Mosi's Math

Amplitude-shift keying sign

Amplitude shift keying is a form of amplitude modulation used to transmit digital data over optical fiber or another media. The digital information is transmitted like a fixed-amplitude carrier for 1 and 0 for 0. A 1, 0 sequence looks like:

(See Amplitude-shift keying - Wikipedia)

For the b_k components:

There are 2 situations:

If k ≠ n

If k = n

For the a_k components:

There are 3 situations:

If k ≠ n

If n = k

If k = 0

And so:

If n is odd

If n is even

With this the reconstructed function is:

If n is even

If n is odd

Put into a function with a_k as real part and b_k as imaginary part:

private void Ask(double peak, int n)
{

int j;
c[0].real = 0;
for (j = 1; j < 500; j++)
{

c[j].imag = 0;
if ((n+j) % 2 == 0)
{

c[j].real = 0;

}
else
{

if (j != n)
c[j].real = peak / Math.PI * 2.0 * (double)n / ((double)n * (double)n - (double)j * (double)j);
else
c[j].real = 0;

}

}
c[n].imag = peak/2;
if (n % 2 != 0)

c[0].real = peak / Math.PI / (double)n;

}

That creates for n=4 a spectrum like:

With the blue bars are the real elements and red the imaginary parts

And if all the harmonics are put together, we get the signal shape:

The brightest line is the base frequency. The next darker is the base frequency plus the first harmonic, then comes the same plus the next harmonic … and so on. The more harmonics are included, the closer the shape approximates the origin shape :-)

C# Demo Project Fourier signals

FourierSignals.zip

Java Demo Project Fourier signals

FourierSignals.zip