Thanks Koji , for your hint for the brain teasing puzzle. I was looking into Filters that are usually used in devices like counters, DSO and other scopes. I found that , to improve the quality of the measurement one of the best approach is averaging. I looked deeper into averaging and found out this:
There are two general use-cases for averaging . The first, successive sample averaging, takes a single acquisition and averages between its samples. The second, successive capture averaging, combines the corresponding samples of multiple captures to create a single capture. Successive sample averaging is also called boxcar filtering or moving average filtering. In an implementation of this type of averaging each output sample represents the average value of M consecutive input samples. This type of averaging **removes noise** (one of the reasons the noise level was not bad: http://nodus.ligo.caltech.edu:8080/40m/10151) by decreasing the device's bandwidth(could be one of the reasons why the FC operates in 4 different frequency ranges). It applies an LPF function with a 3dB point approximated by **0.433 * s / M**, where M is the number of samples to be averaged, and s is the sample rate in samples per second.
Now I tried verifying the 3 dB points in the gain plots I generated :
For 1 s Sampling time : the 3 dB point for such a Boxcar filter should be at 0.433* 1/M. If we assume that it averages for 2 samples, M=2 which gives the 3dB point at **0.288 **Hz but occurs somewhere between **0.3 and 0.4 Hz**. (http://nodus.ligo.caltech.edu:8080/40m/140619_120548/GainVsFreq.png)
For 0.1s Sampling time: the 3dB point should be at **2.17 Hz **and in reality is** 2.5 Hz**(http://nodus.ligo.caltech.edu:8080/40m/140701_211904/gain.png).
Also, This type of filter will have very sharp nulls at frequencies corresponding to signals whose periods are integer sub-multiples of M/s. As seen my previous plots (http://nodus.ligo.caltech.edu:8080/40m/10118 , http://nodus.ligo.caltech.edu:8080/40m/10070) there are sharp nulls at frequencies
0.4 Hz for 1S sampling time and
at 1.5 Hz,3 Hz for 0.1 S sampling time as correctly predicted.
The moving average filter is L-sample moving average FIR, with the frequency response as: *H*(*ω*) = (1/*L*) (1 − *e*^{− jω }^{L})/(1 − *e*^{− jω})..
There is an overall delay of (M - 1)/2 samples from such a length-M causal FIR filter.
The expected bode plots for such a filter with L= 5 is attached(attachment 2). |