# Précis of Danesfahani and Jeans, Optimisation of modified Mueller and Müller algorithm

This post is a summary of the main content of G. R. Danesfahani and T. G. Jeans, Optimisation of modified Mueller and Müller algorithm, Electronics Letters 31(13), 22nd June 1995, pages 1032-1033 (DOI 10.1049/el:19950711). The original paper is copyright IEE, and is available from IEEE or your local library. The algorithm described in that paper is the one used in GNU Radio‘s gr_clock_recovery_mm block.

The paper first recalls the modified Mueller and Müller algorithm: given

• data symbols $a(i)$
• received signal $p(k)$ with real part $p_r(k)$,
• the receiver’s decision on the data symbol $\hat{a}(k)$

then the modified Mueller and Müller algorithm (mM&M) computes an error $\mu_1(k) = (\hat{a}(k-1) - \hat{a}(k+1))p_r(k)$.

The paper then observes that this algorithm contains a self-noise term which can be cancelled by adding $\mu_2(k) = \hat{a}(k)(p_r(k+1) - p_r(k-1))$.

The resulting algorithm, the optimised modified Mueller and Müller algorithm for real symbols (e.g., BPSK), outputs a timing error $\tau = 0.5(\mu_1(k) + \mu_2(k))$.

This can be generalised to the complex domain (e.g., for QPSK) as $\mu(k) = {\bf R}((\hat{c}(k) - \hat{c}(k-2))p^*(k-1) + \hat{c}^*(k-1)(p(k)-p(k-2)))$,

where

• $\hat{c}(k)$ is the receiver’s decision on the (complex) data symbol, and
• $p^*(k)$ is the complex conjugate of $p(k)$.

Figure 1 presents this equation in the form of a block diagram, including a combined filter and interpolator (and presumably sampler) betweeen the input signal and $p(k)$, a decision block between $p(k)$ and $\hat{c}(k)$, and a loop filter driven from the real output of the equation via $\beta$ which influences the filter/interpolator/sampler.

The paper then presents simulation results (using a gain factor of $\beta = 0.18$) which show that while the mM&M algorithm has fast acquisition it has lots of jitter and some symbol slips; by contrast the optimised mM&M algorithm exhibits much less jitter and no symbol slips, while preserving the same fast acquisition characteristics.

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