How to design a filter with custom group delay -- why is it an IIR?
As pointed out by Peter K., it is true that many well-known techniques for designing FIR filters actually only design linear phase filters. However, FIR filters are very well suited for delay equalization, simply because the design process is much simpler than for IIR filters. The reason for this is the fact that the design problem can be formulated in such a way that it is still linear in the unknown filter coefficients, which is usually not the case for IIR filters. This type of design problem is known as complex approximation problem, because in the case of a desired non-linear phase response a complex frequency response must be approximated by the filter, unlike a real one in the case of a linear phase filter. You can find a lot about this topic, including Matlab programs, in this thesis. Have a look at the design example #2 on page 119 ff. where an FIR magnitude and phase equalizer is designed compensating the effects of an analog anti-aliasing filter. The following figure is taken from the thesis and it shows the group delay of the analog low pass filter, the group delay of the equalizer(s) (there are two with different magnitude responses), and the total group delay of the cascade of analog and digital filter, which is supposed to be constant in the passband: