Fourier Filtering 1D

This figure shows 1-D low-pass filters in both time (left) and frequency (right) domains.

A signal $x(t)$, a square impulse train, and its spectrum $X(f)$ are shown in the top row. Following that there are four pairs of plots showing each of the four types of filters and their filtering effects. The filters are given as both the impulse response function $h(t)$ in time domain and its spectrum, the frequency response function, $H(f)$ in frequency domain. The filtering process is a convolution $y(t)=h(t)*x(t)$ in time domain(left) and a multiplication $Y(f)=H(f)X(f)$ in frequency domains (right).

• Moving average: the signal is convolved with a square window in time domain, or its spectrum is multiplied by a sinc function in frequency domain.
• Ideal low-pass filter: The signal spectrum is multiplied by an ideal low-pass filter in frequency domain, the signal is convolved with a sinc window. Note the ringing effect.
• Butterworth filter: The signal spectrum is multiplied by a Butterworth low-pass filter smoother than the ideal filter, the signal is convolved with a smoother window. Now the ringing effect is reduced.
• Gaussian filter: The signal spectrum is multiplied by a smooth Gaussian low-pass filter, the signal is convolved with a smooth Gaussian window. Consequently the ringing effect has totally disappeared.

From top to bottom:

• Daily temperature in Los Angeles, 2010
• The spectrum (magnitude only)
• Frequency components of lower magnitudes
• Modified spectrum with 94 percent of the components suppressed to zero
• Reconstructed time signal with 6 percent of the components carrying 99 percent energy