The autoregressive filter is the simplest IIR filter. It is characterized by a list of numbers '[a_1,a_2,...,a_M]' called the autoregression coefficients and a single number b called the gain. M is the order of the filter. Let '[x_n]' denote the input signal, '[y_n]' denote the ouput signal. The formula for '[y_n]' is $sum_{k=1}^M {a_k*y_{n-k}+b*x_n}$. Although it is an IIR filter, here we only get the same length of ouput signal as the input signal.

The ARMA filter combines the FIR and AR filters. Let '[x_n]' denote the input signal, '[y_n]' denote the ouput signal. The formula for '[y_n]' is $sum_{k=1}^M {a_k*y_{n-k}+b*x_n} + sum_{i=0}^{N-1} b_i*x_{n-i}$. The ARMA filter can be defined as the composition of an FIR filter having the impulse reponse '[b_0,b_1,...,b_N-1]' and an AR filter having the regression coefficients '[a_1,a_2,...,a_M]' and a gain of '1'. Although it is an IIR filter, here we only get the same length of ouput signal as the input signal.

The solver mode.

The general linear filter in S-domain at CT-MoC. As the kernel implementation is in Z-domain, the smapling rate should be specified. It is used on the S-transformation with the following forms, with coefficients for the descending powers of s and m < n.

    b_0*s^m + b_1*s^m-1 + ... + b_m-1*s^1 + b_m*s^0
H(s) = ------------------------------------------------     (Eq 1)
    a_0*s^n + a_1*s^n-1 + ... + a_n-1*s^1 + a_n*s^0

If we multiply both the numerator and the denominator with s^-n, we get another equivelent canonical form

    b_0*s^m-n + b_1*s^m-n-1 + ... + b_m-1*s^1-n + b_m*s^-n
H(s) = -----------------------------------------------------    (Eq 2)
    a_0*s^0 + a_1*s^-1 + ... + a_n-1*s^1-n + a_n*s^-n

To instantiate a filter, with sampling interval 'T ', we use

filter1 = sLinearFilter T [1] [2,1]

to get a filter with the transfer function

      1
H(s) = --------
   2*s + 1

and

filter2 = sLinearFilter T [2,1] [1,2,2]

to get another filter with the transfer function

       2*s +1
H(s) = ----------------
    s^2 + 2*s + 2

There are two solver modes, S2Z and RK4. Caused by the precision problem, the time interval in CT uses Rational data type and the digital data types in all the domains are Double.

The general linear filter in Z-domain.

s2z domain coefficient tranformation with a specified sampling rate. The Tustin transformation is used for the transfer, with

 2(z - 1)  
s = ----------                 (Eq 3)
 T(z + 1)

in which, T is the sampling interval.

The Z-domain to ARMA coefficient tranformation. It is used on the Z-transfer function

    b_0*z^m-n + b_1*z^m-n-1 + ... + b_m-1*z^1-n + b_m*z^-n
H(z) = -----------------------------------------------------    (Eq 4)
    a_0*z^0 + a_1*z^-1 + ... + a_n-1*z^1-n + a_n*z^-n

which is normalized as

    b_0/a_0*z^m-n + b_1/a_0*z^m-n-1 + ... + b_m/a_0*z^-n
H(z) = -------------------------------------------------------  (Eq 5)
    1 + a_1/a_0*z^-1 + ... + a_n-1/a_0*z^1-n + a_n/a_0*z^-n

The implementation coudl be

y(k) = b_0/a_0*x_k+m-n + b_1/a_0*x_k+m-n-1 + ... + b_m/a_0*x_k-n
                        (Eq 6)
           - a_1/a_0*y_k-1 - ... - a_n/a_0*y_k-n

Then, we could get the coefficients of the ARMA filter.