![]() ![]() Let’s add some high frequency components to our signal. In combination, these two filters only allow a “band” of frequencies between a low frequency cutoff and a high frequency cut off to pass. ![]() A bandpass filter is composed of two thresholds: (1) a filter that removes frequencies below some threshold, allowing higher frequencies to pass (i.e., “high pass”) and (2) a filter that removes frequencies above some threshold, allowing lower frequencies to pass (i.e., “low pass”). To touch on one common filtering method, let’s take a look at the bandpass filter. There are lots of other approaches we can use, though! Real-world noise can be lot nastier! In many cases, the noise might be louder than our signal. We usually don’t end up getting quite so lucky. Keeping the values above the threshold ( Figure 1, “noised FFT” in green) and dropping the values before the threshold ( Figure 1, “noised FFT” in red), we get a great reconstruction of the signal without noise ( Figure 1, “Denoised signal”)! In this simple case, we can perfectly recover our noiseless signal pretty easily! Since we have clean separation between the signal and the noise, we can set a hard threshold and drop (i.e. Looking at the magnitude values of our Fourier transform allows us to visualize the amplitude as a function of frequency. The Gaussian noise I added was zero mean additive “white” noise: it had roughly equal intensity across frequencies.Įven though the noise looks pretty grisly in the temporal domain, it is easy to separate out in the frequency domain. Super interesting! In the FFT, we can see some properties of the noise I chose to add. Figure 1: Simple denoising to remove low-power noise from a pure tone signal. ![]()
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