### Abstract

This application note on clock (CLK) signal quality describes the relationship between jitter and phase-noise spectrum and how to convert the phase-noise spectrum to jitter.

Clock (CLK) signals are required in almost every integrated circuit or electrical system. In today's world, digital data is processed or transmitted at higher and higher speeds, while the conversions between analog and digital signals are done at higher resolutions and higher data rates. These functions require engineers to pay special attention to the quality of clock signals.

Clock quality is usually described by jitter or phase-noise measurements. The often-used jitter measurements are period jitter, cycle-to-cycle jitter, and absolute, otherwise known as phase or Time Interval Error jitter. Clock phase-noise measurement examines the spectrum of the clock signal.

This article first briefly reviews the measurement setups for clock jitter and phase noise. The relationship between jitter and the phase-noise spectrum is then described. Finally, a simple equation to convert the phase-noise spectrum to jitter is presented.

### Jitter and Phase Noise: Definition and Measurement

#### Period Jitter

One type of jitter is known as phase or absolute jitter. Other jitter types such as period or cycle-to-cycle jitter can be derived from the absolute jitter. Henceforth in this document, the term jitter is used for brevity with the understanding that it refers to absolute jitter. Jitter is the deviation of clock edges from their ideal instant. If we assume an ideal clock signal and compare it against the actual clock, then the jitter of the n^{th} rising edge is the time deviation of the n^{th} rising edge of the actual clock from the ideal clock. See Figure 1.

Jitter is calculated based on the statistics measured over many clock edges and is specified by its root mean square (RMS) value. The ideal clock is typically generated by postprocessing the captured actual clock signal in high-end oscilloscopes. A so-called Golden PLL is used to phase lock onto the clock signal. Jitter that has a frequency within the PLL loop bandwidth is tracked and shows up on the ideal clock that is output from the PLL. Jitter that is outside of the loop bandwidth is attenuated. Conceptually, the measurement is made as shown in Figure 2. The oscilloscope is set to infinite persistence, and the oscilloscope is triggered with the ideal clock output from the PLL. Any jitter that is common to the actual clock and ideal clock is not displayed by the oscilloscope, hence the measured jitter is effectively highpass filtered. For this discussion, assume a very narrow PLL bandwidth so that all jitter of significance to the system is measured.

### Phase-Noise Spectrum

To understand the definition of the phase-noise spectrum L(f), we first define the power spectrum density of a clock signal as S_{C}(f). The S_{C}(f) curve results when we connect the clock signal to a spectrum analyzer. The phase-noise spectrum L(f) is then defined as the attenuation in dB from the peak value of S_{C}(f) at the clock frequency, f_{C}, to a value of S_{C}(f) at f. Figure 3 illustrates the definition of L(f).

Mathematically, the phase-noise spectrum L(f) can be written as:

Remember that L(f) presents the ratio of two spectral amplitudes at the frequencies, f_{C} and f. The meaning of L(f) will be discussed in next section.

### Phase-Noise Spectrum L(f) Measurement

As Equation 1 showed above, L(f) can be measured with a spectrum analyzer directly from the spectrum, S_{C}(f), of the clock signal. This approach, however, is not practical. The value of L(f) is usually larger than 100dBc which exceeds the dynamic range of most spectrum analyzers. Moreover, f_{C} can sometimes be higher than the input-frequency limit of the analyzer. Consequently, the practical way to measure the phase noise uses a setup that eliminates the spectrum energy at f_{C}. This approach is similar to the method of demodulating a passband signal to baseband. Figure 4 illustrates this practical setup and the signal-spectrum changes at different points in the test setup.

The structure described in Figure 4 is typically called a carrier-suppress demodulator. In Figure 4, n(t) is the input to the spectrum analyzer. We will next show that by scaling down the spectrum of n(t) properly, we can obtain the dBc value of L(f).

### Relation between RMS Jitter and Phase Noise

Using the Fourier series expansion, it can be shown that a square-wave clock signal has the same jitter behavior as its fundamental harmonic sinusoid signal. This property makes the jitter analysis of a clock signal much easier. A sinusoid signal of a clock signal with phase noise can be written as:

and the jitter is:

From Equation 2 we see that the sinusoid signal is phase modulated by the phase noise Θ(t). As the phase noise is always much smaller than π/2, Equation 2 can be approximated as:

The spectrum of C(t) is then:

where S_{Θ}(f) is the spectrum of T(t). Using the definition of L(f), we can find:

This illustrates that L(f) is just S_{Θ}(f) presented in dB. This also explains the real meaning of L(f).

We have now shown that the setup in Figure 4 enables the measurement of L(f). Furthermore, one can see that the signal C(t) is mixed with cos(2πf_{C}t) and filtered by the lowpass filter. Thus, we can express the signal n(t) at the input of the spectrum analyzer as:

The spectrum appears on the spectrum analyzer as:

Therefore we can obtain the phase noise spectrum S_{Θ}(f) and L(f):

Then L(f) can be read in dBc directly from the spectrum of n(t) scaled by 4/A².

From Equation 9, the mean square (MS) of Θ(t) can be calculated by:

Following Equation 3 above, we finally show the relationship between the jitter, J_{ABS} and the phase noise spectrum, L(f), as:

In some applications like SONET and 10Gb, engineers only monitor the jitter in a certain frequency band. In such a case, the RMS J_{ABS} within a certain band can be calculated by:

### Approximation of RMS J_{ABS} from L(f)

The phase noise usually can be approximated by a linear piece-wise function when the frequency axis of L(f) is in log scale. In such a case, L(f) can be written as:

where K-1 is the number of the pieces of the piece-wise function and U(f) is the step function. See Figure 5.

If we substitute L(f) shown in Equation 13 into Equation 12, we have:

To illustrate this, the following table presents a piecewise L(f) function with f_{C} = 155.52MHz.

Frequency (Hz) | 10 | 1000 | 3000 | 10000 |

L(f) (dBc) | -58 | -118 | -132 | -137 |

Next we calculate the a_{i} and b_{i} by:

The results are listed in Table 2.

i | 1 | 2 | 3 | 4 |

f_{i} (Hz) |
10 | 1000 | 3000 | 10000 |

a_{i} (dBc/decade) |
-30 | -29.34 | -9.5 | N/A |

b_{i} (dBc) |
-58 | -118 | -132 | -137 |

Substituting the Table 2 values into Equation 14, we get:

The RMS jitter of the same clock measured by the setup in Figure 4 at the same band is 4.2258ps. Therefore, the proposed approximation approach for converting phase noise to jitter has proved quite accurate. In this example, the error is less than 4%.

Equation 14 can also be used to estimate the required jitter limit when the phase-noise spectrum envelope is given.

A simple spreadsheet file has been posted with the equation coded as an example.

### Summary

This article demonstrates the exact mathematical relationship between jitter measured in time and the phase-noise measured in frequency. Many engineers concerned with signal integrity and system timing frequently question this relationship. The results presented here clearly answer the question. Based on that mathematical relationship, we proposed a method for estimating the jitter from the phase-noise spectrum. Engineers can use this method to quickly establish a quantitative relationship between the two measurements, which will help greatly in the application or design of systems and circuits.

### References

SEMI G80-0200, "Test Method for the Analysis of Overall Digital Timing Accuracy for Automated Test Equipment".

Tektronix Application Note: "https://www.tek.com/jitter-measurement-and-timing-analysis"

LeCroy White Paper: "The Accuracy of Jitter Measurements"

David Chandler, "Phase Jitter-Phase Noise and VCXO", Corning Frequency Control Inc.