## Guessing at the discretization, of the Sallen-Key Filter, with Q-Multiplier.

One concept that exists in modern digital signal processing is, that a simple algorithm can often be written, to perform what old-fashioned, analog filters were able to do.

But then, one place where I find lacking progress – at least, where I can find the information posted publicly – is, about how to discretize slightly more complicated analog filters. Specifically, if one wants to design 2nd-order low-pass or high-pass filters, one approach which is often recommended is, just to chain the primitive low-pass or high-pass filters. The problem with that is, the highly damped frequency-response curve that follows, which is evident, in the attenuated voltage gain, at the cutoff frequency itself.

In analog circuitry, a solution to this problem exists in the “Sallen-Key Filter“, which naturally has a gain at the corner frequency of (-6db), which would also result, if two primitive filters were simply chained. But beyond that, the analog filter can be given (positive) feedback gain, in order to increase its Q-factor.

I set out to write some pseudo-code, for how such a filter could also be converted into algorithms…


Second-Order...

LP:
for i from 1 to n
Y[i] := ( k * Y[i-1] ) + ((1 - k) * X[i]) + Feedback[i-1]
Z[i] := ( k * Z[i-1] ) + ((1 - k) * Y[i])
Feedback[i] := (Z[i] - Z[i-1]) * k * α
(output Z[i])

BP:
for i from 1 to n
Y[i] := ( k * Y[i-1] ) + ((1 - k) * X[i]) + Feedback[i-1]
Z[i] := ( k * (Z[i-1] + Y[i] - Y[i-1]) )
Feedback[i] := Z[i] * (1 - k) * α
(output Z[i])

HP:
for i from 1 to n
Y[i] := ( k * (Y[i-1] + X[i] - X[i-1]) ) + Feedback[i-1]
Z[i] := ( k * (Z[i-1] + Y[i] - Y[i-1]) )
Feedback[i] := Z[i] * (1 - k) * α
(output Z[i])

Where:

k is the constant that defines the corner frequency via ω, And
α is the constant that peaks the Q-factor.

ω = 2 * sin(π * F0 / h)
k = 1 / (1 + ω), F0 < (h / 4)

h   Is the sample-rate.
F0  Is the corner frequency.

To achieve a Q-factor (Q):
α = (2 + (sin^2(π * F0 / h) * 2) - (1 / Q))
'Damping Factor' = (ζ) = 1 / (2 * Q)

Critical Damping:
ζ = 1 / sqrt(2)
(...)
Q = 1 / sqrt(2)



(Algorithm Revised 2/08/2021, 23h40. )

(Computation of parameters Revised 2/09/2021, 2h15. )

(Updated 2/10/2021, 18h25… )

## Some realizations about Digital Signal Processing

One of the realizations which I’ve just come across recently, about digital signal processing, is that apparently, when up-sampling a digital stream twofold, just for the purpose of playing it back, simply to perform a linear interpolation, to turn a 44.1kHz stream into an 88.2kHz, or a 48kHz stream into a 96kHz, does less damage to the sound quality, than I had previously thought. And one reason I think this is the factual realization that to do so, really achieves the same thing that applying a (low-pass) Haar Wavelet would achieve, after each original sample had been doubled. After all, I had already said, that Humans would have a hard time being able to hear that this has been done.

But then, given such an assumption, I think I’ve also come into more realizations, of where I was having trouble understanding what exactly Digital Signal Processors do. It might be Mathematically true to say, that a convolution can be applied to a stream after it has been up-sampled, but, depending on how many elements the convolution is supposed to have, whether or not a single DSP chip is supposed to decode both stereo channels or only one, and whether that DSP chip is also supposed to perform other steps associated with playing back the audio, such as, to decode whatever compression Bluetooth 4 or Bluetooth 5 have put on the stream, it may turn out that realistic Digital Signal Processing chips just don’t have enough MIPS – Millions of Instructions Per Second – to do all that.

Now, I do know that DSP chips exist that have more MIPS, but then those chips may also measure 2cm x 2cm, and may require much of the circuit-board they are to be soldered in to. Those types of chips are unlikely to be built-in to a mid-price-range set of (Stereo) Bluetooth Headphones, that have an equalization function.

But what I can then speculate further is that some combination of alterations of these ideas should work.

For example, the convolution that is to be computed could be computed on the stream before it has been up-sampled, and it could then be up-sampled ‘cheaply’, using the linear interpolation. The way I had it before, the half-used virtual equalizer bands would also accomplish a kind of brick-wall filter, whereas, to perform the virtual equalizer function on the stream before up-sampling would make use of almost all the bands, and doing it that way would halve the amount of MIPS that a DSP chip needs to possess. Doing it that way would also halve the frequency linearly separating the bands, which would have created issues at the low end of the audible spectrum.

Alternatively, implementing a digital 9- or 10-band equalizer, with the
bands spaced an octave apart, could be achieved after up-sampling, instead of before up-sampling, but again, much more cheaply in terms of computational power required.

Dirk