The time it takes a signal to travel from one location to another on a network?

Vernier Mechanism

There are two main components to the uncertainty of signal propagation time. The first is a static component that is due to differences in process technologies, process variations, electrical loading of an unknown number of memory modules, etc. The second is a dynamic component that is typically due to temperature or voltage fluctuations. Some fluctuations change too fast to be handled effectively, but most of the fluctuations in voltage and temperature change over a relatively long time period. Most voltage fluctuations are associated with the 60-Hz AC power cycle, which is a relatively long time-span, and most temperature fluctuations occur on the order of milliseconds or longer [Lee 2002, Macri 2002]. In many DDR systems, the static component is corrected by using a Vernier mechanism in the memory controller at system initialization. It is essentially a trial-and-error method in which the upper and lower limits of timing failure are found and the midpoint is used for signal transmission (analogous to setting the tracking on one's VCR by going forward and backward to the points of noise and then leaving the tracking set somewhere in between) [Lee 2002]. The dynamic component is then corrected by the on-chip DLL.

An alternative to the on-chip DLL is to perform recalibration of the Vernier correction often enough to account for the slower changes in voltage and temperature, as opposed to performing the timing correction only once at system initialization. Assuming that the most significant voltage fluctuations are associated with the 60-Hz AC power supply and that most of the temperature fluctuations occur on the order of milliseconds or longer [Lee 2002, Macri 2002], recalibration would be needed roughly once or twice every millisecond.

Go Back N

A protocol that is more efficient than ABP for long propagation times is Go Back N. The network designer or user selects a window size N. Typically, N is just large enough so that the pipe is full: the sender gets the acknowledgment of the first packet when it finishes transmitting packet number N. The sender numbers the packets sequentially modulo N + 1, that is, 1, 2, …, N, 1, 2, …, N, ….

The receiver acknowledges every correct packet that it receives with an acknowledgment that it numbers as the largest numbered packet that it has received so far. For instance, if the receiver gets the packets 1, 2, 4, 5, then it sends the acknowledgments with numbers 1, 2, 2, 2. Before transmitting packet number n + N (modulo N + 1), the sender waits until it receives the acknowledgment of packet number n. Whenever the sender sends a packet, it starts a timer with a timeout T selected by the network designer. When the sender fails to receive the acknowledgment of a packet before the packet times out, the sender retransmits that packet and all the packets that it has transmitted since it last transmitted that packet. Therefore, the receiver need not store any packet since it eventually gets them correctly in sequence. The sender must be able to store up to N packets. Note that with this protocol the sender may have to retransmit packets that were correctly received by the receiver. Figure 2.21 illustrates the sequence of events in Go Back N with N = 4.

In Figure 2.22 we illustrate the calculation of the efficiency of Go Back N where there is no transmission error. The efficiency is given by

Efficiency=min{NTRANSTRANS+ACK+2PROP,1}.

The efficiency increases linearly with N up to 100%. To appreciate the magnitudes that may be involved in practice, we consider a numerical example. Suppose the packet size is 2,000 bits, the ACK is 80 bits, transmission is over a 30-km fiber link at 155 Mbps. Then

Efficiency=min{NTRANSTRANS+ACK+2PROP,1}=min{N×20002000+80+46,500,1}.

So the efficiency is 4.2% when N = 1, and 100% for N ≥ 24. The network engineer should propose a window size equal to 24 in this example. Note that Go Back N with N = 1 is exactly ABP. Transmission errors reduce the efficiency of Go Back N. The analysis of the efficiency when errors occur is more complicated.

Carry–Look–Ahead Adder

In the carry–look–ahead adder (CLA), the carry propagation time is reduced to O(log2(Wd)) by using a treelike circuit to compute the carry rapidly [5,18,22,44], The area requirement is O(Wd log2(Wd)). The CLA algorithm was first introduced by Weinberger and Smith [44], and several variants have since been developed. Brent and Kung [5] have derived an adder with an addition time and area proportional to 2 log2(Wd) and 2Wd log2(Wd), respectively.

The CLA exploits the fact that the carry generated by a bit-position depends on the three inputs to that position. If xi = yi = 1, a carry is generated independently of the carry from the previous bit-position and if xi = yi = 0, no carry is generated. If xi ≠ yi, a carry is generated if and only if the previous bit-position generates a carry. It is possible to compute all the carries with only two gate delays (although this would require gates with excessive fan-in). The fan-in of logic gates increases linearly with the number of bits in the adder. The high fan-in forces the CLA to be partitioned into blocks with carry–look–ahead. The block size is usually selected in the range three to five.

A 1992 comparison among adders [8] shows that the CLA adder, which also is amenable to pipelining, is one of the fastest adders. The implementation of a CLA adder, using dynamic CMOS logic, was reported in 1991 [41].

II.C.2 Single-Mode Fibers

The pulse dispersion discussed so far is caused by the various rays in the fiber having different propagation times—it is therefore called multipath, multimode, or intermode dispersion. It is by far the strongest dispersion mechanism in a multimode fiber. An obvious remedy is to ensure that only rays having the same propagation time are allowed to exist in the fiber. In a multimode fiber this is impossible because even if the input light could all be channelled into a single ray (mode) it would, in a short distance, be scattered into all possible guided rays by bends, microbends, and other inhomogeneities in the fiber. We have seen that in a well-designed graded-index multimode fiber the dispersion can be reduced to about 1% of that in a step-index multimode fiber, but nevertheless, it remains the factor limiting the bandwidth. The only solution, a very successful one, is to ensure that only a single ray or mode is allowed to propagate. This is done very simply by reducing the core diameter to a value approaching the wavelength of the propagating light. The actual value depends on the refractive indices of the core and cladding and the wavelength. The critical diameter dc for a step-index fiber is given by

(12a)dc=2.4 λπn1(2Δ)1/2

As an example, if n1 = 1.48 (silica), Δ = 1%, and λ = 1.55 μm, then dc = 5.7 μm.

