【正文】 of 10 11 with respect to the reference. This simple example requires only two time interval readings to be made， and △ t is simply the difference between the two readings. Often， multiple readings are taken and the frequency offset is estimated by using least squares linear regression on the data set， and obtaining △ t from the slope of the least squares line. This information is usually presented as a phase plot， as shown in Fig. . The device under test is high in frequency by exactly 110 9， as indicated by a phase deviation of 1 ns/s. Dimensionless frequency offset values can be converted to units of frequency (Hz) if the nominal frequency is known. To illustrate this， consider an oscillator with a nominal frequency of 5 MHz and a frequency offset of + ′ 10 11. To find the frequency offset in hertz， multiply the nominal frequency by the offset： (5 106) (+ 10 11) = 10 5 =+ Hz Then， add the offset to the nominal frequency to get the actual frequency： 5， 000， 000 Hz + Hz = 5， 000， Hz Stability Stability indicates how well an oscillator can produce the same time or frequency offset over a given time interval. It doesn?t indicate whether the time or frequency is “right” or “wrong， ” but only whether it stays the same. In contrast， accuracy indicates how well an oscillator has been set on time or on frequency. To understand this difference， consider that a stable oscillator that needs adjustment might produce a frequency with a large offset. Or， an unstable oscillator that was just adjusted might temporarily produce a frequency near its nominal value. Figure shows the relationship between accuracy and stability. Stability is defined as the statistical estimate of the frequency or time fluctuations of a signal over a given time interval. These fluctuations are measured with respect to a mean frequency or time offset. 中北大學(xué) 2021 屆畢業(yè)設(shè)計(jì)說明書 第 8 頁 共 16 頁 Shortterm stability usually refers to fluctuations over intervals less than 100 s. Longterm stability can refer to measurement intervals greater than 100 s， but usually refers to periods longer than 1 day. Stability estimates can be made in either the frequency domain or time domain， and can be calculated from a set of either frequency offset or time interval measurements. In some fields of measurement， stability is estimated by taking the standard deviation of the data set. However， standard deviation only works with stationary data， where the results are time independent， and the noise is white， meaning that it is evenly distributed across the frequency band of the measurement. Oscillator data is usually no stationary， since it contains time dependent noise contributed by the frequency offset. With stationary data， the mean and standard deviation will converge to particular values as more measurements are made. With no stationary data， the mean and standard deviation never converge to any particular values. Instead， there is a moving mean that changes each time we add a measurement. For these reasons， a nonclassical statistic is often used to estimate stability in the time domain. This statistic is sometimes called the Allan variance， but since it is the square root of the variance， its proper name is the Allan deviation. The equation for the Allan deviation (σy(τ)) is 2M 1y i + ii = 1= y yM+?? ? 11（ ） （ ）（ 21 ） where yi is a set of frequency offset measurements containing y1， y2， y3， and so on， M is the number of values in the yi series， and the data are equally spaced in segments τ seconds long. Or 2N 2x i + 1 i2 i = 1= 2 x + xN+?? ? ? i+21（ ） （ x ）（ 21 ） Where xi is a set of phase measurements in time units containing x1， x2， x3， 中北大學(xué) 2021 屆畢業(yè)設(shè)計(jì)說明書 第 9 頁 共 16 頁 and so on， N is the number of values in the xi series， and the data are equally spaced in segments τ seconds long. Note that while standard deviation subtracts the mean from each measurement before squaring their summation， the Allan deviation subtracts the previous data point. This differencing of successive data points removes the time dependent noise contributed by the frequency offset. An Allan deviation graph is shown in Fig. . It shows the stability of the device improving as the averaging period (τ)