Change formula delimiter to $ and $$.

DSP.md

@@ -28,7 +28,7 @@ Image credits are from Wikipedia.
 
 ## Signals
 
-A *signal* is a function \\(x(t): \mathbb{R} \rightarrow \mathbb{R}\\) of amplitudes (voltages, sound pressure levels, etc.) defined on a time continuum, and a *sequence* is a function \\(x(n): \mathbb{Z} \rightarrow \mathbb{R}\\) defined only at integer points, often written as \\(x_n\\).
+A *signal* is a function $x(t): \mathbb{R} \rightarrow \mathbb{R}$ of amplitudes (voltages, sound pressure levels, etc.) defined on a time continuum, and a *sequence* is a function $x(n): \mathbb{Z} \rightarrow \mathbb{R}$ defined only at integer points, often written as $x_n$.
 In other words, a signal is an infinitely long continuum of values with infinite time detail, and a sequence is an infinitely long stream of samples at evenly-spaced isolated points in time.
 
 Analog hardware produces and processes signals, while digital algorithms handle sequences of samples (often called *digital signals*.)
@@ -36,41 +36,31 @@ Analog hardware produces and processes signals, while digital algorithms handle
 
 ### Fourier analysis
 
-In DSP you often encounter periodic signals that repeat at a frequency \\(f\\), like cyclic waveforms.
-Periodic signals are composed of multiple harmonics (shifted cosine waves) with frequencies \\(fk\\) for \\(k = 1, 2, 3, \ldots\\).
+In DSP you often encounter periodic signals that repeat at a frequency $f$, like cyclic waveforms.
+Periodic signals are composed of multiple harmonics (shifted cosine waves) with frequencies $fk$ for $k = 1, 2, 3, \ldots$.
 In fact, a periodic signal can be *defined* by the amplitudes and phases of all of its harmonics.
-For simplicity, if \\(x(t)\\) is a periodic signal with \\(f=1\\), then
-\\[
-	x(t) = A_0 + \sum_{k=1}^\infty A_k \cos \left( 2\pi k t + \phi_k \right)
-\\]
-where \\(A_0\\) is the *DC component* (the signal's average offset from zero), \\(A_k\\) is the amplitude of harmonic \\(k\\), and \\(\phi_k\\) is its angular phase.
+For simplicity, if $x(t)$ is a periodic signal with $f=1$, then
+$$x(t) = A_0 + \sum_{k=1}^\infty A_k \cos \left( 2\pi k t + \phi_k \right)$$
+where $A_0$ is the *DC component* (the signal's average offset from zero), $A_k$ is the amplitude of harmonic $k$, and $\phi_k$ is its angular phase.
 Each term of the sum is a simple harmonic signal, and when superimposed, they reconstruct the original signal.
 
 As it turns out, not only periodic signals but *all* signals have a unique decomposition into shifted cosines.
-However, instead of integer harmonics, you must consider all frequencies \\(f\\), and instead of a discrete sum, you must integrate over all shifted cosines of real-numbered frequencies.
-Furthermore, it is easier to combine the amplitude and phase into a single complex-valued component \\(X(f) = A(f) e^{i \phi(f)}\\).
+However, instead of integer harmonics, you must consider all frequencies $f$, and instead of a discrete sum, you must integrate over all shifted cosines of real-numbered frequencies.
+Furthermore, it is easier to combine the amplitude and phase into a single complex-valued component $X(f) = A(f) e^{i \phi(f)}$.
 With these modifications, the decomposition becomes
-\\[
-	x(t) = \int_{-\infty}^\infty X(f) e^{2\pi i t f} df
-\\]
-The \\(X(f)\\) component can be calculated with
-\\[
-	X(f) = \int_{-\infty}^\infty x(t) e^{-2\pi i t f} dt
-\\]
+$$x(t) = \int_{-\infty}^\infty X(f) e^{2\pi i t f} df$$
+The $X(f)$ component can be calculated with
+$$X(f) = \int_{-\infty}^\infty x(t) e^{-2\pi i t f} dt$$
 This is known as the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform).
 
