Timing Recovery by Digital Interpolation

August 27, 2009 at 1:35 pm · Filed under Digital Filters ·Tagged , , , ,

Astoundingly enough, J.G. Proakis textbook “Digital Communications” in its 4th edition does not mention how to perform truly digital timing recovery. That is, timing recovery with a sampling clock which is unrelated to the symbol rate, or in other words, timing recovery where symbol rate (1/T) and sampling rate (1/Ts) are incommensurate (T/Ts isn’t an integer). Thus, there is no need to control the sampling instant (no need for a Voltage-Controlled Oscillator, VCO; the receiver uses a free-running oscillator for sampling).

Fortunately, H. Meyr, M. Moeneclaey and S.A. Fechtel textbook “Digital Communication Receiver” comes to the rescue. The trick is to use the shift-property of band-limited functions applied to the sampling theorem,

x(t+\epsilon) = \sum_{n=-\infty}^{\infty}{x(\epsilon+nT_s)\cdot\text{sinc}[\frac{\pi}{T_s}(t-nT_s)] }

= \sum_{n=-\infty}^{\infty}{x(nT_s) \cdot\text{sinc}[\frac{\pi}{T_s}(t+\epsilon-nT_s)]}

with \epsilon an arbitrary time shift. This states that our signal x(t) can be represented by samples either at (nTs+\epsilon) or (nTs)! Thus, we can perform interpolation through sinc filtering to obtain a set of samples at the needed (nTs+\epsilon) instants. Interpolation can be performed by a digital time-variant FIR filter, in most practical cases, a 4-tap FIR filter running at 1/Ts. If oversampling (Ts \geq T) was performed then decimation is needed. The time shift can be estimated using a digital phase-locked loop (DPLL).

Theoretical aspects are discussed in Chapter 4. Chapters 9 and 10 are a must for the practitioner.

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