Signal

The Signal module includes various discrete transforms necessary for signal processing.

Let’s define a two Complex sequences:

#include <Signal.hpp>
#include <Complex.hpp>
#include <iostream>
#include <vector>

int main(){
    std::vector<Complex> X = {1, 2 + 2_j, 3 + 4_j, 5, 6 + 1_j};
    std::vector<Complex> Y = {3, 4, 7, 8, 1 + 1_j};
    return 0;
}

And apply a 1D convolution:

#include <Signal.hpp>
#include <Complex.hpp>
#include <iostream>
#include <vector>

int main(){
    std::vector<Complex> X = {1, 2 + 2_j, 3 + 4_j, 5, 6 + 1_j};
    std::vector<Complex> Y = {3, 4, 7, 8, 1 + 1_j};
    std::vector<Complex> Z = conv(X, Y);

    for(int i = 0; i < Z.size(); i++){
        std::cout << Z[i] << "\n";
    }
   return 0;
}

Output:

3 - 2.21058e-14j
10 + 6j
24 + 20j
49 + 30j
76 + 48j
83 + 40j
81 + 14j
53 + 13j
5 + 7j

The 1D convolution is defined as the following operation:

\[(X * Y)[n] = \sum_{-\infty}^{\infty}X[m]Y[n - m]\]

If an index is less than zero, or greater than the size of the array (meaning that it doesn’t exist!) we can simply set it to zero.

A very similar operation is the cross-correlation operation, defined as:

\[\newcommand{\compconj}[1]{% \overline{#1}% } (X \star Y)[n] = \sum_{-\infty}^{\infty}\compconj{X[m]}Y[n + m]\]

Where the horizontal line notation indicates the conjugate of the complex sequence \(X\). We can implement it in the same way:

#include <Signal.hpp>
#include <Complex.hpp>
#include <iostream>
#include <vector>

int main(){
    std::vector<Complex> X = {1, 2 + 2_j, 3 + 4_j, 5, 6 + 1_j};
    std::vector<Complex> Y = {3, 4, 7, 8, 1 + 1_j};
    std::vector<Complex> Z = crosscorr(X, Y);

    for(int i = 0; i < Z.size(); i++){
        std::cout << Z[i] << "\n";
    }
   return 0;
}

Output:

1 - 1j
12 - 6.35623e-14j
30 + 17j
47 + 41j
79 + 31j
101 + 30j
71 + 19j
39 + 4j
18 + 3j

Finally, the auto-correlation operation, defined as a cross-correlation by a sequence with itself, may be implemented in a similar manner:

#include <Signal.hpp>
#include <Complex.hpp>
#include <iostream>
#include <vector>

int main(){
    std::vector<Complex> X = {1, 2 + 2_j, 3 + 4_j, 5, 6 + 1_j};
    std::vector<Complex> Z = autocorr(X);

    for(int i = 0; i < Z.size(); i++){
        std::cout << Z[i] << "\n";
    }
   return 0;
}

Output:

6 - 1j
19 + 10j
35 + 27j
61 + 11j
96 + 5.88879e-16j
61 - 11j
35 - 27j
19 - 10j
6 + 1j