compute the finite linear convolution of two arrays of samples
Convolution( A, B, options )
Array, Vector, or list of real or complex numeric sample values for first signal
Array, Vector, or list of real or complex numeric sample values for second signal
algorithm : keyword auto, direct, or fft for the algorithm to use for computation; default is direct
container : Array or Vector, predefined container for holding result
shape : keyword full, same, or valid for the shape of the convolution; default is full
subtype : keyword Array, Vector, Vector[column], or Vector[row] for the subtype of the output; default is Array
The Convolution(A, B) command computes the full convolution of the signals A and B of length M and N respectively, storing the result in a container C of length M+N−1 and having datatype float or complex, which is then returned.
The full convolution is defined by the formula
for each k from 1 to M+N−1, with Aj=0 for M<j and Bj=0 for N<j.
For all choices of shape, the full convolution of size P=M+N−1 is computed. When shape=same, the full convolution is trimmed on both sides so that the result is of length Q=M. Note that when the number of elements to be trimmed is odd, one more element will be trimmed from the left side than the right. When shape=valid, the final convolution will be found in a similar manner to the shape=same case, but the value of the size Q will be M when N=0, and max⁡M−N+1,0 otherwise. The valid convolution effectively discards the elements which involve padded zeros for the signals.
Before the code performing the computation runs, A and B are converted to datatype float (if the values are all real-valued) or complex (if all the values are complex-valued, but not all real-valued) if they do not have that datatype already. For this reason, it is most efficient if A and B have one of these datatypes beforehand.
If either A or B is an rtable that is not a 1-D Array, it is accepted by the command and converted to an Array. Should this not be possible, an error will be thrown.
If the container=C option is provided, then the results are put into C and C is returned. With this option, no additional memory is allocated to store the result. The container must be a 1-D rtable of appropriate size having datatype float (for two real signals) or complex (for one or two complex signals).
The algorithm=name option can be used to specify the algorithm used for computing the convolution. Supported algorithms:
auto - automatically choose the fastest algorithm based on input.
direct - use direct convolution formula for computation. This is the default.
fft - use an algorithm based on the Fast Fourier Transform (FFT). This is a much faster algorithm than the direct formula for large samples, but numerical roundoff can cause significant numerical artifacts, especially when the result has a large dynamic range.
The SignalProcessing[Convolution] command is thread-safe as of Maple 17.
For more information on thread safety, see index/threadsafe.
a ≔ Array⁡1,2,3,'datatype'='float'8
b ≔ Array⁡1,−1,1,−1,'datatype'='float'8
c ≔ Array⁡1..numelems⁡a+numelems⁡b−1,'datatype'='float'8:
A ≔ Vectorrow⁡2−I,0,5+3⁢I,0,4⁢I
B ≔ Vectorrow⁡−7,3+10⁢I,9−2⁢I,1
C1 ≔ Convolution⁡A,B,'algorithm'='fft'
C2 ≔ `~`round⁡C1
A ≔ Vector'row'⁡2,3,5,7,11
B ≔ Vector'row'⁡13,17,19,23
The SignalProcessing[Convolution] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
The A and B parameters were updated in Maple 2020.
The algorithm option was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
The SignalProcessing[Convolution] command was updated in Maple 2023.
The shape and subtype options were introduced in Maple 2023.
For more information on Maple 2023 changes, see Updates in Maple 2023.
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