 # Introduction to Jitter, Noise, and Signal Integrity at High-Speed

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## 1.2 Sources of Timing Jitter, Amplitude Noise, and Signal Integrity

Jitter and noise are deviations from an ideal signal. Jitter and noise can have many causes. The physical nature of various noise and jitter sources for a communication system can be classified into two major classes: intrinsic and nonintrinsic. The intrinsic type has to do with the physical properties of electrons and "holes" in electrical or semiconductor devices. The nonintrinsic type are design-related and may be eliminated. These types are discussed in detail in the following sections.

### 1.2.1 Intrinsic Noise and Jitter

Intrinsic noise is fundamentally caused by the randomness and fluctuation of electrons and "holes" existing in all the electronic/optical/semiconductor circuits/devices. Intrinsic noise can be minimized but cannot be completely removed from devices or systems. Therefore, this kind of noise puts a fundamental limit on device and system performance and dynamic range. Typical intrinsic noises in electrical-optical devices include thermal noise, shot noise, and flick noise.

#### 1.2.1.1 Thermal Noise

Thermal noise is caused by the random motion of charge carriers under the thermal equilibrium condition. The kinetic energy of those randomly fluctuating charge carriers is proportional to their temperature, as well as to their mean-square velocity. The power spectrum density (PSD) of the thermal noise is white and apparently proportional to its temperature. Thermal noise places a fundamental limit on the signal-to-noise ratio (SNR) performance because it exists in all electric/optical/semiconductor devices having a nonzero absolute temperature. Johnson4 first discovered that the noise in a conductor depends on temperature and resistor under the thermal equilibrium condition. Nyquist5 shortly after developed a theory to explain Johnson's discovery based on the second law of thermodynamics. Because of their pioneering contributions, thermal noise is sometimes called Johnson noise or Nyquist noise.

#### 1.2.1.2 Shot Noise

Shot noise is produced by individual quantized carrier flow (current) in a potential barrier with a random generation time or spatial distribution. In other words, shot noise is basically due to random flow fluctuation. Schottky6 first studied shot noise in vacuum tube diodes and later it was also found in P-N junction in a semiconductor transistor. Shot noise is directly proportional to DC bias current, as well as the charge of the carrier. Shot noise is typically larger than thermal noise in semiconductor devices.

#### 1.2.1.3 Flick Noise

Flick noise is a phenomenon that is found to have a noise power spectrum inversely proportional to the frequency over a wide range of frequencies. Johnson was the first to observe flick noise in an electronic system.7 Flick noise can be found in all active devices, and some passive devices such as carbon resistors. DC current is necessary to produce flick noise. No universally accepted theory explains the cause and mechanism of flick noise, unlike the causes of thermal and shot noise. As a result, the quantitative study of flick noise is mostly empirical. It has been found that the PSD of flick noise is proportional to 1/fα, where α is around 1. Because of this reason, flick noise is also called 1/f noise. One common interpretation of flick noise is the "trap and release" theory. It is believed that the flow of carriers due to the DC current can be trapped due to contamination and defects in devices. However, the "trap and release" process is random, giving rise to the flick noise that is most significant at low frequencies.8

### 1.2.2 Translation of Noise to Timing Jitter

Noise is typically described using physical quantities or parameters. In communication, computer, and electronic systems, those quantities may include voltage, current, or power. We use the generic term of amplitude to represent those physical quantities. Assuming that the amplitude noise ΔA(t) is superimposed on the amplitude waveform of A0(t) so that the total waveform has the following form:

#### Equation 1.1 the corresponding timing jitter can be estimated through the linear small-signal perturbation theory as the following:

#### Equation 1.2 where k = (dA0(t)/dt) is the slope or slew rate of the waveform.

This linear amplitude noise to timing jitter conversion is shown in Figure 1.5. Figure 1.5 Amplitude noise to timing jitter conversion through the linear perturbation model.