-For a digital periodic signal with period \\(N\\), no harmonics above \\(N/2\\) can be represented, so its decomposition is truncated.
-\\[
-	x_k = \sum_{n=0}^{N/2} X_n e^{2\pi i kn / N}
-\\]
+For a digital periodic signal with period $N$, no harmonics above $N/2$ can be represented, so its decomposition is truncated.
+$$x_k = \sum_{n=0}^{N/2} X_n e^{2\pi i kn / N}$$
 Its inverse is
-\\[
-	X_n = \sum_{k=0}^{N} x_k e^{-2\pi i kn / N}
-\\]
-Since the sequence \\(x_k\\) is real-valued, the components \\(X_n\\) are conjugate symmetric across \\(N/2\\), i.e. \\(X_n = X_{N-n}^\ast\\), so we have \\(N/2 + 1\\) unique harmonic components.
+$$X_n = \sum_{k=0}^{N} x_k e^{-2\pi i kn / N}$$
+Since the sequence $x_k$ is real-valued, the components $X_n$ are conjugate symmetric across $N/2$, i.e. $X_n = X_{N-n}^\ast$, so we have $N/2 + 1$ unique harmonic components.
 This is the (real) [discrete Fourier transform](https://en.wikipedia.org/wiki/Discrete_Fourier_transform) (DFT).
 
-The [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) is a method that surprisingly computes the DFT of a block of \\(N\\) samples in \\(O(N \log N)\\) steps rather than \\(O(N^2)\\) like the DFT by using a recursive [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) algorithm, while being more numerically stable than the naive sum.
+The [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) is a method that surprisingly computes the DFT of a block of $N$ samples in $O(N \log N)$ steps rather than $O(N^2)$ like the DFT by using a recursive [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) algorithm, while being more numerically stable than the naive sum.
 This makes the DFT feasible in high-performance DSP when processing blocks of samples.
 The larger the block size, the lower the average number of steps required per sample.
 FFT algorithms exist in many libraries, like [pffft](https://bitbucket.org/jpommier/pffft/) used in VCV Rack itself.
@@ -78,11 +68,11 @@ FFT algorithms exist in many libraries, like [pffft](https://bitbucket.org/jpomm
 
 ### Sampling
 
-The [Nyquist–Shannon sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) states that a signal with no frequency components higher than half the sample rate \\(f_{sr}\\) can be sampled and reconstructed without losing information.
-In other words, if you bandlimit a signal (with a brick-wall lowpass filter at \\(f_{sr}/2\\)) and sample points at \\(f_{sr}\\), you can reconstruct the bandlimited signal by finding the unique signal which passes through all points and has no frequency components higher than \\(f_{sr}/2\\).
+The [Nyquist–Shannon sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) states that a signal with no frequency components higher than half the sample rate $f_{sr}$ can be sampled and reconstructed without losing information.
+In other words, if you bandlimit a signal (with a brick-wall lowpass filter at $f_{sr}/2$) and sample points at $f_{sr}$, you can reconstruct the bandlimited signal by finding the unique signal which passes through all points and has no frequency components higher than $f_{sr}/2$.
 
-In practice, analog-to-digital converters (ADCs) apply an approximation of a brick-wall lowpass filter to remove frequencies higher than \\(f_{sr}/2\\) from the signal.
-The signal is integrated for a small fraction of the sample time \\(1/f_{sr}\\) to obtain an approximation of the amplitude at a point in time, and this measurement is quantized to the nearest digital value.
+In practice, analog-to-digital converters (ADCs) apply an approximation of a brick-wall lowpass filter to remove frequencies higher than $f_{sr}/2$ from the signal.
+The signal is integrated for a small fraction of the sample time $1/f_{sr}$ to obtain an approximation of the amplitude at a point in time, and this measurement is quantized to the nearest digital value.
 
