Written by Travis M. Moore
Last edited 2-Oct-2019

Frequency Response

If a transfer function attenuates (makes softer) certain frequencies, we can say that the function is filtering out those frequencies. In fact, a system with a transfer function that passes some frequencies and attenuates others is called a filter. Just like any other system, we can characterize a filter by measuring the output for a range of inputs. A common way to refer to the transfer function of a filter is as its frequency response.

Specifically, we can measure the output amplitude (y axis) for a number of input frequencies (x axis). As long as we keep the input level of each frequency we put through the system constant, the shape of the transfer function will reveal what the system does. Figure 1 shows the brute force method of measuring the transfer function: input one frequency at a time, and record the output.

Measuring a transfer function frequency-by-frequency
FIG. 1. © Travis M. Moore (2019)

You can image how tedious this can get if you're measuring a filter's frequency response in real life. Luckily, there is a shortcut. As long as the filter is linear, we can simply put all the frequencies through the filter at once. What type of signal contains a wide range of frequencies that all have the same amplitude? White noise.

Let's look at a Fourier Transform to confirm the spectrum of white noise is flat (the same amplitude across all frequencies).

Spectrum of white noise is flat
FIG. 2. © Travis M. Moore (2019)

Figure 2 shows a white noise is the time and frequency domains (upper and lower panels, respectively). The time domain shows a jumble of non-repeating frequencies (about as far from a pure tone as you can get!), and the frequency domain shows a nice, flat spectrum. If we pass this noise through a filter, we will be testing every frequency at once, and the shape of the white noise spectrum will change to take on the shape of the filter. Figure 3 shows the time and frequency domains of the white noise after passing it through a filter that attenuates frequencies above 4 kHz.

Low-Pass Filter
FIG. 3. © Travis M. Moore (2019)

The filter in Figure 3 is called a low-pass filter. After looking at the filter's frequency response, the name makes sense. The filter in Figure 3 is passing (i.e., not attenuating) low frequencies, and attenuating high frequencies. There are four main filter shapes to consider, and each one is named after the shape of its frequency response. Let's take a look at them.

Filter Shapes
FIG. 4. © Travis M. Moore (2019)

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