You can see that for amplitude noise ΔA, the corresponding timing jitter decreases as the slope increases, and vice versa. To maintain a smaller timing jitter conversion, a large slope or fast slew rate is favored. In the context of a digital signal, this implies a small rise/fall time.

### 1.2.3 Nonintrinsic Noise and Jitter

Nonintrinsic jitter and noise are design-related deviations. In other words, those types of jitter and noise can be controlled or fixed with appropriate design improvements. Commonly encountered nonideal design-related noise and jitter include periodic modulation (phase, amplitude, and frequency), duty cycle distortion (DCD), intersymbol interference (ISI), crosstalk, undesired interference such as electromagnetic interference (EMI) due to radiation, and reflection caused by unmatched media. The following sections discuss these noise sources and their root causes.

#### 1.2.3.1 Periodic Noise and Jitter

Periodic noise or jitter is a type of signal that repeats every time period. It can be described mathematically by the following general equation:

#### Equation 1.3 where T0 is the period, t is the time, and φ0 is the phase of the periodic signal. The period T0 and frequency f0 satisfy the reciprocal relationship of T0 = 1/f0. Although the notation and discussion are based on timing jitter, the same type of discussion can be applied to amplitude noise. The frequency-domain periodic function can be obtained through Fourier Transformation (FT), a subject that is discussed in Chapter 2, "Statistical Signal and Linear Theory for Jitter, Noise, and Signal Integrity."

Periodic jitter can be caused by various modulation mechanisms, such as amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM). Moreover, the modulation function can have various shapes. Typical modulation shapes include sinusoidal, triangular, and sawtooth. It is apparent that a periodic amplitude noise causes period timing jitter, with the amplitude proportional inversely to the slope or slew rate of the edge transition, as discussed in section 1.3.2. In the computer environment, period noise/jitter can be caused by switching power supply, spread-spectrum clock (SSC), and period EMI sources.

#### 1.2.3.2 Duty Cycle Distortion (DCD)

DCD is defined as the deviation in duty cycle from its normal value. Mathematically, a duty cycle is the ratio of pulse width to its period for a clock signal, as shown in Figure 1.6. Figure 1.6 Illustration of period (T0), pulse width PW+/PW- (either positive or negative), and reference level for a periodic signal.

Duty cycle is defined as follows:

#### Equation 1.4 Most clocks have a nominal duty cycle of 50%. So either shorter pulse width or longer pulse width causes DCD. DCD can be caused by pulse width deviation, period deviation, or both. Furthermore, pulse width deviation can be caused by the deviation of reference signal level. Another DCD-causing mechanism is propagation delay if the clock is formed from rising and falling edges of two half-rate clocks and those two half-rate clocks undergo different propagation delays. Because a clock can have many periods, DCD must be looked at from the distribution point of view with many samples considered, and the average period should be used for the overall DCD estimation.

#### 1.2.3.3 Intersymbol Interference (ISI)

ISI is related to data signal, but a clock signal does not have ISI by definition. A data signal is a generic digital signal form that does not have to have an edge transition in every UI or bit period, like the clock signal. The data signal can be kept at the same amplitude level for many UIs without an edge transition, whereas a clock signal cannot be. The type of data pattern used in digital communication critically depends on the coding scheme of the communication architecture.9 An important parameter for digital pattern is the run length, which is defined as the maximum length of consecutive 1s or 0s within a pattern. The run length determines the lowest frequency of the data pattern spectrum and therefore governs the frequency range for the test coverage. The long-haul fiber-optic communication standard SONET uses a scramble code scheme and can have a much longer run length (such as a run length of 23, 31) and therefore relatively low-frequency spectral content. A short-haul data communication standard such as Fibre Channel or Gigabit Ethernet uses block code (e.g, 8B10B coding) that has a shorter run length (e.g., a run length of 5) and relatively high-frequency spectral content.

