# 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.

# SDR and weather stations

I’ve been having fun recently learning about software defined radio (SDR). The impetus is this weather station, which Aidan got for Christmas. We’ve all very much enjoyed the information it provides, and it’s been a great talking point with the receiver on the family dining table. But when we’re at work or on holiday, wouldn’t it be nice to know what the weather’s like back home? Is it raining? Has the fence blown over? For that reason, in May I bought one of these:

The mast transmits the weather data to the display by radio in the 433 MHz band. This NooElec NESDR Mini is a software-controlled digital radio receiver which covers 25MHz to 1750MHz. So it should work!
Since I got it, I’ve been busy learning how to make it go. This has turned out to be more complex than I realised – I finally got it all working in October! – so expect a series of follow-up blog posts.