 Digital-to-analog converters (DACs) convert a digital value to an amplitude and hold it for a fraction of the sample time.
 A [reconstruction filter](https://en.wikipedia.org/wiki/Reconstruction_filter) is applied, producing a signal close to the original bandlimited signal.
@@ -95,7 +85,7 @@ Fortunately, modern DACs and ADCs as cheap as $2-5 per chip can digitize and rec
 
 ### Aliasing
 
-The Nyquist–Shannon sampling theorem requires the original signal to be bandlimited at \\(f_{sr}/2\\) before digitizing.
+The Nyquist–Shannon sampling theorem requires the original signal to be bandlimited at $f_{sr}/2$ before digitizing.
 If it is not, reconstructing will result in an entirely different signal, which usually sounds ugly and is associated with poor-quality DSP.
 
 Consider the high-frequency sine wave in red.
@@ -104,16 +94,16 @@ If correctly bandlimited, the original signal would be zero (silence), and thus
 
 [![](https://upload.wikimedia.org/wikipedia/commons/2/28/AliasingSines.svg)](https://en.wikipedia.org/wiki/File:AliasingSines.svg)
 
-A square wave has harmonic amplitudes \\(\frac{1}{k}\\) for odd harmonics \\(k\\).
-However, after bandlimiting, all harmonics above \\(f_{sr}/2\\) become zero, so its reconstruction should look like this.
+A square wave has harmonic amplitudes $\frac{1}{k}$ for odd harmonics $k$.
+However, after bandlimiting, all harmonics above $f_{sr}/2$ become zero, so its reconstruction should look like this.
 
 [![](https://upload.wikimedia.org/wikipedia/commons/b/b3/Gibbs_phenomenon_50.svg)](https://en.wikipedia.org/wiki/File:Gibbs_phenomenon_50.svg)
 
 The curve produced by a bandlimited discontinuity is known as the [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon).
 A DSP algorithm attempting to model a jump found in sawtooth or square waves must include this effect, such as by inserting a minBLEP or polyBLEP signal for each discontinuity.
-Otherwise, higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below \\(f_{sr}/2\\).
+Otherwise, higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below $f_{sr}/2$.
 
-Even signals containing no discontinuities, such as a triangle wave with harmonic amplitudes \\((-1)^k / k^2\\), must be correctly bandlimited or aliasing will occur.
+Even signals containing no discontinuities, such as a triangle wave with harmonic amplitudes $(-1)^k / k^2$, must be correctly bandlimited or aliasing will occur.
 One possible method is to realize that a triangle wave is an integrated square wave, and an integrator is just a filter with a -20dB per [decade](https://en.wikipedia.org/wiki/Decade_%28log_scale%29) slope.
 Since linear filters commute, a bandlimited integrated square wave is just an integrated bandlimited square wave.
 
@@ -127,136 +117,104 @@ It is sometimes *not* required for reverb, linear filters, audio-rate FM of sine
 ## Linear filters
 
 A linear filter is a operation that applies gain depending on a signal's frequency content, defined by
-\\[
-	Y(s) = H(s) X(s)
-\\]
-where \\(s = i \omega\\) is the complex angular frequency, \\(X\\) and \\(Y\\) are the [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) of the input signal \\(x(t)\\) and output signal \\(y(t)\\), and \\(H(s)\\) is the [transfer function](https://en.wikipedia.org/wiki/Transfer_function) of the filter, defining its character.
+$$Y(s) = H(s) X(s)$$
+where $s = i \omega$ is the complex angular frequency, $X$ and $Y$ are the [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) of the input signal $x(t)$ and output signal $y(t)$, and $H(s)$ is the [transfer function](https://en.wikipedia.org/wiki/Transfer_function) of the filter, defining its character.
 Note that the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) is not used because of time causality, i.e. we do not know the future of a signal.
 The filter is "linear" because the filtered sum of two signals is equal to the sum of the two individually filtered signals.
 
-A log-log plot of \\(H(i \omega)\\) is called a [Bode plot](https://en.wikipedia.org/wiki/Bode_plot).
+A log-log plot of $H(i \omega)$ is called a [Bode plot](https://en.wikipedia.org/wiki/Bode_plot).
 
