Frequency domain understanding of filters

When we start the digital filter, it is easy to understand the concept of the filter from the frequency domain, and in the application of signal processing, the filter is also described by some characteristics of the frequency domain.

If an analogy can be made, a digital filter can be likened to a sieve of sand. We all know the following screening process: First put a pile of sand on the sieve, this heap of sand has large sand, fine sand, and possibly stone. The process of sifting the sand is shaking the sieve. The result of the sieve is a fine sand below the sieve, while the stone and the large sand remain on top of the sieve. Back to the digital filter, it is also an input signal, through a filter, to get an output. In the frequency domain, the output signal is equal to the product of the input signal and the filter response. Sand containing large sand, fine sand and stone is the equivalent of the input signal, the sieve out of the sand is equivalent to the output signal.

Ideally, all sieve eyes are the same size, and the diameter of the sand through the sieve will not exceed the sieve eye. The case that corresponds to the filter is that only a signal less than the cutoff frequency of the filter can pass the filter. But the reality is that because sand is subjected to a different pressure during the screening process, some sand that is slightly larger than the sieve eye may also pass through the sieve. In the actual filter, there is also a small part larger than the cutoff frequency of the signal through the filter, which is called the transition zone in the filter term.

In the process of screening sand, must first put in the sieve in the sand, after a period of time, there will be fine sand out. Also in the filter, the input signal enters the filter, it may take some time before the output signal. This is called delay in the filter term. More generally, the larger sand is harder to pass through the sieve than the finer sand, and that is to say, the big sand usually has a sieve after the fine sand. This is reflected in the filter is different frequency delay time may not be the same, the term is called phase nonlinearity.

Of course, there are many analogies to be made. This simple analogy allows a very intuitive understanding of how digital filters work in the frequency domain. It is important to note that the most scientific description of how the digital filter works must also be aided by mathematical tools, and the content of the digital filter is far richer and more complex than this simple analogy.

Frequency domain understanding of filters