exp
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Hi,
Suppose I have an analog filter operating at 5kHz which I want to identify. Usually, the analog filter is now approximated as FIR filter with M taps and the output signal sampled at >10kHz to obtain y[n] (suppose the input to the analog filter is already available as x[n]).
The action of the system can not be described as \[\mathbf{y} = \mathbf{X}\mathbf{c}\] where the matrix \[\mathbf{X}\] contains the the tap input vectors of x[n]. Without using LMS, the coefficients could be found by
\[\mathbf{c} = (\mathbf{X}^H \mathbf{X})^{-1} \mathbf{X}^H \cdot \mathbf{y}\]
This system is heavily underdetermined and finds a solution as long as there are M linearly independend rows. However, using fewer rows (e.g. every 2nd row) would correspond to undersamlping the output signal which contradicts my intuition.
Is there a definite sampling requirement for system identification? If yes, why do I still get the correct coefficients when taking e.g. only each 4th row?
I know that the input signal must have "persistent excitation", i.e., it must contain all 5kHz components in the example above. But I can't find anything about the definite sampling requirement of the output signal ...
Thanks!
Suppose I have an analog filter operating at 5kHz which I want to identify. Usually, the analog filter is now approximated as FIR filter with M taps and the output signal sampled at >10kHz to obtain y[n] (suppose the input to the analog filter is already available as x[n]).
The action of the system can not be described as \[\mathbf{y} = \mathbf{X}\mathbf{c}\] where the matrix \[\mathbf{X}\] contains the the tap input vectors of x[n]. Without using LMS, the coefficients could be found by
\[\mathbf{c} = (\mathbf{X}^H \mathbf{X})^{-1} \mathbf{X}^H \cdot \mathbf{y}\]
This system is heavily underdetermined and finds a solution as long as there are M linearly independend rows. However, using fewer rows (e.g. every 2nd row) would correspond to undersamlping the output signal which contradicts my intuition.
Is there a definite sampling requirement for system identification? If yes, why do I still get the correct coefficients when taking e.g. only each 4th row?
I know that the input signal must have "persistent excitation", i.e., it must contain all 5kHz components in the example above. But I can't find anything about the definite sampling requirement of the output signal ...
Thanks!