 [![](https://upload.wikimedia.org/wikipedia/commons/6/60/Butterworth_response.svg)](https://en.wikipedia.org/wiki/File:Butterworth_response.svg)
 
-To be able to exploit various mathematical tools, the transfer function is often written as a rational function in terms of \\(s^{-1}\\)
-\\[
-	H(s) = \frac{\sum_{p=0}^P b_p s^{-p}}{\sum_{q=0}^Q a_q s^{-q}}
-\\]
-where \\(a_q\\) and \\(b_p\\) are called the *analog filter coefficients*.
-With sufficient orders \\(P\\) and \\(Q\\) of the numerator and denominator polynomial, you can approximate most linear analog filters found in synthesis.
+To be able to exploit various mathematical tools, the transfer function is often written as a rational function in terms of $s^{-1}$
+$$H(s) = \frac{\sum_{p=0}^P b_p s^{-p}}{\sum_{q=0}^Q a_q s^{-q}}$$
+where $a_q$ and $b_p$ are called the *analog filter coefficients*.
+With sufficient orders $P$ and $Q$ of the numerator and denominator polynomial, you can approximate most linear analog filters found in synthesis.
 
-To digitally implement a transfer function, define \\(z\\) as the operator that transforms a sample \\(x_n\\) to its following sample, i.e. \\(x_{n+1} = z[x_n]\\).
-We can actually write this as a variable in terms of \\(s\\) and the sample time \\(T = 1/f_{sr}\\).
-(Consider a sine wave with angular frequency \\(\omega\\). The \\(z\\) operator shifts its phase as if we delayed by \\(T\\) time.)
-\\[
-	z = e^{sT}
-\\]
+To digitally implement a transfer function, define $z$ as the operator that transforms a sample $x_n$ to its following sample, i.e. $x_{n+1} = z[x_n]$.
+We can actually write this as a variable in terms of $s$ and the sample time $T = 1/f_{sr}$.
+(Consider a sine wave with angular frequency $\omega$. The $z$ operator shifts its phase as if we delayed by $T$ time.)
+$$z = e^{sT}$$
 
 A first order approximation of this is
-\\[
-	z = \frac{e^{sT/2}}{e^{-sT/2}} \approx \frac{1 + sT/2}{1 - sT/2}
-\\]
+$$z = \frac{e^{sT/2}}{e^{-sT/2}} \approx \frac{1 + sT/2}{1 - sT/2}$$
 and its inverse is
-\\[
-	s = \frac{1}{T} \ln{z} \approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}
-\\]
+$$s = \frac{1}{T} \ln{z} \approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}$$
 
 This is known as the [Bilinear transform](https://en.wikipedia.org/wiki/Bilinear_transform).
 In digital form, the rational transfer function is written as
-\\[
-	H(z) = \frac{\sum_{n=0}^N b_n z^{-n}}{\sum_{m=0}^M a_m z^{-m}}
-\\]
-Note that the orders \\(N\\) and \\(M\\) are not necessarily equal to the orders \\(P\\) and \\(Q\\) of the analog form, and we obtain a new set of numbers \\(a_m\\) and \\(b_n\\) called the *digital filter coefficients*.
+$$H(z) = \frac{\sum_{n=0}^N b_n z^{-n}}{\sum_{m=0}^M a_m z^{-m}}$$
+Note that the orders $N$ and $M$ are not necessarily equal to the orders $P$ and $Q$ of the analog form, and we obtain a new set of numbers $a_m$ and $b_n$ called the *digital filter coefficients*.
 
-The *zeros* of the filter are the nonzero values of \\(z\\) which give a zero numerator, and the *poles* are the nonzero values of \\(z\\) which give a zero denominator.
-A linear filter is stable (its [impulse response](https://en.wikipedia.org/wiki/Impulse_response) converges to 0) if and only if all poles lie strictly within the complex unit circle, i.e. \\(|z| < 1\\).
+The *zeros* of the filter are the nonzero values of $z$ which give a zero numerator, and the *poles* are the nonzero values of $z$ which give a zero denominator.
+A linear filter is stable (its [impulse response](https://en.wikipedia.org/wiki/Impulse_response) converges to 0) if and only if all poles lie strictly within the complex unit circle, i.e. $|z| < 1$.
 
