|newtype T filter i t a v = C (Map i [(i, filter t a v)])|
|newtype Signal list i v = Signal (Map i (list v))|
|fromList :: Ord i => [(i, [(i, filter t a v)])] -> T filter i t a v|
|toList :: T filter i t a v -> [(i, [(i, filter t a v)])]|
|signalFromList :: Ord i => [(i, list v)] -> Signal list i v|
|signalToList :: Signal list i v -> [(i, list v)]|
|lookupSignal :: Ord i => Signal list i v -> i -> Maybe (list v)|
|apply :: (Ord i, Show i, C t, C t, C t, C a v, C a (list v), Filter list filter) => T filter i t a v -> Signal list i v -> Signal list i v|
|transferFunction :: (Ord i, Show i, C t, Fractional (T t), Scalar (T t), C a t, Filter list filter) => T filter i t a v -> t -> [[T t]]|
|newtype T filter i t a v ||Source|
A filter network is a graph consisting
of nodes (input and output signals)
and edges (filter processes).
Output signals can be taken from every node,
inputs can be injected in some nodes
which means that the signal at this node is superposed with
the injected signal.
The same can be achieved by duplicating the network,
one duplicate per input,
and superposing the corresponding outputs.
It is also sensible to setup a graph without inputs,
e.g. a recursive filter with some initial conditions
that works independent from any input.
In opposition to electric filter networks
digital filter networks must be directed.
Test-case: leap-frog filters like
(u) -+-> [a] (v) -+-> [b] (w) -+-> [c] (y) -+->
v = a (u + d w)
w = b (v + e y)
y = c w
We model the general network by a list of nodes,
where each node is an adder that holds a list of its inputs.
Each input of a node is an output
of another node that has gone through a processor.
Additionally there may be one input from outside.
In principle a processor could be a simple filter network
as defined by the structure Filter.
The network is an applyable filter
whenever each circle contains a delay.
To compute the transfer function
we have to solve a system of linear equations
which we can construct quite straight forward
from the processors' input lists.
The current design can be abstractly seen
as the system of linear equations:
y = A*y + u
where A is a matrix containing the edges hosting the filters,
y the vector of output signals,
u the vector of input signals.
In this formulation the number of inputs and outputs must match
but since you are free to ignore some of the inputs and outputs
you can use nodes for pure output, pure input or both of them.
|C (Map i [(i, filter t a v)])|
|newtype Signal list i v ||Source|
|fromList :: Ord i => [(i, [(i, filter t a v)])] -> T filter i t a v||Source|
|toList :: T filter i t a v -> [(i, [(i, filter t a v)])]||Source|
Compute a matrix that tells how an input frequency
is mapped to the various output nodes.
According to the formulation given above
we have to invert the matrix (I-A).
Currently this is done by a QR decomposition for each frequency.
It would be probably faster if we decompose
the matrix containing polynomial elements.
Then the inverted matrix would consist of some
polynomial ratios which can be evaluated for each frequency.
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