Frequency-shift keying signal

Frequency shift keying is used to transmit digital data for instance in amateur radio technology. The digital information is transmitted by shifting the frequency of the carrier signal. The frequency for a logical 1 is usually higher than the frequency for a logical 0. A 1, 0 sequence looks like:


Fsk
Fsk

For the bk components:

There are 4 situations:

If k ≠ n and k ≠ 2

Fsk

Fsk


If k = n and k ≠ 2

Fsk

Fsk

Fsk



If k = n and k = 2

Fsk



If k ≠ n and k = 2

Fsk

Fsk

Fsk

Fsk



For the ak components:

Fsk




We first take a look onto the first part:

If k ≠ n

Fsk


(See ASK signal)



Now the second part:

If k ≠ n

Fsk
Fsk


as
Fsk

and
Fsk


Fsk



A special case is n = 2
Fsk


In this case all a elements are 0.




If n = k

Fsk
Fsk




All put together there are 2 different cases:

n ≠ k and k ≠ 2

Fsk




If n = k or if n = 2

Fsk



In code with ak as real parts and bk as imaginary parts:


private void Fsk(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;
}
if ((j != 2) && ((j) % 2 != 0))
c[j].real = c[j].real - peak / Math.PI * (4.0 / (4.0 - (double)j * (double)j));
}
c[2].imag = peak / 2.0;
if (n == 2)
{
c[n].imag = peak;
c[n].real = 0;
}
else
{
c[n].imag = peak / 2.0;
}
}



That creates for n=4 a spectrum like:

Fsk



The blue bars are the real elements and red the imaginary parts.


And with all the harmonics 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 quite soon :-)

Fsk



C# Demo Project Fourier signals
  • FourierSignals.zip


    • Java Demo Project Fourier signals
  • FourierSignals.zip