-We should now have all the tools we need to digitally implement any linear analog filter response \\(H(s)\\) and vise-versa.
+We should now have all the tools we need to digitally implement any linear analog filter response $H(s)$ and vise-versa.
 
 ### IIR filters
 
 An infinite impulse response (IIR) filter is a digital filter that implements all possible rational transfer functions.
-By multiplying the denominator of the rational \\(H(z)\\) definition above on both sides and applying it to an input \\(x_k\\) and output \\(y_k\\), we obtain the difference relation
-\\[
-	\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}
-\\]
-Usually \\(a_0\\) is normalized to 1, and \\(y_k\\) can be written explicitly.
-\\[
-	y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1}^M a_m y_{k-m}
-\\]
+By multiplying the denominator of the rational $H(z)$ definition above on both sides and applying it to an input $x_k$ and output $y_k$, we obtain the difference relation
+$$\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}$$
+Usually $a_0$ is normalized to 1, and $y_k$ can be written explicitly.
+$$y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1}^M a_m y_{k-m}$$
 This iterative process can be computed for each sample
 
-For \\(N, M = 2\\), this is a [biquad filter](https://en.wikipedia.org/wiki/Digital_biquad_filter), which is very fast, numerically stable (assuming the transfer function itself is mathematical stable), and a reasonably good sounding filter.
+For $N, M = 2$, this is a [biquad filter](https://en.wikipedia.org/wiki/Digital_biquad_filter), which is very fast, numerically stable (assuming the transfer function itself is mathematical stable), and a reasonably good sounding filter.
 
-\\[
-	H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}
-\\]
+$$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}$$
 
 ### FIR filters
 
-A finite impulse response (FIR) filter is a specific case of an IIR filter with \\(M = 0\\) (a transfer function denominator of 1). For an input and output signal, the difference relation is
-\\[
-	y_k = \sum_{n=0}^N b_n x_{k-n}
-\\]
+A finite impulse response (FIR) filter is a specific case of an IIR filter with $M = 0$ (a transfer function denominator of 1). For an input and output signal, the difference relation is
+$$y_k = \sum_{n=0}^N b_n x_{k-n}$$
 They are computationally straightforward and always stable since they have no poles.
 
-Long FIR filters (\\(N \geq 128\\)) like FIR reverbs (yes, they are just linear filters) can be optimized through FFTs.
-Note that the above formula is the convolution between vectors \\(y\\) and \\(b\\), and by the [convolution theorem](https://en.wikipedia.org/wiki/Convolution_theorem),
-\\[
-	y \ast b = \mathcal{F}^{-1} \{ \mathcal{F}\{y\} \cdot \mathcal{F}\{b\} \}
-\\]
-where \\(\cdot\\) is element-wise multiplication.
+Long FIR filters ($N \geq 128$) like FIR reverbs (yes, they are just linear filters) can be optimized through FFTs.
+Note that the above formula is the convolution between vectors $y$ and $b$, and by the [convolution theorem](https://en.wikipedia.org/wiki/Convolution_theorem),
+$$y \ast b = \mathcal{F}^{-1} \{ \mathcal{F}\{y\} \cdot \mathcal{F}\{b\} \}$$
+where $\cdot$ is element-wise multiplication.
 
-While the naive FIR formula above requires \\(O(N)\\) computations per sample, the FFT FIR method requires an average of \\(O(\log N)\\) computations per sample when processing blocks of \\(N\\) samples.
-A disadvantage of the FFT FIR method is that the signal must be delayed by \\(N\\) samples before any output can be determined.
-You can combine the naive and FFT methods into a hybrid approach with the [overlap-add](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) or [overlap-save](https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods, allowing you to process smaller blocks of size \\(M < N\\) at a slightly worse \\(O((N/M) \log M)\\) average computations per sample.
+While the naive FIR formula above requires $O(N)$ computations per sample, the FFT FIR method requires an average of $O(\log N)$ computations per sample when processing blocks of $N$ samples.
+A disadvantage of the FFT FIR method is that the signal must be delayed by $N$ samples before any output can be determined.
+You can combine the naive and FFT methods into a hybrid approach with the [overlap-add](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) or [overlap-save](https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods, allowing you to process smaller blocks of size $M < N$ at a slightly worse $O((N/M) \log M)$ average computations per sample.
 
