Consider the M-phase sampling system of Figure 1.1. It samples the x(t) reference and generates the xj[k] of Equation (3.4), where 1 ≤ j ≤ M. Its nominal sampling interval is Ts. However, each φj clock has its own timing offset, τj. If two adjacent clocks, φj and φj+1, have different timing offset, i.e., τj 6= τj+1, then a timing skew occurs. The sampling interval between xj[k] and xj+1[k] becomes Ts− τj+ τj+1. Figure 1.3 shows the proposed multi-phase timing-skew calibration architecture. A multi-phase timing-skew calibration processor (TSCP) is used to detect the timing skew between every adjacent clock pairs.
For each j, the delay of the Bj clock buffer is controlled by the Tj[k] output from the TSCP, such that
τj[k]= τj,0+ µt× Tj[k] (4.1) where µtis the step size for the timing control and τj,0 is the timing offset of the φjclock when Tj[k]= 0. The TSCP measures the timing skew between φjand φj+1, then adjusts Tj+1[k] to minimize the skew.
Figure 4.1 shows the TSCP’s block diagram. It includes M ZC detectors, which are deployed to measure the sampling intervals. Either the ZCD1 of Figure 3.11 or the ZCD2 of Figure 3.14 can be used as the ZC detectors. The x(t) reference is assumed to be a narrow-band signal. Its center frequency fi is near 0.25fc in order to establish a low-fi scenario. For the j-th calibration channel, its ZC detector senses any ZC be-tween xj[k] and xj+1[k], and generates a binary output, zj[k] ∈ {0, 1}. The probability of zj[k] = 1 is Pj,jz+1, which can be calculated from the ZC density ZR(t0, Tc) using
Equa-33
Σ Σ
Figure 4.1: A multi-phase timing-skew calibration processor (TSCP).
35
Figure 4.2: The ZC recorder.
tion (A.65). From Equation (A.67), a narrow-band asynchronous x(t) reference has an uniform ZR(t0, Tc) close to 2fi. Thus, the Pj,jz+1 probability is proportional to the sam-pling interval between xj[k] and xj+1[k], which is denoted the j-interval. In Figure 4.1, the zj[k] sequence is integrated onto an ACC1 accumulator. The accumulator’s output represents the average of zj[k], which is also proportional to the j-interval.
For each j, the TSCP compares the j-interval with the nominal sampling interval. The difference between the two intervals is the timing skew. The TSCP then adjusts Tj+1[k]
to minimize the skew. The nominal sampling interval is defined as the average of all j-interval where 1 ≤ j ≤ M. In Figure 4.1, the timing skew is calculated as the difference between the accumulation of zj[k] and the accumulation of m[k]. The m[k] sequence represents the average of the ZC occurrences among all sampling intervals. The m[k] is generated from the ZC recorder shown in Figure 4.2. The recorder accumulates every ZC from all ZC detectors. A comparator compares the accumulation result a[k] with M, and generates a binary m[k] ∈ {0, 1} for every clock cycle. Whenever a[k] ≥ M, the comparator issues m[k] = 1, and an amount of M is subtracted from the accumulation result during the following clock cycle. Note that m[k] is a sequence of 0 and 1. Its mean value represents the nominal sampling interval. The operation of m[k] averaging is provided by the ACC1 accumulator in each calibration channel. The proposed ZC recorder is simple and its hardware cost is low.
In the j-th calibration channel, the timing skew is calculated as U [k]= m[k] − zj[k].
Whenever R[k] ≥ +NC, S[k] = +1. Whenever R[k] ≤ −NC, S[k] = −1. Otherwise, S[k] = 0. In addition, the ACC1 accumulator is reset to zero whenever S[k] = +1 or S[k] = −1. Thus, −NC ≤ R[k] ≤ +NC. Each time S[k] is either +1 or −1, it can retain in such state for only one clock cycle. Finally, the ACC2 accumulator integrates the S[k] sequence, and generates the Tj+1[k] output. By adding the integration-and-dump operation of ACC1 and BPD, the fluctuation in Tj+1[k] is reduced.
In the proposed calibration scheme, the φ1clock in Figure 1.3 is a designated reference phase. It is no need to adjust the corresponding T1[k] control. Thus, T1[k] is preset to 0 in the TSCP of Figure 4.1.
