x$$). Then $$f$$ can be expressed as an infinite series
+
+$$
+f(x) = \sum_{n \in \mathbb{Z}} A_n e^{2 \pi i n x / L}
+$$
+
+The values of the coefficients $$A_n$$ can be found using the orthogonality of basic exponential
+sinusoids. For any $$m \in \mathbb{Z}$$,
+
+$$
+\begin{align*}
+\int_{-L/2}^{L/2} f(x) e^{-2 \pi i m x / L} dx
+&= \int_{-L/2}^{L/2} \sum_{n \in \mathbb{Z}} A_n e^{2 \pi i n x / L} e^{-2 \pi i m x / L} dx \\
+&= \sum_{n \in \mathbb{Z}} A_n \int_{-L/2}^{L/2} e^{2 \pi i (n - m) x / L} dx \\
+&= \sum_{n \in \mathbb{Z}} A_n L \delta_{nm} \\
+&= L A_m
+\end{align*}
+$$
+
+where the second to last line uses [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta)
+notation. Thus by the periodicity of $$f$$ and the exponential function
+
+$$
+\begin{align*}
+A_n &= \frac{1}{L} \int_{-L/2}^{L/2} f(x) e^{-2 \pi i n x / L} dx \\
+&= \frac{1}{L} \int_{0}^{L} f(x) e^{-2 \pi i n x / L} dx
+\end{align*}
+$$
+
+
+## Relation to the Fourier Transform
+
+Now suppose we have a function $$f: \mathbb{R} \rightarrow \mathbb{C}$$, and we construct an
+$$L$$-periodic version
+
+$$
+f_L(x) = \sum_{n \in \mathbb{Z}} f(x + nL)
+$$
+
+The coefficients of the Fourier Series of $$f_L$$ are
+
+$$
+\begin{align*}
+L A_n &= \int_0^L \sum_{m \in \mathbb{Z}} f(x + mL) e^{-2 \pi i n x / L} dx \\
+&= \sum_{m \in \mathbb{Z}} \int_0^L f(x + mL) e^{-2 \pi i n x / L} dx \\
+&= \sum_{m \in \mathbb{Z}} \int_{mL}^{(m + 1)L} f(x) e^{-2 \pi i n (x - mL) / L} dx \\
+&= e^{-2 \pi i n m} \sum_{m \in \mathbb{Z}} \int_{mL}^{(m + 1)L} f(x) e^{-2 \pi i x n / L} dx \\
+&= \int_\mathbb{R} f(x) e^{-2 \pi i x n / L} dx \\
+&= \hat{f}(n/L)
+\end{align*}
+$$
+
+so
+
+$$
+f_L(x) = \frac{1}{L} \sum_{n \in \mathbb{Z}} \hat{f}(n / L) e^{2 \pi i n x / L}
+$$
+
+Thus the periodic summation of $$f$$ is completely determined by discrete samples of $$\hat{f}$$.
+This is remarkable in that an uncountable set of numbers (all the values taken by $$f_L$$ over one
+period) can be determined by a countable one (the samples of $$\hat{f}$$). Even more incredible, if
+$$f$$ has finite bandwidth then only a finite number of the samples will be nonzero. So the
+uncountable set of numbers is determined by a finite one.
+
+
+## Derivation of the Discrete Time Fourier Transform
+
+We can apply the result above to $$\hat{f}$$ as well. Recall that the Fourier transform of
+$$\hat{f}$$ is $$f(-x)$$. Thus the periodic summation of the Fourier transform of $$f$$ is
+
+$$
+\begin{align*}
+\hat{f}_L(x) &= \frac{1}{L} \sum_{n \in \mathbb{Z}} f(-n / L) e^{2 \pi i n x / L} \\
+&= \frac{1}{L} \sum_{n \in \mathbb{Z}} f(n / L) e^{-2 \pi i n x / L}
+\end{align*}
+$$
+
+This is precisely the definition of the discrete time Fourier transform.
+
+If $$f$$ is time-limited, then we'll have only a finite number of nonzero samples. But then
+$$\hat{f}$$ is necessarily not bandwidth limited, so the tails of $$\hat{f}$$ will overlap in the
+periodic summation. On the other hand, if $$f$$ is bandwidth limited, for sufficiently large $$L$$
+we can recover $$\hat{f}$$. To do so perfectly requires an infinite number of samples, but in
+practice reasonably bandwidth limited signals can still be recovered quite well from a finite number
+of samples.
+
+
+## Poisson Resummation
+
+By taking $$x = 0$$ we see that
+
+$$
+\sum_{n \in \mathbb{Z}} f(nL) = \frac{1}{L} \sum_{n \in \mathbb{Z}} \hat{f}(n / L)
+$$
+
+This is known as the
+[Poisson summation formula](https://en.wikipedia.org/wiki/Poisson_summation_formula).
diff --git a/_notes/fourier_transform.md b/_notes/fourier_transform.md
index b6ac0dd08d3f0b2261a9c58bb7557b3f534a03d5..8f4b45244caee7ccffa4b4844bb9cd86191ca3b7 100644
--- a/_notes/fourier_transform.md
+++ b/_notes/fourier_transform.md
@@ -2,7 +2,7 @@
 title: Fourier Transforms
 ---
 
-To really make sense of chapter 2 I needed to review the properties of Fourier Transforms. These
+To really make sense of chapter 2 I needed to review the properties of Fourier transforms. These
 notes are based on my prior knowledge and some helpful websites:
 
 - [Properties of Fourier Transform](http://fourier.eng.hmc.edu/e101/lectures/handout3/node2.html)
@@ -177,7 +177,7 @@ for spectral power analysis).
 
 ## Transforms of Gaussians
 
-A Fourier Transform that comes up frequently is that of a Gaussian. It can be calculated by
+A Fourier transform that comes up frequently is that of a Gaussian. It can be calculated by
 completing a square.
 
 $$
@@ -214,5 +214,5 @@ $$
 \end{align*}
 $$
 
-This depends on the variance, which is inverted by the Fourier Transform. So since the power is
+This depends on the variance, which is inverted by the Fourier transform. So since the power is
 invariant, the normalization cannot in general be conserved.