 
 ### Impulse responses
 
 Sometimes we need to simulate non-rational transfer functions.
-Consider a general transfer function \\(H(f)\\), written in terms of \\(f\\) rather than \\(s\\) for simplicity.
-Our original definition of a transfer function for an input \\(x(t)\\) and output \\(y(t)\\) is
-\\[
-	Y(f) = H(f) X(f)
-\\]
+Consider a general transfer function $H(f)$, written in terms of $f$ rather than $s$ for simplicity.
+Our original definition of a transfer function for an input $x(t)$ and output $y(t)$ is
+$$Y(f) = H(f) X(f)$$
 and by applying the convolution theorem again, we obtain
-\\[
-	y(t) = (h \ast x)(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) d\tau
-\\]
-where \\(h(t)\\) is the *impulse response* of our filter.
+$$y(t) = (h \ast x)(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) d\tau$$
+where $h(t)$ is the *impulse response* of our filter.
 
-The signal \\(h(t)\\) is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since \\(\delta(t)\\) is the "identity" function of convolution, i.e. \\(h(t) = (h \ast \delta)(t)\\).
-Clapping your hands or popping a balloon (both good approximations of \\(\delta\\)) in a large cathedral will generate a very sophisticated impulse response, which can be recorded as \\(h(t)\\) and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral.
+The signal $h(t)$ is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since $\delta(t)$ is the "identity" function of convolution, i.e. $h(t) = (h \ast \delta)(t)$.
+Clapping your hands or popping a balloon (both good approximations of $\delta$) in a large cathedral will generate a very sophisticated impulse response, which can be recorded as $h(t)$ and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral.
 
 Repeating this process in the digital realm gives us the discrete convolution.
-\\[
-	y_k = \sum_{n=-\infty}^\infty h_n x_{k-n}
-\\]
+$$y_k = \sum_{n=-\infty}^\infty h_n x_{k-n}$$
 
 
 ### Brick-wall filter
 
-An example of a non-rational transfer function is the ideal lowpass filter that fully attenuates all frequencies higher than \\(f_c\\) and passes all frequencies below \\(f_c\\).
+An example of a non-rational transfer function is the ideal lowpass filter that fully attenuates all frequencies higher than $f_c$ and passes all frequencies below $f_c$.
 Including [negative frequencies](https://en.wikipedia.org/wiki/Negative_frequency), its transfer function is
-\\[
 
-H(f) = \begin{cases}
+$$H(f) = \begin{cases}
 	1 & \text{if } -f_c \leq f \leq f_c \\\\
 	0 & \text{otherwise}
-\end{cases}
+\end{cases}$$
 
-\\]
-The inverse Fourier transform of \\(H(f)\\) is
-\\[
-	h(t) = 2 f_c \operatorname{sinc}(2 f_c t)
-\\]
-where \\(\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)\\) is the [normalized sinc function](https://en.wikipedia.org/wiki/Sinc_function).
+The inverse Fourier transform of $H(f)$ is
+$$h(t) = 2 f_c \operatorname{sinc}(2 f_c t)$$
+where $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)$ is the [normalized sinc function](https://en.wikipedia.org/wiki/Sinc_function).
 
 
 ### Windows
 
-Like the brick-wall filter above, many impulse responses \\(h_n\\) are defined for all integers \\(n\\), so they are both non-causal (requires future knowledge of \\(x(t)\\) to compute \\(y(t)\\)) and infinitely long.
+Like the brick-wall filter above, many impulse responses $h_n$ are defined for all integers $n$, so they are both non-causal (requires future knowledge of $x(t)$ to compute $y(t)$) and infinitely long.
 
