Looking for chebychev equation

i think it might be here:

rots-o-ruk decoding it.

r b-j

alright, the reference is too bitchy to decode, here it is for normalized analog s:

|H(jw)|^2 = 1/(1 + (e*T(w))^2)

when w = 1, you are at the edge of the passband ripple just before the gain takes a dive.

T(x) is abreviated from T_N(w), the Nth order Tchebyshev polynomial

{ cos(N*arccos(x)) |x| = 1

"epsilon" = e = sqrt( 10^(dBripple/10) - 1 )

-3 dB frequency: w1 = cosh((1/N)*arccosh(1/e))

no analog zeros, the analog poles are at:

p = -cos(theta)*sinh(phi) +/- j*sin(theta)*cosh(phi)

where phi = (1/N)*arcsinh(1/e)

{ pi/N*(n + 1/2) 0

Thank you rbj But I am looking for a set of equations for a CASCADED structure of biquad filters.

I.E. If the biquad 2nd order (chebychev) IIR filters form a cascaded higher order IIR filter structure, as to obtain a very sharp & immediate response at the cutoff frequency, how would the filter coeffcients of one filter be related to the other filters? I know from online applets that for a 4x2nd order filter, that all 4 filters are different.

Can you advise me?

-Roger

Roger, the information provided is for exactly that.

OK, "4x2nd order" means N = 8.

perhaps i should have subscripted each pole pair:

p_n = -cos(theta)*sinh(phi) +/- j*sin(theta)*cosh(phi)

where phi = (1/N)*arcsinh(1/e)

{ pi/N*(n + 1/2) 0

The equations look rather daunting, but I'll take a crack at them. However, I want to ask you again this question: Q: The equations ARE for a cascaded structure of biquad 2nd order (chebychev) IIR filters, Rigth ? The reason why I am asking you the question is that upon scanning the equations, I cannot seem to locate the different DCgains of the individual biquad filters in the cascade structure on the analog set of equations...According to my intuition, it is the fact that some of the filters (in the cascacded structure) start to attenuate sooner (than other filters) and the fact that some of the filters have a positive DCgain and others negative DCgain and the fact that some (almost all) of the filters have some kind of shaping (underdamping) about them THAT we observe on the output of the cascaded structure a very sharp & immediate & steady reponse at the intended cutoff frequency (or maybe a little bit higher than the intended higher frequency). I know, from online-applet experience, that for N=8, the 4 filters obtained were all different and gave a great output response.

However, I have faith that it is my newbieism that is probably the center of the misunderstanding.

I also want to confirm the following: Is the basic procedure for obtaining coeffcients the following (at least for analog-prototype equivalent filters, Chebbychev, Cauer, Butterworth....):

1. Obtain Filter Analog Equation
2. Go through some normalization process of the analog equation
3. Perform the Bilinear Transform

-Roger

pretend that you're an electrical engineering student taking a class in digital and analog filtering.

sure. when you express transfer functions that multiply each other, that means the filters that are represented by such transfer functions are in cascade.

the analog filter prototype for each 2nd order stage that was spelled out in the previous posts all have a DC gain of 1 (or 0 dB). when you're at DC, w=0, so s= jw = 0 also. plug s=0 into the filter prototype H(s) and see what you get.

perhaps. to get as far as you have, have you had any formal training in linear system theory (where one learns about transfer functions, frequency response, and the like)

yeah, Cauer (or "elliptical") is a complete bitch. i do not have closed form expressions for those. Butterworth, Tchebyshev (type 1), Inverse Tchebyshev (type2), all have closed form expressions.

usually it is in normalized form when you first create it. if not, and the "significant frequency" (usually the corner frequency or the passband edge or stopband edge), is "W0", then arrange the equation so that every occurance of s is divided by W0 and replace "s/W0" with just "s" to normalize.

when your analog prototype is normalized the BLT mapping that also maps the analog "significant frequency" (which is w=1 for normalized analog) to w0 = 2*pi*f0/Fs is:

The bilinear transform (with compensation for frequency warping) substitutes:

1 1 - z^-1 (normalized) s

Roger Bourne skrev:

The usual way of designing discrete-time IIR filters from analog prototypes goes as follows:

1) Formulate a filter specification in discrete-time domain 2) Transform the discrete-time spec to continuous time 3) Transform general spec to normalized LP spec 4) Determine minimum order required to meet normalized spec 5) Compute coefficients in cascaded 1st and 2nd order sections 6) Transform from continuous time to discrete time 7) Transform from normalized LP filter to desired filter

There are plenty of details to keep track of. The one recent text I am aware of that reviews this process in any detail for Butterworth, Cebyshev 1 and 2, as well as elliptic filters, is this one:

Rune