# What is the purpose of the recurrence relation in low and high pass (audio) filters?

Newbie here. I'm familiar with time and frequency domains from math. I think I understand what low and high pass filters do, sonically. But digging into an implementation of tone in csounds, I think I'm missing something fundamental. Take this description:

A first-order recursive low-pass filter with variable frequency response.

tone is a 1 term IIR filter. Its formula is:

yn = c1 * xn + c2 * yn-1

where

• b = 2 - cos(2 π hp/sr);

• c2 = b - sqrt(b2 - 1.0)

• c1 = 1 - c2

I see that there is a recurrence relation. But I'm reading about IIRs and filters in general I don't see any explanation of why it's there or what it does.

• Would performing a single pass of this formula not be considered performing a "true" low or high pass filter?

• Or is that it takes multiple passes (infinite, in theory) to completely "remove" the frequencies desiring to be removed, so it's a tradeoff between iterations versus computation?

I'm gravitating toward the latter, but I don't see where that is being said. Can someone point me to the essential statement of it on Wikipedia or somewhere?

An IIR filter is a recursive one. It means, current value of output $$y[n]$$ is computed based on its previous value(s).

Consider a simple 1st order IIR filter definition:

$$y[n] = a \cdot y[n-1]+ b \cdot x[n]$$

$$a$$ and $$b$$ are the filter coefficients.

Now assume you have a long block of N samples input data. If you want to process that block of input data with this filter, you can do it like in this matlab code :

N = 1024;       % number of input samples
x = randn(N,1); % some random input.
a = 0.5;        % IIR filter coefficients
b = 0.9;

yi = 0;         % initial value for output
y = zeros(N,1); % matrix (array) to hold processed output samples

y(1) = a*yi + b*x(1)   % assign first output sample based on y(-1)
for n=2:N              % assign rest
y(n) = a*y(n-1) + b*x(n);
end

• Yes, but why? I guess to anyone in the field it might sound like I'm asking, "Why does addition work?" Maybe I'm asking: How is the sound changing in each iteration? Is it improving the filtering? How is it determined how many iterations to compute, since obviously we can't compute infinity? Commented Oct 3, 2020 at 20:03
• Oh, maybe your code does clarify things. Is it doing a pass over the input samples in time, e.g. y(t)? So a previous calculation in time is affecting the next? Commented Oct 3, 2020 at 20:06
• Ah, "successive approximation" is exactly what I thought it was doing. And I thought that because it felt unbelievable to me that an algorithm can remove frequencies in a single pass over the time domain... but apparently it can. That's crazy. Commented Oct 3, 2020 at 20:29
• @AndrewCheong yes that's a good question and so is this answer link Commented Oct 3, 2020 at 20:47
• Holy crap, I can't wait to read all these. Thanks very much. I would even say that it's valid to close this question as a duplicate of that last one. I just didn't know how to ask what I was confused about. Commented Oct 3, 2020 at 21:00