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| ForSyDe.Shallow.FilterLib | | Portability | portable | | Stability | experimental | | Maintainer | forsyde-dev@ict.kth.se |
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| Description |
This is the filter library for ForSyDe heterogeneous MoCs - CT-MoC, SR-MoC,
and Untimed-MoC.
The filters at CT-MoC are based on the filters implemented at SR-MoC and Untimed-MoC,
which means a signal in CT-MoC is always digitalized by a A/D converter, processed by
digital filters at SR or Untimed domain, and converted back into a CT domain signal by
a D/A converter. A CT-filter is composed of one A/D converter, one digital filter, and
one D/A converter.
The implementation of the filters at untimed and synchronous domains follows the
way in a paper /An introduction to Haskell with applications to digital signal processing,
David M. Goblirsch, in Proceedings of the 1994 ACM symposium on Applied computing./,
where the details of the FIR/IIR, AR, and ARMA filters are introduced. The ARMA filter
is the kernel part of our general linear filter zLinearFilter in Z-domain at SR/Untimed
MoC, and is also the kernel digital filter for the linear filter sLinearFilter in
S-domain at CT-MoC.
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| Synopsis |
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| FIR filter
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| :: Num a | | | => [a] | Coefficients of the FIR filter
| | -> Signal a | Input signal
| | -> Signal a | Output signal
| | The FIR filter. Let '[x_n]' denote the input signal, '[y_n]' denote the ouput
signal, and '[h_n]' the impulse response of the filter. Suppose the length of
the impulse responses is M samples. The formula for '[y_n]' is
$sum_{k=0}^{M-1} h_k*x_{n-k}$.
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| AR and ARMA filter trim
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| :: (Num a, Fractional a) | | | => [a] | Coefficients of the AR filter.
| | -> a | The gain
| | -> Signal a | Input signal
| | -> Signal a | Output signal
| | 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.
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| :: (Num a, Fractional a) | | | => [a] | Coefficients of the FIR filter
| | -> [a] | Coefficients of the AR filter.
| | -> Signal a | Input signal
| | -> Signal a | Output 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.
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| The solver mode
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| The solver mode.
| | Constructors | | S2Z | Tustin tranfer from s-domain to z-domain
| | RK4 | Runge Kutta 4 with fixed simulation steps
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|  Instances | |
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| The general linear filter in S-domain
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| :: (Num a, Fractional a) | | | => SolverMode | Specify the solver mode
| | -> Rational | Fixed simulation interval
| | -> [a] | Coefficients of the polynomial numerator in s-domain
| | -> [a] | Coefficients of the polynomial denominator in s-domain
| | -> Signal (SubsigCT a) | Input CT-signal
| | -> Signal (SubsigCT a) | Output CT-signal
| 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.
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| The general linear filter in Z-domain
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| The general linear filter in Z-domain.
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| s2z domain coefficient tranformation
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| :: (Num a, Fractional a) | | | => Rational | Sampling rate in Z-domain
| | -> [a] | Coefficients of the polynomial numerator in s-domain
| | -> [a] | Coefficients of the polynomial denominator in s-domain
| | -> ([a], [a]) | Tuple of the numerator and denominator
coefficients 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.
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| The Z-domain to ARMA coefficient tranformation
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| :: (Num a, Fractional a) | | | => ([a], [a]) | Coefficients in Z-domain
| | -> ([a], [a]) | Coefficients of the ARMA filter
| 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.
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| Produced by Haddock version 2.1.0 |
