Keywords: random noise, jitter, timing jitter, rms noise, phase noise, rms phase noise
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TUTORIALS 3631

Abstract: Numerous factors affect random timing jitter, including noise sources such as phase noise, broadband noise, and spurs, as well as slew rate and bandwidth. This article explores these sources, and also provides equations to translate noise into timing jitter.

The purpose of this article is to both explain and demonstrate the direct relationship between timing jitter and these three noise sources.

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 a 40GHz bandwidth, a 10dB noise figure, a 20dB small-signal gain, and an output power of 0dBm, generates a -38dBm noise output (-174dBm + 10dB + 20dB + 10log10(40GHz)). This results in a signal-to-noise ratio (SNR) of 38dB. 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.

Mathematically, you can represent a sinusoid containing broadband white noise with the following equation:

where A is the amplitude, ω is the angular frequency, and

where

Equation 3 can be simplified by assuming that Δ

Equation 3 then becomes:

Dividing the numerators and denominators of various terms within Equation 4 by

Equation 5 is a jitter distribution function similar to the Gaussian distribution shown in Equation 2, except for the scale factor 1/

The test setup shown in

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

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

The test setup shown in Figure 4 was again used to verify Equation 9. The sinusoid was replaced by a variable-slew-rate square wave. Jitter was measured at the 50% point of the rising edge of the square wave. The results shown in

The information presented in Figure 7 raises 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. Because the RMS noise is directly proportional to the bandwidth, system designers must carefully choose the slew rate and bandwidth to minimize jitter.

Due to the limitations of most jitter-measuring equipment, it is often easier to characterize the purity of a low-noise signal by measuring its phase noise in the frequency domain, rather than measuring jitter in the time domain. For example, most jitter-measuring oscilloscopes are only capable of measuring jitters down to 1ps

The translation between phase noise and timing jitter has been explored in previous articles [1-2]. To derive the necessary equations relating phase noise to jitter, consider Equation 10 as a sinusoid containing phase noise:

where A is the amplitude, f

where t

The time between the two zero-crossings is the number of periods plus the jitter:

T

Rearranging Equation 15 and cancelling out the 2πN terms yields the jitter:

The squared RMS jitter is:

Because Φ(t) is a stationary process:

where

where

Recalling the algebraic identity 1 - cos(2Φƒτ = 2sin²(Φƒτ) and assuming the phase noise is close to the carrier and symmetrical (meaning the integration from -f

A phase-modulating circuit [4] was used as part of the test setup shown in

Again, τ ≅

Component | Noise Specification | Unit |

Amplifier | Residual noise-floor power density | dBm/Hz |

Noise figure | dB | |

Input referred noise density | nV/ | |

Oscillator | Phase noise floor | dBc/Hz |

If the noise density is given, you can estimate the total RMS noise by integrating the noise density over the effective bandwidth, as shown in the following equations:

The system load impedance Z

Noise figure (NF) is often used to characterize the noise performance of low-noise amplifiers and power amplifiers. You can derive the noise-floor density from the noise figure by summing it with the thermal noise of a 50Ω resistor and the system gain, as shown in the following equation:

For example, an amplifier with a 10dB noise figure, and a 20dB small-signal gain, has a noise floor density of -144dBm/Hz:

Knowing the noise density allows the derivation of the total noise voltage.

Op amp noise performance, on the other hand, is usually given in the form of input referred noise in nV/. Assuming the noise current is negligible and the source impedance is much smaller than the amplifier's input impedance, the total RMS noise can be calculated with the following equation:

For example, an op amp with an input-referred noise density of 8nV/, a small-signal gain of 20dB, and a bandwidth of 1GHz, generates 800µV

The phase noise of oscillators is usually given in dBc/Hz. The unit dBc indicates the normalization of the output noise to the desired signal power. The following equation can be used to obtain the total RMS noise voltage:

where P

- Ali Hajimiri et. al., "Jitter and Phase Noise in Ring Oscillators," IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, pp. 790-804.
- Boris Drakhlis, "Calculate Oscillator Jitter By Using Phase-Noise Analysis," Microwaves & RF, Jan. 2001 pp. 82-90 and p. 157.
- W. F. Egan, Frequency Synthesis by Phase Lock. New York: Wilen, 1981.
- Enrico Rubiola et. al., "The ±45° Correlation Interferometer as a Means to Measure Phase Noise of Parametric Origin" IEEE Transactions On Instrumentation and Measurement, Vol. 52, No. 1, pp. 182-188.