The proposed calibration scheme contains two system parameters, µt and NC. To simplify analysis, we assume each calibration channel in the TSCP of Figure 4.1 employs identical µtand NC. Together with the ZRof x(t), they affect calibration behaviors, such as calibration converging speed and timing fluctuation. In general, large µt and small NC result in fast converging speed but large timing fluctuation. On the other hand, small µt and large NC result in small timing fluctuation but also slow converging speed. The following two subsections give detailed analyses.
4.1 Convergent Speed
Consider the j-th calibration channel in Figure 4.1. Its ZC detector measures the sampling interval between the φj and φj+1 clocks. According to Equation (3.4), the φj and φj+1
clocks have the timing offsets τj and τj+1 respectively. In Figure 4.1, the U [k] signal is the difference between zj[k] and m[k], representing the timing skew τj− τj+1. The U [k] is used to update the Tj+1[k] signal, which controls the τj+1[k] timing offset. In most cases, this calibration loop can be modeled as a continuous-time single-pole feedback system
4.2. TIMING FLUCTUATION 37
like
dτj+1 = µt× (τj− τj+1) · ZR· dk NC
The above equation states that, to update τj+1by one µtstep, it takes dk sampling intervals during which the τj − τj+1 timing skew causes NC zero crossings. Thus, we obtain the following differential equation for τj+1[k]
dτj+1[k]
dk = −τj+1[k] − τj[k]
τc (4.2)
where the system time constant τcis
τc = NC
µt × 1
ZR (4.3)
For a M-channel system, Equation (4.2) with 1 ≤ j ≤ M − 1 can be expanded into M − 1 coupled equations. In most practical case, τc is much larger than 1. Thus, by treating τj[k] as a constant, the transient behavior of τj+1[k] can be approximated by a simple exponential function with the τc time constant.
4.2 Timing Fluctuation
Consider the τj[k] of Equation (A.62). The TSCP measures the sampling interval be-tween the φj−1 and φj clocks, and then adjusts Tj[k]. Assume Tj−1[k] remains constant and φj−1 is fixed. The TSCP adjusts only Tj[k] to force τj[k] moving toward 0. As the process converges, the behavior of τj[k] becomes a discrete random fluctuation around zero. Figure 4.3 illustrates a probability mass function for τj, M (τj). The discrete values for τjis τj,0, τj,±1, τj,±2, . . . , with τj,0being closest to zero. The distance between two adja-cent discrete values is µt. The value of τj,0is between −0.5µtand+0.5µt. The calibration loop forces the maximum value of M (τj) to occur at τj,0. A mathematical treatment of M(τj) is included in Appendix A.3. The resulting standard deviation of τj, averaged over possible value of τj,0, can be found as
τ j τ j,−2 τ j,−1 τ j,0 τ j,+1 τ j,+2
0
Figure 4.3: Probability mass function of τj, M (τj).
For the multi-phase calibration system shown in Figure 1.3 and Figure 4.1, the φ1 clock with T1[k] = 0 is chosen as the designated reference phase. All other clocks are adjusted by the TSCP to achieve uniform phase spacing. The timing skew between φ1
and φ2 is minimized by adjusting the delay of the φ2 clock buffer through T2[k]. The timing skew between φ2and φ3is minimized by adjusting the delay of the φ3clock buffer through T3[k]. This calibration arrangement repeats for φ4, φ5, etc, and is referred as the linear referencing arrangement illustrated in Figure 4.4. Note that φ1 does not fluctuate.
The timing fluctuation of φ2 is summarized by Equation (4.4). The timing fluctuation of φ3 is larger than Equation (4.4), since it uses the fluctuating φ2 as its phase reference.
In fact, the timing fluctuation is accumulated along the reference chain. The standard deviation of the φj’s timing fluctuation can be expressed as
σ2(τj) = (j − 1) × σ2(τ) j ≥2 (4.5) To reduce the overall timing fluctuation, the circular referencing arrangement shown in Figure 4.4 is suggested. In this scheme, both φ2and φ8use φ1as the reference for timing-skew calibration. Then φ3 and φ7 use φ2 and φ8 as the references respectively. In this arrangement, the maximum σ2(τj) is reduced by half. The overall averaged timing
fluctu-4.2. TIMING FLUCTUATION 39