Compensation of Two-periodic Nonuniform Holding Signal Distortions by Using a Variable FIR Filter Christian Vogel Signal and Information Processing Laboratory ETH Zurich email: c.vogel@ieee.org Abstract—We introduce a Farrow-based variable FIR filter to compensate distortions in digital-to-analog converters (DACs) that are due to two-periodic nonuniform holding signals. With- out compensation such holding signals introduce distortions that reduce the DAC performance considerably. By utilizing a discrete-time model of a zero-order hold, we first derive the ideal frequency responses of the compensation filters. Then we exploit the spectral properties of the ideal frequency responses to introduce the design of a variable FIR filter that approximates the ideal response with low implementation complexity. Finally, a design example is provided. Index Terms—compensation, nonuniform holding signals, digital-to-analog converters, variable FIR filter, time-interleaved I. I NTRODUCTION In this paper we investigate the design of a variable FIR filter to compensate the distortions that are due to two-periodic nonuniform holding signals as shown in Fig. 1. Such holding signals can be found at the output of time-interleaved digital- to-analog converters (DACs) [1]–[3] and in DACs driven by clock signals with periodic phase-errors (deterministic jitter) [4]. In a previous paper [5], we have shown that we can compensate the distortions caused by M -periodic nonuniform holding signals by employing an M -periodic time-varying digital filter before digital-to-analog conversion. However, the general solution involves a quite complex filter design that has do be redone each time the hold times change. The complexity of the filter design makes it difficult to find the hold times on- line during normal operation of the DAC. In this paper we focus on the two-periodic case, where we can use a Farrow- based [6] variable FIR filter to compensate the distortions. The design of the filter is off-line and spectral characteristics can be tuned by a single parameter. Therefore, the filter can be particularly useful for blind identification and tracking of the hold times. For the derivations in this paper, we do not consider the analog reconstruction filter, which attenuates all C. Vogel was supported by the Austrian Science Fund FWF’s Erwin Schroedinger Fellowship J2709-N20. 0T + r 0 T 1T + r 1 T 2T + r 2 T 3T + r 3 T 4T + r 4 T x(0T ) x(1T ) x(2T ) x(3T ) x(4T ) t ˆ y(t) T (1 + λ)T (1 − λ) ×T Fig. 1. Two-periodic nonuniform holding signal. The solid arrows are the two-periodically shifted sampling instants nT + rnT , where rn = r n+2 , that lead to the nonuniform holding signal. The dash-dotted arrows show the ideal sampling instants nT . out-of-band signal energy including the additional distortions of the nonuniform holding signals. II. SYSTEM MODEL The zero-order hold (ZOH) output of a DAC can be mathematically represented by an impulse train modulator ∑ ∞ n=-∞ x(nT )δ(t − nT ), where x(nT )= x[n] and T is the nominal sampling period, followed by a filter h(t) [7]. In the uniform case, the impulse response of the filter h(t) is one for 0 ≤ t<T and zero for all other times. Contrary, in the two-periodic nonuniform case, the hold time T of the ZOH changes periodically over time, where the relative timing offsets r n are periodic with 2, i.e., r n = r n+2 for all n. Such holding signals are illustrated in Fig. 1 and can be represented as [8], [9] ˆ y(t)= ∞  n=-∞ x(nT )h n (t − (n + r n )T ) (1) where the impulse responses of the filters are given by h n (t)= u(t) − u(t − (1 + r n+1 − r n )T ). (2) After substituting (2) in (1) and defining r n+1 − r n =(−1) n λ and r n + r n+1 = φ, we can rearrange (1) as ˆ y(t)= y(t) ∗ δ(t − (1 + φ) T 2 ) (3)