Your requirements are contadictory: you want design a FIR filter with equiripple filter with low group delay.This means that you FIR filters will have many taps and high group delay. FIR filters have linear phase if their impusle response is symmtric with respec to the middle point. I would consider releasing your requirements: e.g. a IIR filter with almost linear filter,  which will have much lower group delay.

I saw a mention of this in section 6.6 of a book "practical DSP" by Edmund Lai.

Parks-McClelland method.

It seems to be adaptation in digital domain of checbychev approximation used in many analog filters.  the author mentions that:

"Closed form design formulas are not available for these filters, however. An iterative

algorithm has to be used. A very efficient one is the Remez exchange algorithm. It was first developed in the early 1970s."     the author does give some more data in subsequent sections. It may be worthwhile looking at these. In section 6.6.3 he discusses the REMEZ exchange algorithm

 While I am an expert in analog filters, I have not yet reached that status in digital filters.

Dear Dr.Kennedy 

I worked on resizing "pass-band ripple" of FIR filter[https://www.researchgate.net/publication/281584541_The_Importance_of_Precise_sizing_of_Passband_Ripple_in_Specifying_Order_of_FIR_Filter] . I worked in matlab, so it seems possible to help you. give me your filter's specifications and i will design filter and resize "ripple" and give you results.

Conference Paper The Importance of Precise sizing of Passband Ripple in Speci...

Hello.

Have you ever tried to cast your problem as optimization problem, which the requeriments are written in the frequency domain?

I'll send give you an article that may you can get the ideia, how to cast your problem as an optimization problem. The context is related to control a lauch vehicle, where the requirements of the filter is to gain and phase the bending modes of a lauch vehicle and minimize the phase lag introduced to the system.

I think you can extrapolate it to other domains.

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20130011343.pdf

Thank you for your responses so far!

@ V.s.V Mani, that is an interesting book. I hadn't seen it before, and it does have some fresh perspectives. But  chapter 6 on FIR filters does not deal with equiripple filters with arbitrary phase.

@ Mohammadreza Meidani, thanks for the paper, but it too only looks at the magnitude aspect of the problem.

@ Adolfo Silva, that is a very challenging control problem! And I am always interested in learning how engineers go about designing digital filters for controls. This paper does consider phase, but it looks at the IIR case, not the FIR equiripple case. It looks like a variation of Antoniou's approach (attached, and see refs within).

The phase response of FIR filter is linear. If you want to control the group delay of a low-pass filter with good amplitude response, your filter must be an IIR filer.  The phase response of an IIR filter is nonlinear, there exist a big space for the trade-off  between phase response and amplitude response. An IIR filter with minimum group delay can be designed with specified amplitude response. There is an example in the datasheet of CS5676A IC, which is digital filter chip with linear phase and non-linear phase. 

@ Rujun Chen, unlike IIR filters, FIR filters may have a perfectly linear phase across the entire frequency range, but that doesn't mean that they must be linear phase. In applications where filter delay is not critical, this would be a gift that is too good to pass up; however for healthy stability margins in feedback control loops, having approximately linear phase in the pass band, for a reduced phase delay may be an acceptable trade-off   

Have a look at Fig. 5 in the attached conference paper. I have already uploaded the FIR-WISE code to RG, if you would like to take a look (C++ source, see the OptimalFilter class). That code minimizes the weighted integral squared error (WISE) for a specified magnitude & delay response. I would like to find analogous code that minimizes the maximum error (equiripple).

Conference Paper Improved IIR Low-Pass Smoothers and Differentiators with Tunable Delay

Data Recursive DFT - C++ Source Code

There is an IEEE Circuits and Device II paper of 1997 on optimized group delay IIR filter. The second type uses equiripple for group delay in the pass-band of the low pass filter. The group delay variation is significantly smaller than those of ordinary IIR filters can achieve. The paper is titled "Group Delay Optimized IF Filters for Narrowband PCS".

@Liping Ai, thanks for reference. I already have quite a few papers on reduced group delay IIR filters designed using an equiripple optimization criterion. But I hadn't seen that one before. I have added it to my collection.

