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The Q value about the cascading BPF filter.

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I have read some books and found the Q value of the whole filter can be expressed as formula.jpg,Q1 is the Q value of 2rd BPF，n is the number of 2rd BPF，my problem is how to get the formula,how to derive the formula,is there any material to explain this formula?
 

Q is a property of a second order filter (or "biquad"). There's no thing like a "Q value of the whole filter". Instead, the combination of individual Q and center frequency values makes the filter characteristic.

Can you clarify, which meaning of the "overall Q value" is discussed in these books?
 

what you said is true,actually,my question is clear,Q and Q1 in that formula just a value,i don't think it take any matter,Q1 is that formula represents the value of a second order filter,Q1 represents the value of the combination of several 2rd filter.
FvM said:
Q is a property of a second order filter (or "biquad"). There's no thing like a "Q value of the whole filter". Instead, the combination of individual Q and center frequency values makes the filter characteristic.

Can you clarify, which meaning of the "overall Q value" is discussed in these books?
Click to expand...
 

It may be a way to describe looking at the individual Q factors of all stages of the filter, and seeing whether this or that stage is centered on a different frequency than the desired frequency. If a stage turns out to have a different center frequency, then you have compromised your resulting Q.

A high resulting Q can be maintained by using matching capacitor and resistor values. The goal is to have matching time constants through all stages of the filter.
 

I was asking for a clear definition of the "resulting Q", respectively questioning if it exists at all?

Q1 represents the value of the combination of several 2rd filter
Click to expand...
What is Q1? How do you measure it?
If a stage turns out to have a different center frequency, then you have compromised your resulting Q.
Click to expand...
I assume, that you translate "resulting Q" to overall filter bandwidth. Actual filter design works just the other way around. Individual filter stages are intentionally tuned to different center frequencies to achieve an intended frequency characteristic. That's one reason, why you can't conclude from filter stage Q to overall bandwidth in general.

If you are discussing a particular filter prototype, e.g. 6th order Butterworth with 10% relative bandwidth, the Q and center frequency numbers are well defined.
 
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