
Keywords: CLK jitter, clock jitter, phase noise, phase noise conversion, phase noise measurement, jitter measurement, jitter and phase noise, phase noise calculation, timing error
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Clock (CLK) Jitter and Phase Noise Conversion

Abstract: This application note on clock (CLK) signal quality describes how to measure jitter and phasenoise, including period jitter, cycletocycle jitter, and accumulated jitter. It describes the relationship between period jitter and phasenoise spectrum and how to convert the phasenoise spectrum to the period 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 phasenoise measurements. The oftenused jitter measurements are period jitter, cycletocycle jitter, and accumulated jitter. Among these jitters, period jitter is most often encountered. Clock phasenoise measurement examines the spectrum of the clock signal.
This article first briefly reviews the measurement setups for clock period jitter and phase noise. The relationship between the period jitter and the phasenoise spectrum is then described. Finally, a simple equation to convert the phasenoise spectrum to the period jitter is presented.
Period Jitter and Phase Noise: Definition and Measurement
Period Jitter
Period jitter (J
_{PER}) is the time difference between a measured cycle period and the ideal cycle period. Due to its random nature, this jitter can be measured peaktopeak or by root of mean square (RMS). We begin by defining the clock risingedge crossing point at the threshold V
_{TH} as T
_{PER}(n), where n is the time domain index, as shown in
Figure 1. Mathematically, we can describe J
_{PER} as:
where T
_{0} is the period of the ideal clock cycle. Since the clock frequency is constant, the random quantity J
_{PER} must have a zero mean. Thus the RMS of J
_{PER} can be calculated by:
where <•> is the expected operation. Figure 1 shows the relation between J
_{PER} and T
_{PER} in a clock waveform.
Figure 1. Period jitter measurement.
PhaseNoise Spectrum
To understand the definition of the phasenoise 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 phasenoise 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 2 illustrates the definition of L(f).
Figure 2. Definition of phasenoise spectrum.
Mathematically, the phasenoise 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.
Period Jitter (J_{PER}) Measurement
There are different instruments used to measure the period jitter. People most commonly use a highprecision digital oscilloscope to conduct the measurement. When the clock jitter is more than 5 times larger than the oscilloscope's triggering jitter, the clock jitter can be acquired by triggering at a clock rising edge and measuring it at the next rising edge.
Figure 3 shows a splitter generating the trigger signal from the clock under test. This method eliminates the internal jitter from the clock source in the digital oscilloscope.
Figure 3. Selftrigger jitter measurement setup.
It is possible for the duration of scope triggerdelay to be longer than the period of a highfrequency clock. In that case, one must insert a delay unit in the setup that delays the first rising edge after triggering so that it can be seen on the screen.
There are more accurate methods for measuring jitter. Most of these approaches use a postsampling process of the data sampled from highspeed digital oscilloscopes to estimate the jitter according to the definitions in Equations 1 or 2. This postsampling approach provides highprecision results, but it can only be performed with highend digital oscilloscopes [2, 3].
PhaseNoise Spectrum L(f) Measurement
As Equation 3 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 inputfrequency 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 signalspectrum changes at different points in the test setup.
Figure 4. Practical phasenoise measurement setup.
The structure described in Figure 4 is typically called a carriersuppress 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 Period Jitter and Phase Noise
Using the Fourier series expansion, it can be shown that a squarewave clock signal has the same jitter behavior as its base 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 period jitter is:
From Equation 4 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 4 can be approximated as:
The spectrum of C(t) is then:
where S
_{Θ}(f) is the spectrum of q(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(2
πf
_{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) after scaled down by A²/4.
From Equation 11, the mean square (MS) of Θ(t) can be calculated by:
Following Equation 5 above, we finally show the relationship between the period jitter, J
_{PER}, and the phase noise spectrum, L(f), as:
In some applications like SONET and 10Gb, engineers only monitor the jitter at a certain frequency band. In such a case, the RMS J
_{PER} within a certain band can be calculated by:
Approximation of RMS J_{PER} from L(f)
The phase noise usually can be approximated by a linear piecewise function when the frequency axis of L(f) is in log scale. In such a case, L(f) can be written as:
where K1 is the number of the pieces of the piecewise function and U(f) is the step function. See
Figure 5.
Figure 5. A typical L(f) function.
If we substitute L(f) shown in Equation 15 into Equation 14, we have:
To illustrate this, the following table presents a piecewise L(f) function with f
_{C} = 155.52MHz.
Table 1. Measurements of Function L(f).
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.
Table 2. Parameters to Present L(f) as a Piecewise Function.
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 16, 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 16 can also be used to estimate the required jitter limit when the phasenoise 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 phasenoise 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 period jitter from the phasenoise 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.
Reference
 SEMI G800200, "Test Method for the Analysis of Overall Digital Timing Accuracy for Automated Test Equipment".
 Tektronix Application Note: "https://www.tek.com/jittermeasurementandtiminganalysis"
 LeCroy White Paper: "The Accuracy of Jitter Measurements"
 David Chandler, "Phase JitterPhase Noise and VCXO," Corning Frequency Control Inc.
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© Dec 10, 2004, Maxim Integrated Products, Inc.

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APP 3359: Dec 10, 2004
APPLICATION NOTE 3359,
AN3359,
AN 3359,
APP3359,
Appnote3359,
Appnote 3359
