It is well known that the formula for the bit error rate of 2-FSK using non-coherent demodulation is \(\exp(-\frac{E_b}{2N_0})/2\). However, I can never quickly find a source where this formula is derived, so I decided to figure this out and write down the derivation. I will use my post about m-FSK symbol error rate as a starting point.
From that post we know that the symbol error rate of non-coherent demodulation of 2-FSK is\[1-\int_{-\infty}^{+\infty} G(t) g_{\lambda}(t)\, dt,\]where \(G(t)\) is the cumulative distribution function of the chi-squared distribution with two degrees of freedom, \(g_\lambda(t)\) is the probability density function of the non-central chi-squared distribution with two degrees of freedom and non-centrality parameter \(\lambda = 2E_s/N_0\). For 2-FSK, the symbol error rate and the bit error rate are the same, and \(E_s = E_b\).
We have \(G(t) = 1 – e^{-t/2}\) for \(t \geq 0\) and \(G(t) = 0\) for \(t < 0\). Therefore, the bit error rate is\[\mathrm{BER} = 1 – \int_0^{+\infty} (1 – e^{-t/2}) g_\lambda(t)\, dt = \int_0^{+\infty} e^{-t/2} g_\lambda(t)\, dt,\]where we have used\[\int_0^{+\infty} g_\lambda(t)\, dt = 1,\]which comes from the fact that \(g_\lambda(t)\) is a probability density function in \([0, +\infty)\).
The probability density function \(g_\lambda(t)\) can be written as\[g_\lambda(t) = \frac{1}{2}e^{-(t + \lambda)/2} I_0(\sqrt{\lambda t}),\]where \(I_0\) denotes the modified Bessel function of the first kind. Therefore,\[\mathrm{BER} = \frac{e^{-\lambda/2}}{2} \int_0^{+\infty} e^{-t} I_0(\sqrt{\lambda t})\, dt.\]This integral can be computed by using the power series expansion of the Bessel function,\[I_0(x) = \sum_{n=0}^\infty \frac{x^{2n}}{4^n (n!)^2}.\]Replacing this expansion in the above equality and exchanging the summation and integral, we have\[\mathrm{BER} = \frac{e^{-\lambda/2}}{2} \sum_{n=0}^\infty \frac{\lambda^n}{4^n (n!)^2} \int_0^{+\infty} e^{-t} t^n\,dt.\]
Let us denote\[A_n = \int_0^{+\infty} e^{-t} t^n\, dt.\]These integrals can be computed by induction on \(n\). We have \(A_0 = 1\). For \(n\geq 1\), applying integration by parts we get\[A_n = -e^{-t}t^n\Big]_{0}^{+\infty}+n\int_{0}^{+\infty}e^{-t}t^{n-1}\,dt = nA_{n-1}.\]Therefore, we see that \(A_n = n!\).
This implies that\[\mathrm{BER} = \frac{e^{-\lambda/2}}{2} \sum_{n=0}^\infty \frac{\lambda^n}{4^n n!}.\]The series is just the power series of \(e^{\lambda/4}\), so using \(\lambda = 2E_b/N_0\) we get\[\mathrm{BER} = \frac{e^{-\lambda/4}}{2} = \frac{1}{2}e^{-E_b/(2N_0)},\]as we wanted to show.
I should admit that when writing this post I have used some help from Gemini (version 2.5 Pro). When I encountered the integral involving the Bessel function, I wasn’t sure if this integral could be solved, or if a completely different approach was needed to arrive to the well known closed form expression for the BER. I find Bessel functions somewhat scary, because I don’t known many of the identities involving these and other special functions.
After looking through the Wikipedia pages about Bessel functions and chi-squared distributions, for useful formulas without success, I tried asking Gemini to write a mathematical derivation of the FSK BER formula. It gave an argument which is perhaps not perfectly clean, but which is completely correct. As a key step in the argument, it claimed that it would use the known identity\[\int_0^{+\infty} e^{-ax} I_0(b\sqrt{x})\,dx = \frac{1}{a}e^{b^2/(4a)}.\]This formula is a variation on what I have used implicitly above, with \(a = 1\) and \(b = \sqrt{\lambda}\). It actually follows from the above because it is easy to reduce to the case \(a = 1\) with a change of variables. When I asked Gemini to provide references for this formula, it pointed me to Gradshteyn and Ryzhik, Table of Integrals, Series, and Products, Section 6.623, formula 1, and to a more general formula in the NIST Digital Library of Mathematical Functions, Section 11.4, formula 29. I haven’t checked these references, because I quickly realized that I knew how to prove this formula by using the power series expansion of the Bessel function, which is what I have done above.
I have been using Wolfram Alpha for a very long time to figure out how to compute integrals for which I don’t know which tricks can be used, or whether it is at all possible to compute them. Using an LLM in this way feels a similar extension of this approach. Being aware that it sometimes may produce wrong results, it becomes a very useful tool to discover tricks and formulas and quickly check if an approach is reasonable.
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