 *TODO*
 
@@ -285,27 +243,24 @@ This allows you to easily omit large sections of the circuit that offer nothing
 ### Numerical methods for ODEs
 
 A system of ODEs (ordinary differential equations) is a vector equation of the form
-\\[
-	\vec{x}'(t) = f(t, \vec{x}).
-\\]
-Higher order ODEs like \\(x'\'(t) = -x(t)\\) are supported by defining intermediate variables like \\(x_1 = x'\\) and writing
-\\[
-	\begin{bmatrix} x \\\\ x_1 \end{bmatrix}'(t)
-	= \begin{bmatrix} x_1 \\\\ -x \end{bmatrix}.
-\\]
+$$\vec{x}'(t) = f(t, \vec{x}).$$
+Higher order ODEs like $x''(t) = -x(t)$ are supported by defining intermediate variables like $x_1 = x'$ and writing
+
+$$
+\begin{bmatrix} x \\ x_1 \end{bmatrix}'(t)
+= \begin{bmatrix} x_1 \\ -x \end{bmatrix}.
+$$
 
 The study of numerical methods deals with the computational solution for this general system.
 Although this is a huge field, it is usually not required to learn beyond the basics for audio synthesis due to tight CPU constraints and leniency for the solution's accuracy.
 
 The simplest computational method is [forward Euler](https://en.wikipedia.org/wiki/Euler_method).
-If the state \\(\vec{x}\\) is known at time \\(t\\) and you wish to approximate it at \\(\Delta t\\) in the future, let
-\\[
-	\vec{x}(t + \Delta t) \rightarrow \vec{x}(t) + \Delta t f(t, \vec{x}).
-\\]
+If the state $\vec{x}$ is known at time $t$ and you wish to approximate it at $\Delta t$ in the future, let
+$$\vec{x}(t + \Delta t) \rightarrow \vec{x}(t) + \Delta t f(t, \vec{x}).$$
 More points can be obtained by iterating this approximation.
-It is first-order, so its error scales proportionally with \\(\Delta t\\).
-The [Runge-Kutta](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) methods are common higher-order approximations at the expense of evaluating \\(\vec{f}(t, \vec{x})\\) more times per timestep.
-For example, the fourth-order RK4 method has an error proportional to \\((\Delta t)^4\\), so error is drastically reduced with smaller timesteps compared to forward Euler.
+It is first-order, so its error scales proportionally with $\Delta t$.
+The [Runge-Kutta](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods) methods are common higher-order approximations at the expense of evaluating $\vec{f}(t, \vec{x})$ more times per timestep.
+For example, the fourth-order RK4 method has an error proportional to $(\Delta t)^4$, so error is drastically reduced with smaller timesteps compared to forward Euler.
 
 
 ## Optimization
@@ -379,7 +334,7 @@ A few important ones including their first CPU introduction date are as follows.
 - [SSE3](https://en.wikipedia.org/wiki/SSE3) (2004) Slightly extends SSE2 functionality.
 - [SSE4](https://en.wikipedia.org/wiki/SSE4) (2006) Extends SSE3 functionality.
 - [AVX](https://en.wikipedia.org/wiki/Advanced_Vector_Extensions) (2008) For processing up to 256 bits of floats.
-- [FMA](https://en.wikipedia.org/wiki/FMA_instruction_set) (2011) For computing \\(ab+c\\) for up to 256 bits of floats.
+- [FMA](https://en.wikipedia.org/wiki/FMA_instruction_set) (2011) For computing $ab+c$ for up to 256 bits of floats.
 - [AVX-512](https://en.wikipedia.org/wiki/AVX-512) (2015) For processing up to 512 bits of floats.
 
 You can see which instructions these extensions provide with the [Intel Intrinsics Guide](https://software.intel.com/sites/landingpage/IntrinsicsGuide/) or the complete [Intel Software Developer’s Manuals](https://software.intel.com/en-us/articles/intel-sdm) and [AMD Programming Reference](https://developer.amd.com/resources/developer-guides-manuals/).