dealing with Huge Length (Order) FIR filters

Question

I'm studying a project.

My signal is audio 8KHz Mono, 8 bits in Low frequencies 70Hz until 1200Hz.

When I perform the calculus with the transition band is very narrow (Transition Frequency) my Order results are (according to proposed window):

For this Example the worst case:

DeltaFrequency = 1.060180596543088 Hz
dNormalizedDeltaFreq= 1.060180596543088/8000

How is calculated the Order? My Answer is below, check the PD!

• Rectangular: 6793
• Hamming:24903
• Hanning:23393
• Kaiser:27353
• Blackman:41503

If I try to use 16KHz, before orders will be doubled!

Then, I want to use Hardware only to perform the filtering FIR process, and I'm worry about of Time needed to perform this functionalities.

COMMENT1:

Reduce Sampling Frequency is not optional, I need to deal with 1200Hz signals too!

PD: (This is not question!)

    if (windowType.toLowerCase().equals("rectangular")) {
      Size = (int)Math.ceil(0.9/dNormalizedDeltaFreq);
    }
    if (windowType.toLowerCase().equals("hamming")) {
      Size = (int)Math.ceil(3.3/dNormalizedDeltaFreq);
    }
    if (windowType.toLowerCase().equals("hanning")) {
      Size = (int)Math.ceil(3.1/dNormalizedDeltaFreq);
    }
    if (windowType.toLowerCase().equals("kaiser")) {
//      Mitra's book Handbook for Digital Signal Processing quotes Kaiser with a 
//      simple estimate: N = [-20*log10(sqrt(dp*ds)) - 13]/[14.6(ws - wp)/2pi]
       double Att = 60.0;
      Size = (int)Math.ceil((Att - 7.95)/(14.36*dNormalizedDeltaFreq));
    }
    if (windowType.toLowerCase().equals("blackman")) {
      Size = (int)Math.ceil(5.5/dNormalizedDeltaFreq);
    }

QUESTION 1: it is feasible to use a FIR filter of this Order?

QUESTION 2: Which single device (hardware) help me perform this function?

QUESTION 3: How could be calculated the delay time?

Answer 1

There are three different implementations of FIR filters that I am aware of.

The first is the basic "transversal" FIR in which you sum up the terms of the summation directly. It gets very expensive if $L$, the FIR length, gets long.

The second is the so-called "fast convolution" in which the FIR impulse response is zero-padded to a longer length and the DFT is applied to it once. Then there are two slightly different procedures, "overlap-add" (OLA) or "overlap-save" (OLS, sometimes called "overlap-scrap" which I think is more descriptive) to process blocks of the input into blocks of the output.

Both of these methods are well covered in textbooks.

A third method, not really covered in textbooks at all, is the "Truncated IIR" filters (TIIR) and this might apply in your case. The simplest example of a TIIR filter is the moving-sum or moving-average that you might find in cascade-integrate-comb (CIC) filters. They have recursion in them, but they are not, mathematically, IIR. Their IIR impulse response gets cancelled or truncated.

If you're trying to implement a long FIR that is a sliding rectangular window (that's a moving sum) or a sliding Hann window or sliding Hamming window, you can do that very efficiently with a TIIR. A sliding Blackman is a little harder and I don't think TIIR will work for a sliding Kaiser.

If you know your 1st or 2nd-order IIR filters and want to truncate their impulse response, this shows you how to do it. Higher order IIRs can be factored into 2nd-order sections, but the finite lengths of the TIIR will add. If you want a higher order IIR with a given truncation length, the thing to do is to parallelize the 2nd-order sections (using partial fraction expansion) and then truncate each parallel 2nd-order section to the same common FIR length.