Audio equalizer formula question

Hallo everyone

i have pdf which inside has some formoulas .But i cand undestand why these formoulas mapping the cutoff frequency and the range is [ 0 pi/2] .But i know in digital domain the mapping is [ 0 pi] why the mapping is [0 pi/2] in what help us???/
the formula is tan(pi*fc/fs)-1/tan(pi*fc/fs)+1 probably hier the cutoff frequency maps into the
range[0 pi/2]

I don't understand where you see a "mapping". It's just a formula for the filter coefficients with a useable fc range of 0..fs/2. What problem are you having with the calculation particularly?

To understand what the calculation is about, one should review at least the preceding page of section 2.3.1 or better the complete 2.3 paragraph about equalizers. You should also mention the book title Zoelzer, Digital Audio Effects.

let me expalin better my problem
tan(pi*fc/fs)-1/tan(pi*fc/fs)+1 is equal =tan(pi*Wc/2)-1/tan(pi*Wc/2)+1 where Wc=2*fc/fs
Now i know fc/fs is bound to 0.5, as fc can never be higher than fs/2 (Nyquist).
Therefore pi*fc/fs is a normalisation of the frequency fc in Hz to a range 0...pi/2
but why the range is [0 pi/2] ? shound be [0 pi] ? as it is the range for digital frequency?
probably this formoula above is way to express the cutoff frequency in digital domain? but why the range is [0 pi/2] probably the the -1 and +1 is zplane(-1 +1) or for w=[pi 0]
my problem is why the formoula is tan(pi*fc/fs)-1/tan(pi*fc/fs)+1 but i i thing it should be
tan(2*pi*fc/fs)-1/tan(2*pi*fc/fs)+1?
thanks

There's no problem at all.

You have just a filter coefficient calculation formula, pi*fc/fs is an argument of a tan() function without further implications.

Why don't you simply apply the calculation and verify the result?

yes i get the corect result but i have to know why is like that the formoula if someone ask me what i have to say to him
if ask me why the range of frequency is [ 0 pi/2] and not [0 pi] what i have to ask him

Again, [ 0 pi/2] not the "range of frequency", it's the argument range applied to a tan() function in some place of the calculation.

thank you for your time to answer my questions
1)so with this fraction tan(pi*fc/fs)-1/tan(pi*fc/fs)+1 is a way to express the cutoff frequency in digital domain???
2)what do you mean """"it's the argument range applied to a tan() function in some place of the calculation."""""
if we we do tan(2*pi*fc/fs)-1/tan(2*pi*fc/fs)+1 whats the diference??
thanks

tan(pi*fc/fs)-1/tan(pi*fc/fs)+1 is a way to express the cutoff frequency in digital domain

Click to expand...

No. It's a formula to calculate filter coefficients used in the implementation.

if we we do tan(2*pi*fc/fs)-1/tan(2*pi*fc/fs)+1 whats the difference?

Click to expand...

The coefficient is different, resulting in a wrong cut-off frequency ≠ intended fc.

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I see that the term tan(ωc/(2fs)) or tan(pi*fc/fs) is often used in frequency axis warping of digital filters, in so far it has probably some general meaning. Notice that tan(pi*fc/fs) ≈ fc/fs for small values of fc << fs.

so with your last comment you tell me the same that i say before: that the formoula is used for digitizing the cutoff frequency, to express the cutoff frequency in digital domain
let me help you.....because
look these formoulas tan(pi*fc/fs)-1/tan(pi*fc/fs)+1=tan(pi*wc/2)-1/tan(pi*wc/2)+1(where wc=normalized=2*fc/fs)=tan(wa*Ts/2)-1/tan(wa*Ts/2)+1 wa=2*pi*fc rad/sec (analog frequency)
obviously NOW we understand both that each of these 3 formoula above their porpuse is to specify the cutoff frequency in digital domain with 3 diferent ways but i see and i say it again I KNOW DIGITAL FREQUENCY RANGE IS [-Π Π] why hier the the usable range is [0 0.5] i say above """""""""" fc/fs is bound to 0.5, as fc can never be higher than fs/2 (Nyquist). Therefore pi*fc/fs is a normalisation of the frequency fc in Hz to a range 0...pi/2 IM SURE FOR THIS I CAN PROVE this
the point is this why the range of the normalisation frequency the range is [0 pi/2]???????/why??????????

Hi,

This unformatted text with the formulas is really hard to read.

Klaus

hi Klaus

the formoulas are these but we have to know wc=2*fc/fs and wa=2*pi*fc rad/sec

Now the formoulas are
tan(pi*fc/fs)-1/tan(pi*fc/fs)+1=tan(pi*wc/2)-1/tan(pi*wc/2)+1=tan(wa*Ts/2)-1/tan(wa*Ts/2)+1
lets look an example if we have a cutoff frequency 1000 hz digital frequency=2*pi*fc/fs=0.1425 but with these formoulas is 0.0712 rad/sample
all these 3 formoulas above are equivalent, just using different ways to specify the bandwidth in digital domain i.e rad/sample

i know as i say before fc/fs is bound to 0.5, as fc can never be higher than fs/2 (Nyquist). Therefore pi*fc/fs is a normalisation of the frequency fc in Hz to a range 0...pi/2
My question is why we specify the bandwidth in range [0 pi/2] since we know digital frequency [0 pi]???????????

tan(pi*fc/fs) is a prewarping function, not a normalization. You may compare with implementations of first order allpass filter in digital filter designs text books.

.

About the meaning of tan(pi*fc/fs), compare https://en.wikipedia.org/wiki/Bilinear_transform#Frequency_warping

yes you are correct it comes from the allpass implementation but this formoula c=tan(pi*fc/fs)-1/tan(pi*fc/fs+1) IS USED FOR WHAT ????IS FOR PREWARP ????THIS IS THE WAY TO DIGITISE THE FILTER???????