Frequency Domain Low-pass Filters

Frequency domain low-pass filters mainly consist of three types: Ideal Low-pass Filter, Butterworth Low-pass Filter, and Gaussian Low-pass Filter. These filters are commonly used in signal processing to attenuate high-frequency components while preserving low-frequency elements. The Ideal Low-pass Filter serves as a theoretical model that can completely eliminate high-frequency components, but practical implementations often introduce some distortion due to the Gibbs phenomenon. In code implementation, this filter typically uses a sharp cutoff frequency with a binary transfer function (1 inside cutoff, 0 outside). Butterworth Low-pass Filter is a widely used filter featuring flat passband characteristics and steep stopband attenuation, making it suitable for applications requiring strict frequency response specifications. Implementation involves calculating filter coefficients using the Butterworth polynomial, with the filter order determining the steepness of the roll-off. Gaussian Low-pass Filter, based on the Gaussian function, exhibits smooth frequency characteristics and a wide transition band, making it appropriate for applications with less stringent frequency response requirements. The implementation typically uses a Gaussian kernel whose standard deviation parameter controls the cutoff frequency and transition width, often implemented through frequency domain multiplication or spatial convolution.