Timing jitter and noise are among the least understood engineering concepts, yet they are among the most critical parameters in analog and digital designs. Especially in high-speed communications systems, poor jitter performance can cause increased bit-error rates and limit the system speed. Among the many factors that can influence timing jitter, this article will focus on the effects of random noise.

Timing jitter is generally defined as the short-term variation of a significant instant of a digital signal from its ideal position in time. There are a number of factors that contribute to random timing jitter, including broadband noise, phase noise, spurious signals, slew rate, and bandwidth. Both phase and broadband noise are random, whereas spurious signals are deterministic responses caused by various identifiable interference signals, such as crosstalk and power supply coupling. Slew rate and bandwidth also affect jitter. Figure 1 shows a nonideal sinusoid that contains the three noise sources; in a digital clock signal, these sources contribute to jitter that accumulates over time.

All electronic components, especially amplifiers and logic devices, generate broadband noise. Also known as the noise floor, broadband noise is a combination of shot noise and thermal noise. Commonly found in diodes and transistors, shot noise is caused by random hopping of charges across a potential barrier within a semiconductor junction. Thermal noise, on the other hand, is unaffected by current flow. It is caused by the random thermal motion of carriers, for example within a MOSFET's gate and channel resistance. The thermal noise power is directly proportional to the resistance and temperature.

The effect of broadband noise on timing jitter becomes significant as the operating bandwidth of modern components pushes into the multi-GHz range. For example, a broadband amplifier driver with 40-GHz bandwidth, 10-dB noise figure, 20-dB small-signal gain, and 0-dBm output power generates a ­38-dBm noise output [­174 dBm + 10 dB + 20 dB + 10log₁₀(40 GHz)]. This results in a signal-to-noise ratio (SNR) of 38 dB. At this SNR level, the broadband noise is a significant contributor to timing jitter. The total root-mean-square (RMS) noise voltage is the integral of the noise floor over the bandwidth.

Figure 2 illustrates how RMS noise is translated into timing jitter.

A sinusoid containing broadband white noise can be represented by:

where:

A = the amplitude,

v_n(t) = the noise voltage at time t.

The random noise v_n(t) has a Gaussian (normal) distribution. The probability distribution f(v_n) of the noise voltage v_n is:

where:

v_nRMS = the RMS noise voltage.

To understand how noise voltage is translated into timing jitter, consider applying y(t) into the input of a jitter-measuring instrument, such as a sampling oscilloscope with a histogram function. Each time y(t) crosses the 0-V threshold, a data point is added to the histogram. Just as depicted in Fig. 2, at time Δt, there is a probability that the noisy signal Δy will reach the threshold; thus the jitter is added to the histogram at Δt sooner or later than the anticipated sampling point. The probability density as a function of the timing jitter Δt is calculated by setting v_n = Δy = Asin(2πfΔt) in Eq. 2. The result is the jitter distribution function shown in the histogram.

Equation 3 can be simplified by assuming that Δt is small when compared to the period of the sinusoid; therefore, A sin(2πfΔt) ≅ A(2πΔt) = AωΔt:

Equation 3 then becomes:

Dividing the numerators and denominators of various terms within Eq. 4 by Aω yields:

Equation 5 is a jitter distribution function similar to the Gaussian distribution shown in Eq. 2, except for the scale factor 1/Aω. Thus, the RMS jitter is found as:

The test setup shown in Fig. 3 was used to verify Eq. 6. Both a clean sinusoid and a broadband noise signal were combined and injected into a sampling oscilloscope where jitter was measured at the zero crossing. To ensure meaningful results, the input broadband noise was set higher than the noise floor of the oscilloscope. Figures 4 and 5 show the results of the experiment. Figure 4 shows jitter as a function of frequency with constant RMS noise injected, while Fig. 5 shows jitter as a function of RMS noise voltage with constant frequency. The similarity between the measured and calculated jitter curves proves that Eq. 6 can be used to translate broadband noise into timing jitter.

With slight modification, Eq. 6 can also accommodate the jitter translation of other waveforms. By definition, the Aω term of Eq. 6 is the slew rate S at the 0-V threshold. Any waveforms with a known slew rate at the threshold can be used to relate Δt to Δy, because v_n = Δy = S Δt (see Fig. 2). Substituting this into Eq. 2 yields Eq. 7:

Dividing the numerators and denominators of various terms within Eq. 7 by S yields:

Equation 8 is similar to the Gaussian distribution shown in Eq. 2 except for the scale factor of 1/S. Thus, the RMS jitter is:

The test setup shown in Fig. 3 was again used to verify Eq. 9. The sinusoid was replaced by a variable-slew-rate square wave. Jitter was measured at the 50-percent point of the rising edge of the square wave.

The results of those measurements (not shown) raise an interesting point. It appears that a faster slew rate waveform results in lower jitter. However, a faster slew rate requires a higher operating bandwidth, which increases the RMS noise of the system. The RMS noise is directly proportional to the bandwidth. With this relationship in mind, system designers must carefully choose the slew rate and bandwidth to minimize jitter.