# Digital Signal Processing and FIR filters

## FHT library

The Hartley transform is an integral transform closely related to the Fourier transform, but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse. Fast Hartley transformation can do the same operations as Fourier transformation. It is useful for spectrum analysis and filtering signals in DSP. Fast Hartley transformation can be two times faster than fast Fourier transformation for real numbers data.

- Sine and cosine table
- Hartley transformation
- Time domain to frequency domain
- Frequency domain to time domain
- Spectrum
- Energy spectrum
- Magnitude spectrum
- Real spectrum
- Signal manipulations
- Signal reflection
- Signal derivative
- Signal antiderivative
- Signal time and frequency shift
- Magnitude scaling
- Autocorrelation of signal
- Cross-correlation of signals
- Convolution of signals
- Mixing signals

## Filter library

A digital filter is simply a discrete-time, discrete-amplitude convolver. Basic Fourier transform theory states that the linear convolution of two sequences in the time domain is the same as multiplication of two corresponding spectral sequences in the frequency domain. Filtering is in essence the multiplication of the signal spectrum by the frequency domain impulse response of the filter. For an ideal low-pass filter the pass band part of the signal spectrum is multiplied by one and the band-stop part of the signal by zero.

## Window library

In signal processing, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or a signal (data) is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap; the "view through the window". Applications of window functions include spectral analysis, filter design, and beamforming.