I was recently documenting some DSP equations in Latex and couldn't find any suitable examples so I've put them together and thought I'd share.

## The Fourier Transform

\large \mathscr{F} \normalsize (x(t))=\int_{-\infty}^{\infty}x(t)e^{-j \omega t}dt

### Rendering

$$\large \mathscr{F} \normalsize (x(t))=\int_{-\infty}^{\infty}x(t)e^{-j \omega t}dt$$

## The Discrete Time Fourier Transform

X(k) = \sum_{n=0}^{N-1}x(n).e^{-j 2 \pi nk/N},\; for\; 0 \le k \le N-1

### Rendering

$$X(k) = \sum_{n=0}^{N-1}x(n).e^{-j 2 \pi nk/N},\; for\; 0 \le k \le N-1$$

## The Inverse Discrete Time Fourier Transform

x(n) = \frac{1}{N} \sum_{k=0}^{N-1}X(k).e^{j 2 \pi nk/N},\; for\; 0 \le n \le N-1

### Rendering

$$x(n) = \frac{1}{N} \sum_{k=0}^{N-1}X(k).e^{j 2 \pi nk/N},\; for\; 0 \le n \le N-1$$

## FIR Filter

y(n) = \sum_{n=0}^{N-1}h(k).x(n-k)

### Rendering

$$y(n) = \sum_{n=0}^{N-1}h(k).x(n-k)$$

## IIR Filter

y(n) = \sum_{k=0}^{N}a_{k}.y(n-k) + \sum_{r=0}^{M}b_{r}.x(n-r)

### Rendering

$$y(n) = \sum_{k=0}^{N}a_{k}.y(n-k) + \sum_{r=0}^{M}b_{r}.x(n-r)$$

## Roots Of Quadratic Equation $ax^{2} + bx + c = 0$

x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

### Rendering

$$x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

## One-Pole Filter

### Difference Equation

y(n)=x(n)+\alpha.y(n-1)

### Rendering

$$y(n)=x(n)+\alpha.y(n-1)$$

*z* Transform

Y(z)=X(z)+\alpha z^{-1}Y(z)

### Rendering

$$Y(z)=X(z)+\alpha z^{-1}Y(z)$$

### Transfer Function

H(z)=\frac{1}{1-\alpha z^{-1}}

### Rendering

$$H(z)=\frac{1}{1-\alpha z^{-1}}$$

### Frequency Response

H(e^{j\omega T})=\frac{1}{1-\alpha e^{-j\omega T}}

### Rendering

$$H(e^{j\omega T})=\frac{1}{1-\alpha e^{-j\omega T}}$$

## M-Point Moving Average Filter

### Frequency Response

H[f]=\frac{sin(\pi f M)}{M sin(\pi f)} \: for \: 0 < f \leqslant 0.5

### Rendering

$$H[f]=\frac{sin(\pi f M)}{M sin(\pi f)} \: for \: 0 < f \leqslant 0.5$$

## Notes

\begin{equation*}

\begin{align*}

\end{align*}

\end{equation*}

VSCode doesn't support equation numbering auto increment and Colab doesn't support "equation*" or "aligned*" for disabling equation numbers. Colab requires "aligned" not "align".