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| WaveWarp 2.0 Example DrawingBoard
ConvolutionEducationalExample1
Description
Convolution Educational Example 1
Illustrates how the "Convolution" block can be used to calculate the exact convolution of two input vectors.
NOTE: THIS DRAWINGBOARD WILL NOT RUN IN THE DEMO VERSION SINCE THE OUTPUT FILE
FUNCTIONALITY HAS BEEN DISABLED.
As a first example, compute the convolution of the sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] with itself. (This is the default
setting of this Drawing Board). The input sequence is stored in the ascii file "input1to10.txt" (verify by opening it with any
text editor). By feeding this same file into both inputs of the "Convolution" block (with the Data buffer length set to 10 --
corresponding to the length of the input sequence, the output gain set to 0dB, and the "Scaling" option de-selected), the
first 10 elements of the output sequence of the convolution should be: [1, 4, 10, 20, 35, 56, 84, 120, 165, 220].
Run the Drawing Board. Check the contents of the destination text file "output.txt". The first 10 elements will be equal to 0 (due to latency associated with the
algorithm -- see note below). The next 10 elements will be the sequence: [1, 4, 10, 20, 35, 56, 84, 120, 165, 220] as
expected. The exact convolution has thus been computed. Note that, by definition, the convolution of two sequences
should have a length equal to: [length of first sequence] + [length of second sequence] -1. In order to operate in real time
with selectable buffer length without incurring a data explosion, only the first half of the total sequence is computed by the
Convolution block. There is no loss of generality since the total sequence can be obtained by appropriate arrangement of
the input sequences. For example, consider again the sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] convolved with itself. The
complete resulting sequence is: [1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 264, 296, 315, 320, 310, 284, 241, 180, 100]. This
can be obtained by arranging the input sequences to the Convolution blocks as follows: first input sequence [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; second input sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. For
convenience these files have been included on the DrawingBoard ("input1to10repeated.txt" and "input1to10padded.txt").
Try connecting these two into the Convolution block and checking that the resulting output is the full sequence as
described above.
For the next example, try feeding "impluse.txt" into one of the inputs of the "Convolution" block with "input1to10.txt" in the
other. This "impluse.txt" file contains the "unit impulse" sequence (defined as: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0] for 10 elements).
By convolving this with the original sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] (in "input1to10.txt") the expected result is
simply the original sequence [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] in accordance with a key property of convolution, namely that the
convolution of any sequence with the "unit impulse" should yield the original sequence. Run the Drawing Board. Check the
contents of the destination text file "output.txt". The first 10 elements should be equal to 0 (due to latency associated with the
algorithm -- see note below). The next 10 elements should be the original sequence: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] as expected.
The exact convolution has thus been computed.
The Convolution block performs pure convolution of the two inputs. Although it uses the Fast Fourier Transform (FFT) for
efficiency, appropriate zero-padding and overlapping is applied to eliminate "circular convolution", such that the output is
exactly the direct convolution of the two inputs, as if the computations were carried out in the time domain. The only tell-tale
sign that the FFT has been used is the inherent "latency". In other words, the output is delayed by the length of the input
data buffer.
Components used:
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