How does the Meyr-Oerder Timing Algorithm work?

Question

I`ve came across a Paper for this Algorihm for estimating the Symbol-Timing from Meyr and Oerder, which I want to understand. The Algorithm uses blocks from the samples and calculates the Timing Estimation for the difference between the Transmitter-Clock and the Receivers clock. By squaring the blocks we are getting a peak in the spectrum which is our symbol timing.

My questions are:

• Why are we getting the symbol timing by squaring the signal block?
• What does the Fourier-transformation do to the squared signal-block?

Answer 1

For PAM signals, the squaring creates a spectral component at $\dfrac{1}{T}$. Consider the following trigonometric identity to understand why this is the case:

$$\cos^2\left(2\pi \dfrac{1}{2T}\right) = \dfrac{1}{2}+\dfrac{1}{2}\cos\left(2\pi \dfrac{1}{T}\right)$$

The phase of that spectral component at frequency $\dfrac{1}{T}$ is a measure of the evenness (or oddness) of the symbol relative to the reference time $t=0$ of the fourier transformed squared samples. A phase of zero corresponds to the symbol being centered at that reference time.

The Fourier transform is used to find the phase of the $\dfrac{1}{T}$ spectral component, which is an unfiltered, scaled estimate of the symbol timing offset.

Answer 2

Trace the following steps.

1. Squaring removes the modulation in a binary PAM case, giving rise to a DC and a periodic component, as stated by Andy.
2. Periodicity implies that the spectrum is discrete, remember sampling in time domain creates periodicity in the form of aliases in frequency domain, so periodicity in time domain creates sampling of the spectrum in frequency domain.
3. The data sequence is bandlimited by a pulse shape within $(1+\alpha)/2T$, where $T$ is the symbol time and $\alpha$ is excess bandwidth.
4. Bandlimitation applies that all discrete spectral components beyond $1/T$ are zero (not $1/2T$).
5. The DC component has nothing to do with the period of the modulation removed sinusoid like signal.
6. The phase of the 1st harmonic, or the $1/T$ component then, exhibits the timing phase. Fourier transformation is just one way to access that phase. Otherwise, many timing recovery schemes effectively employ the same in time domain.