Wonderful! Digital Filter (Part Four)
In designing FIR filters, we do not adopt the same method as IIR design in which the analog model is determined at first and then the poles and zeros are mapped to the Z-plane. Instead, we determine a desired frequency characteristic and calculate the filter coefficient directly from the characteristic. The tap(*) coefficients are allocated symmetrically on either side of the center tap. As a result, the filter has a linear phase characteristic. The linear phase means that the phase delay of the filter is proportional to the frequency. In other words, the output has only a constant delay time (called a "group delay") to the input and there is no relative phase delay among the frequencies.
(*) Each step in digital filters is called "tap" and the coefficient is called the "tap coefficients."
Since the phase characteristic is predetermined as the linear phase, gain characteristics are enough for defining the model in designing. This means that you can realize a "pie in the sky." In addition, it has no phase distortion but a group delay. The following figure shows an attempt to obtain a pie in the sky by using TA Signal Processings FIR Design Tool.
The yellow line represents the desired frequency characteristic (the pie in the sky) and the aqua line represents the characteristic of the FIR filter designed base on it. The FIR has 349 taps but there is still some difference. More taps reduce the difference but it is hard to completely match because the original characteristic consists of straight lines linking points.
Another notable point is that the lines in the higher frequency area match well while they do not match so well in the lower frequency area. This shows that FIR filter is not so good at handling lower frequency signals. You may have found there are some ripples in the higher frequency area. The ripples are caused by using the rectangular window (I will make a page for window functions in future) to improve the frequency characteristic in the lower frequency area. Would you try to design a FIR filter by using the trial version of FIR Design Tool?
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