Filters
21/10/2004
Summary:
- Spirals become spirals
- The eigenvalue is the Fourier Transform of the filter kernel
at that frequency
- Complex-rate spirals are also eigenvectors (this is one way
of looking at the z and Laplace transforms)
- Deconvolution of a finite causal filter (FIR/moving average(/all-zero)) is a
recursive/feedback/autoregressive(/all-pole) signal network
- Inverse of linear system is linear
- Inverse of shift invariant system is shift invariant
- Inverse of LSI system is LSI
- An autoregressive filter is LSI, having an
infinite filter kernel (IIR)
- If you have some filters, you can do two things -
- Series (cascade)
- Parallel (addition of output = addition of impulse response)
- FIR filters in cascade - impulse responses convolve
- FIR filters in parallel - impulse responses add up
- Autoregressive filter in cascade - impulse response of inverse system
convolve
- Autoregressive filters in parallel - give a somewhat complicated
autoregressive filter
- The most general filter implementable on a computer is a cascade
of FIR and autoregressive filter
- Called autoregressive-moving average (ARMA), (or pole-zero,
or rational) filters
- Implemented as direct form I (cascaded MA-AR) or direct
form II (cascaded AR-MA) signal network on a computer
- Polynomial multiplication is convolution of coefficients
- Autoregressive inverse filter of an FIR filter can be thought
of as a reciprocal of a polynomial function
- The filter kernel coefficients are the Taylor series expansion
of this "rational function". Turns out we never actually have to
evaluate the Taylor series, because the autoregressive system
can directly be implemented as a feedback network
- All the complicated signal-network manipulations are equivalent
to simple algebraic manipulations of rational functions (another
way of looking at the z and Laplace transforms)
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