Hello.

I have a filter design problem. I have a 1.5Hz 18 db/octave Butterworth LC digital filter that is applied twice....once forward, and once backward. I then have a 0.5 Hz 6dB/oct analog filter that is applied on top of that. I want to confirm the overall filter being applied. My support tells me it should be 1.53Hz 42 db/oct. I completely understand the 42 db/oct on the slope. (18+18+6) However, I do not mathematically understand why it is 1.53 Hz and not SQRT(1.5^2+1.5^2+0.5^2) = 2.179 Hz. Everybody keeps trying to explain it with a graph of the 3 filters and how they move with respect to db/oct and I get that. What I want to see is the math. Can anybody walk me through the math on this one? If not, can you help me explain WHY math alone cannot provide an explanation?

 

Not totally clear from your post, but it sounds like you are surprised that cascading two filters moves the corner frequency?

 

Apologies for the ambiguity. The root of my problem is I thought it was a simple formula to convolve filters. Summing slopes and doing the square root of the frequency squared and summed was what I was taught in my filter design 101 back in high school. I would suspect that relationship between filters was extremely simplistic and does not take into account intricacies between convolving digital and analog, for example. Using a map of db/oct, I can follow how the 3 slopes convolve to arrive at 1.53Hz. However, I want to see the same result via a mathematical formula rather than graphing. I have asked around on several forums with the expectation someone will share a straight forward formula that I can use which helps me arrive at the final frequency of 1.53 Hz.
Thanks for replying to the thread so quickly, btw. Much appreciated.

 

I get a headache thinking about the math...

I use a free download called AADE Filter Design. It helps visualize what happens in various filters. FilterPro from TI is good, too.

 

Maybe these should provide some answers:

http://www.ti.com.cn/cn/lit/an/sloa049b/sloa049b.pdf

http://users.ece.gatech.edu/phasler/Courses/ECE6414/Unit1/Discrete_02.pdf

 

Those .pdf's were very good and I thought the situation I needed was present in the "Discrete_02.pdf" but I am either missing a constant someplace or am not fully understanding that formula. I am downloading the AADE Filter Design now. I think I am getting the picture . . . while summing the slopes is straight forward, handling the frequency part is not.

 

Some follow up to my original question....
Someone took the time to explain how the magnitude responses of the analog and digital filters interact. They went on to derive where the -6dB cut off and -3dB cut off frequencies are.


This is all fine and good, but what I STILL do not comprehend is how a 1.5 Hz filter (x2 due to it being zero-phase....forward-backward filtering) interacting with a 0.5Hz analog filter, results in a filter of 1.53Hz. In their explanation of the cut off frequencies they "prove" it is correct by substituting 1.5276 into the formula at the end and getting the right answer for the cut off. While that is lovely, it still doesn't explain where the 1.5276 came from!!

Again, my filters are:
1.5 Hz 18 dB/oct (digital - zero phase, applied twice, forward-backward)
0.5 Hz 6 dB/oct (analog)

Apparently the result of convolving them is 1.53 Hz 42 db/oct. I completely understand summing the slopes. I just still do not understand the derivation of 1.53Hz. I have attached the work done to derive the magnitude of responses. All I need now is to understand this 1.53. Can anyone explain the 1.53 derivation? Thanks.