In a lossy medium, the previous bits can cause both transition timing and amplitude level off the ideal values. In copper-based communication systems, this is due to the "memory" characteristics of the electronic devices used to switch bits between 1s and 0s. One example of this "memory" nature is the capacitive effect. Due to capacitive effect, each transition has a finite charge or discharge time. If the transition happens such that the next transition occurs before the previous transition reaches the designated level, deviation of both time and level occurs for the current bit. Such an effect can be cascaded. The ISI effect is shown in Figure 1.7.

#### 1.2.3.4 Crosstalk

Two types of crosstalk are discussed here. One is associated with copper cables, and the other is associated with optical fibers.

##### 1.2.3.4.1 Copper-Based Crosstalk

Crosstalk is basically an interference phenomenon. Crosstalk is generally involved in a parallel channel system in which signals are propagated concurrently and affect each other. For copper-based communication channels, crosstalk is caused by electromagnetic coupling. For integrated circuits (ICs) where the geometry and space between connects is relatively small, the capacitive coupling is the dominant mechanism.11, 12 When a signal transition happens in one channel, some of its energy leaks to the neighboring or adjacent channel through charge flow due to capacitive coupling, causing the signal level in that channel to fluctuate. For board-level circuits where the geometry is relatively large, inductive and capacitive coupling are both important. Inductive coupling follows Lentz's Law, in which changing the magnetic field flux generates an electrical field, and that electrical field, coupled with electrical charge, causes voltage fluctuation. In general, the effect of crosstalk can be modeled primarily as the voltage fluctuation or noise. However, it can affect the timing jitter directly as well. When two transmission lines are coupled capacitively, and when digital transitions occur simultaneously on two lines from the same end (the near end), the slew rate of the signals at the other end (the far end) is larger if the two transitions at the near end are in phase (have the same polarity) or is smaller if the two transitions are out of phase (have the opposite polarity). Figure 1.9 shows the capacitive and inductive coupling mechanisms for crosstalk. Figure 1.9 Schematic drawing of crosstalk caused by capacitive and inductive coupling. The crosstalk due to the simultaneous steps' response with opposite polarities slows down the slew rate of the step signals at the far end.

From the definitions of mutual capacitive and inductive constants Cm and Lm, the voltage noises due to capacitive and inductive coupling can be calculated according to the following equation:

#### Equation 1.5 where Zv is the impendence of the impacted or victim line and dVd/dt is the time derivative of the driving voltage. For inductive-induced voltage noise, we have

#### Equation 1.6 where Zd is the impendence of the driving line, and dId/dt and dVd/dt are driver current and voltage time derivatives or change rate, respectively.

You can see that crosstalk is proportional to the voltage or current slew rate. As the date rate or frequency keeps increasing, the rise time of the digital signal becomes smaller. Therefore, the slew rate and crosstalk-induced noise increase. As mentioned in previous sections, timing jitter due to crosstalk can be estimated through division of appropriated far-end signal slew rate.

##### 1.2.3.4.2 Fiber-Based Crosstalk

Crosstalk can also happen in optical fiber-based communication systems, particularly in multiple-channel systems such as wavelength division multiplexing (WDM) systems.13 In a WDM or dense WDM (DWDM) system, crosstalk can happen through linear and/or nonlinear effects. Linear effects often refer to the leaking of photon energy from neighboring channels that have different wavelengths to the concerned channel in the optical filters or demultiplexers, causing the amplitude noise fluctuation. Nonlinear effects include the following:

• Stimulated Raman Scattering (SRS), in which short-wavelength channels can amplify long-wavelength channels over a wide wavelength range
• Stimulated Brillouin Scattering (SBS), in which short-wavelength channels can amplify long-wavelength channels over a narrow wavelength range
• Four-wave mixing (FWM), in which a new wave or signal, or the fourth wave, is generated when three wavelengths from three WDM channels satisfy a certain relationship

Like copper-based crosstalk, fiber-based crosstalk causes amplitude noise for the transmitting signal and subsequently causes timing jitter through the slew rate conversion, in turn degrading system performance.