Click here to purchase the entire book in PDF format. Low-pass Filter One of the conceptually simplest filters is known as a low-pass filter because it allows low frequencies to pass through it. The question, of course, is ``how low is low?'' The answer lies in a single frequency known as the cutoff frequency or $f_{c}$. This is the frequency where the output of the filter is 3.01 dB lower than the maximum output for any frequency (although we normally round this off to -3 dB which is why it's usually called the 3 dB down point). ``What's so special about -3 dB?'' I hear you cry. This particular number is chosen because -3 dB is the level where the signal is at one half the power of a signal at 0 dB. So, if the filter has no additional gain incorporated into it, then the cutoff frequency is the one where the output is exactly one half the power of the input. (Which explains why some people call it the half-power point.) As frequencies get higher and higher, they are attenuated more and more. This results in a slope in the frequency response graph which can be calculated by knowing the amount of extra attenuation for a given change in frequency. Typically, this slope is specified in decibels per octave. Since the higher we go, the more we attenuate in a low pass filter, this value will always be negative. Figure 6.1: The frequency response of a first-order low pass filter with a cutoff frequency of 1 kHz. Note that the cutoff frequency is where the response has dropped in level by 3 dB. The slope can be calculated by dividing the drop in level by the change in frequency that corresponds to that particular drop. The slope of the filter is determined by its order. If we oversimplify just a little, a first-order low-pass filter will have a slope of -6.02 dB per octave above its cutoff frequency (usually rounded to -6 dB/oct). If we want to be technically correct about this, then we have to be a little more specific about where we finally reach this slope. Take a look at the frequency response plot in Figure 6.1. Notice that the graph has a nice gradual transition from a slope of 0 (a horizontal line) in the really low frequencies to a slope of -6 dB/oct in the really high frequencies. In the area around the cutoff frequency, however, the slope is changing. If we want to be really accurate, then we have to say that the slope of the frequency response is really 0 for frequencies less than one tenth of the cutoff frequency. In other words, for frequencies more than one decade below the cutoff frequency. Similarly, the slope of the frequency response is really -6.02 dB/oct for frequencies more than one decade above (ten times) the cutoff frequency. If we have a higher-order filter, the cutoff frequency is still the one where the output drops by 3 dB, however the slope changes to a value of $-6.02 n$ dB/oct, where $n$ is the order of the filter. For example, if you have a 3rd-order filter, then the slope is $$\textrm{slope} = \textrm{order } * -6.02 \textrm{ dB/octave}$$ (7.1) $$= 3 * -6.02 \textrm{ dB/octave}$$ (7.2) $$= -18.06 \textrm{ dB/octave}$$ (7.3)