# Visualizations on Statistics and Signal Processing

Below you find a collection of widgets to illustrate key concepts in statistics and signal processing.

- Bayes Theorem / Dependent Probabilities
- A geometric interpretation of Bayes Theorem showing how dependent probabilties relate to each other.
- Gaußian Estimator
- An interactive visusalization that shows how the parameters of a bivariate Gaußian Distribution can be estimated based on a given set of samples.
- Binary Hypothesis Test
- An interactive visusalization that shows how an optimal binary classifier can be derived from two given hypothesis.
- Parameter Estimation via Likelihood and Prior
- An interactive visusalization that shows how estimate parameters of a distribution by maximizing likelihood, optionally including priors. (WIP)
- Function applied to a Random Variable
- An interactive example of a function applied to a random variable showing the resulting distribution.
- Fourier Cuboid
- Interactive exploration of the dualities between signals and their corresponding Fourier Transform
- Fractional Fourier Transform Cuboid
- Alternative implementation of the Fourier Cube above, featuring some improvements including Fractional Fourier Transform to interpolate between time and frequency domain.
- Complex plane
- Explore the geometry of complex numbers in the complex plane. See you complex number multiplication corresponds to two-dimensional scaling and rotation.
- Complex exponential
- Explore how the exponential function can be extended for complex numbers.
- Sum of two complex exponentials
- Explore how the sum of two complex exponentials may result in a real valued Oscillation and how each of the complex exponentials influence the sum.
- Signal Transformer
- Explore how a signal can be transformed by only delay and scaling.
- Signal Generator
- Generate digital 2d signals by moving your mouse and explore the effect of applying and LTI filter.
- Time Frequency Analysis
- A tool for understanding the trade-off between the time and frequency resolution when applying a filter to a signal.
- Beam Former
- Explore how multiple microphones can be used to capture signals from different directions.
- Linear Quantization
- Explore how quantization introduces noise and how the signal to noise ratio depends on the quantization strategy.
- LTI System impulse reponse
- See how the the impulse reponse of a linear time-invariant system determindes the output signal for any input signal.
- 2D Convolution
- See the effects of convolving a 2D image signal with a 2D kernel and applying element-wise non-linear functions
- 2D DFT
- Explore the relation between space and frequency domain of images to understand how the basis functions of the two-dimensional discrete fourier transform (DFT) make up an image.
- Einsum Notation
- Explore how the
`numpy.einsum`

function can be used to avoid writing custom for-loops in python while working with multidimensional arrays. - Generate Blue Noise
- Explore how the Void and Cluster algorithm for generating blue noise.
- Nyquist–Shannon sampling theorem
- Explanation of how the minimum sample frequency results from from the convolution of the spectra.