Filter design & application
Post date: Nov 07, 2016 1:59:5 PM
Q: What are the main FIR filter design methods?
A:
Window design method: Ideal filter is a rectangle in the frequency domain, hence their time-domain counterpart is a sinc. By windowing this time-domain IIR filter with an appropriate window, one can get an time-domain FIR filter [1]. Common windows are [2]
Hanning window: not sharp transition between signal/interference and noise, but lower side lopes/noise
Hamming window: sharp transition between signal/interference and noise, but higher side lopes/noise
Parametric family of Kaiser windows: closed form relationships between the time-domain and frequency domain parameters, useful for dynamic filter design.
Frequency sampling method: Sample the ideal filter in the frequency domain and take the inverse transform to get the time-domain filter, see [3].
Weighted least square design: gradient-based improvement of the weights to minimize a MSE, see [4]. This is also related to adaptive filter design:
Least mean square (LMS) [6]: The gradient is the product of the error and the input. Extension: normalize the input in the gradient update to make optimal choice of the learning rate => Normalized LMS. LMS filter converges to the Wiener filter [11], provided that the unknown system is LTI and the noise is stationary.
Recursive least square (RLS) [7]: Expend the error term in the gradient update to get a matrix linear equation with the sample correlation (i.e. covariance with zero mean [8]) matrix and the cross-correlation matrix between the desired and the input signals. RLS's recursion for R^{-1} the follows the algebraic Riccati equation, and hence is related to the Kalman filter.
Parks-McClellan method: Equi-ripple, minimax design.
Q: How about IIR filter design?
A: Pole-zeros placement [10]
Q:Why FIR over IIR?
A: Linear phase: Same delay for all frequency components, no distortion, see [12].
Q: An application for adaptive filter?
A: Acoustic echo cancellation [9].
Model:
where
is the local speech, is the local room impulse response.
Objective:
when
For more details, see [1,5].
[1] https://en.wikipedia.org/wiki/Finite_impulse_response#Window_design_method
[2] https://en.wikipedia.org/wiki/Window_function#Generalized_Hamming_windows
[3] https://www.dsprelated.com/freebooks/sasp/Frequency_Sampling_Method_FIR.html
[4] https://en.wikipedia.org/wiki/Least_squares#Weighted_least_squares
[5] https://www.mathworks.com/help/signal/ug/fir-filter-design.html
[6] https://en.wikipedia.org/wiki/Least_mean_squares_filter
[7] https://en.wikipedia.org/wiki/Recursive_least_squares_filter
[8] https://en.wikipedia.org/wiki/Cross-covariance
[9] https://en.wikipedia.org/wiki/Echo_suppression_and_cancellation#Operation
[10] http://www.earlevel.com/main/2013/10/28/pole-zero-placement-v2/
[11] https://en.wikipedia.org/wiki/Similarities_between_Wiener_and_LMS
[12] https://ccrma.stanford.edu/~jos/filters/Linear_Phase_Filters_Symmetric_Impulse.html