I am specifically looking for reduced group-delay FIR filters, designed using an equiripple optimization criterion. Does anything come to mind?

Window functions can be used to design equiripple filtler. Chebyshev window functions provide equiripple response in passband and stopband. For more details, you can refer my paper:

Design of SAW bandpass filters using new window functions

AR Reddy -  IEEE Transactions on UFFC, 1988.

Thanks  @AR Reddy but it looks like that is for the linear-phase case.

OK. I think I have found a few papers on this ...

Xiaoping Lai, "Optimal Design of Nonlinear-Phase FIR Filters With Prescribed Phase Error," in Signal Processing, IEEE Transactions on , vol.57, no.9, pp.3399-3410, Sept. 2009

Xiaoping Lai; Chunlu Lai; Ruijie Zhao, "An Iterative Approach to Near-Uniform Group-Delay Error Design of FIR Filters," in Signal Processing Letters, IEEE , vol.18, no.2, pp.107-110, Feb. 2011

Wei Rong Lee; Caccetta, L.; Kok Lay Teo; Rehbock, V., "Optimal design of complex FIR filters with arbitrary magnitude and group delay responses," in Signal Processing, IEEE Transactions on , vol.54, no.5, pp.1617-1628, May 2006

http://s5electronicsandcommunication.blogspot.co.uk/2012/11/matlab-program-for-equiripple-fir-filter.html

http://uk.mathworks.com/help/dsp/examples/arbitrary-magnitude-and-phase-filter-design.html

Your requirements are contadictory: you want design a FIR filter with equiripple filter with low group delay.This means that you FIR filters will have many taps and high group delay. FIR filters have linear phase if their impusle response is symmtric with respec to the middle point. I would consider releasing your requirements: e.g. a IIR filter with almost linear filter,  which will have much lower group delay.

Tadeusz is correct. You can't have it both ways. Usually you don't need it both ways, so perhaps you should start with your application and see what exactly your application needs.

I've done something along those lines. The important bit is of course (see Tadeusz' reply) that you can't get entirely arbitrary group delay over the pass band if your magnitude response is a given. What you can get is reduced group delay over a fraction of the pass band. The shorter you want that group delay, the narrower the fraction of the pass band over which you can realise it. The extreme case is obviously minimum phase where you get the shortest delay but only at infinitesimal frequencies.

If you can live with this caveat here's a rather dirty but nevertheless useful trick. First, an ordinary linear phase FIR filter is made using the standard tool set. The result has alternating conjugate pairs of zeros inside and outside the unit circle over the pass band and then a string of zeros on the unit circle over the stop band.

If you were to flip all the zeros that are outside the unit circle inside (by reciprocating) you'd get the minimum phase equivalent, which has minimum group delay but of course a large deviation from linear phase. But there are a large number of intermediates between minimum phase and linear phase and they're simply all possibilities of zeros flipped inside and zeros left outside. Suppose you have (just to name a number) 8 conjugate pairs of zeros outside the unit circle, that gives you 256 possible combinations ranging from minimum to linear phase. So you write a script that tries them all out exhaustively and see which matches your group delay requirement most closely.

If minimum group delay is a paramount, a crude running average filter is the best. Concatenating few of these you get a nice approximation of Gauss, again with low delay. It can't get any simpler than that.

Filters with a long impulse response and a short delay may be useful in some cases - for attenuation of sensor noise in feedback control loops for instance.

For a (very) low-pass filter, increasing the length of the impulse response can give you a little bit more high-frequency attenuation. Of course, this becomes a bit pointless for an FIR filter, because you have lost your phase linearity and IIR filters are much more efficient, they also don't have the same stop-band ripple as the 'equivalent' FIR filters. I know how to design these IIR filters OK.

I can do this too for non-linear-phase FIR filters, by minimizing the weighted-least-squares-error (WISE), but I would like to compare a WISE design with an equiripple design (if such a thing exists). I also know that the WISE method can be tuned to give a reasonably equiripple response in the stop-band (by manually adjusting the error weights). So an 'exactly' equiripple response should be do-able.