For a core diameter d dc, more than one mode (ray) can propagate and the fiber becomes a multimode fiber, but for d ≤ dc only a single mode is possible (although this mode can have two orthogonal polarizations).

Equation (12a) can also be used to define the cut-off wavelength of a fiber having a diameter d, namely

(12b)λc=(π2.4) dn1(2Δ)1/2

For wavelengths λ λc only a single mode can propagate, while for wavelengths progressively less than λc an increasing number of modes will exist.

Another widely used parameter is the normalized frequency (or normalized waveguide width) V, defined as

(12c)V=π(dλ )n1(2Δ)1/2

It can be deduced from Eqs. (12a and b) that the cut-off condition may be expressed simply as Vc = 2.4. For V < 2.4 (d<dc;λ λc) only one mode can propagate; all others are “cut off.”

So far propagation has been described in terms of simple rays at various angles. For many purposes this description is quite adequate, especially for multimode fibers. However, it is not possible for rays to propagate at any angle even though in a multimode fiber the number of possible angles can be large, with each angle corresponding to an equivalent mode. The number of modes in a step-index fiber can be estimated quite simply, by V2 / 2, where V ⪢ 1. For d = 50 μm, n1 = 1.48, λ = 1.5 μm, and Δ = 1%, the number of propagating modes is 240.

The condition for the existence of a mode is that there should be a stable energy/field distribution in the core with no radial energy propagation into the cladding. The conditions for this can be shown as follows. The propagation constant of a plane wave in a vacuum is k0 = 2π/λ and in the core of a fiber is n1k0. For a ray at an angle θ to the axis, the propagation constant β in the axial direction is β = n1k0cos θ and the radial component γ of the propagation constant is γ = n1k0 sin θ. The radial component is reflected near the core/cladding boundary so that when the total phase change after two successive reflections at the upper and lower boundaries becomes 2mπ, where m is any integer, a standing wave is established in the radial (transverse) direction, and there is a stable field distribution; that is, a mode exists. Near the core/cladding boundary the electric field becomes zero, whereas in the central region of the core there are one or more maxima. The total phase change Φ of the radial component after two complete reflections is approximately

(13)Φ=4ay−2φ=4an1k0sinθ−2φ

and must equal 2mπ. The phase change ϕ occurs at each reflection and depends on the angle of the propagating ray. It should be noted that m denotes the number of zeros in the intensity distribution of the mode in the transverse direction and is known as the mode number. The lowest-order mode has m = 0 and is the only mode that propagates when V < 2.4 in a single-mode fiber. Figure 5^ shows the ray angles and the electric field distributions for the three lowest-order modes of a fiber, corresponding to m = 0, 1, 2. It is very important to note that a portion of the mode field (energy) distribution penetrates into the cladding so that the losses of this part of the cladding must be as low as that of the core. There is no radial loss propagation of energy into the cladding, providing it has zero loss. In a multimode fiber the degree of penetration of the field into the cladding is less than 1 μm but in a single-mode fiber it can be many micrometers.

Multimode dispersion cannot exist in a single-mode fiber, but two other mechanisms, material dispersion and waveguide dispersion, now come into play in limiting the bandwidth. Material dispersion arises because the refractive index of glass changes with wavelength, and waveguide dispersion is caused by the fact that the waveguide (i.e., the fiber) causes the propagation constant β = n1k0 cos θ = (2π n1/λ) cos θ to change with wavelength. Any signal, such as a single pulse, consists of a spread of wavelengths (frequencies) about the central, or carrier, frequency; the shorter the pulse, the wider the spread of wavelengths. In a vacuum or free space the refractive index is a constant independent of wavelength, and the pulse propagates without distortion. On the other hand, in a dispersive medium such as a fiber waveguide the various wavelength components travel at different velocities, and the pulse is broadened. When bits of information are transmitted in the form of a train of pulses, the spreading of the pulses causes them to overlap one another, and the information is lost. For maximum transmission distance and bandwidth (rate of information transmission) the effects of dispersion must be minimized.

The material dispersion coefficient is defined as

(14)M=−(λc)( d2n1dλ2)

and represents the delay time per unit wavelength spread for unit transmission distance. When d2n1/dλ2 is positive the longer wavelength components of a pulse Δλ travel faster than the shorter ones, and vice versa. The material dispersion of silica glass is shown in Fig. 6^ as a function of wavelength and represents the amount by which a short pulse of 1 nm spectral width is spread by material dispersion when traveling over 1 km of fiber. For a single-mode fiber the material dispersion of both the core and the cladding must be taken into account, depending on the proportion of energy which flows in each. Figure 6 shows that the material dispersion becomes zero at a wavelength near 1.3 μm, leaving waveguide dispersion as the major bandwidth-limiting factor in a single-mode fiber. At a wavelength of 0.85 μm, when the linewidth of the source is small, the bandwidth is limited, by material dispersion alone, to 21 GHz over 10 km transmission distance. In a practical fiber the small additions of other oxides to raise/lower the refractive indices of the core/cladding change the wavelength of zero material dispersion, but not significantly.

The expression for waveguide dispersion is more complicated, but for a normalized frequency between 2.0 and 2.4, which is the case in practice, the spread of a short input pulse in time is approximately

(15)ΔT= KL/cn1ΔVβ/f

where K is of the order 0.1 to 0.2, L is the propagation distance, and f is the optical frequency (c/λ).

The crucial factor is that at wavelengths shorter than 1.3 μm, the material and waveguide dispersions are of the same sign and are additive, but at longer wavelengths they are of opposite signs and tend to cancel each other. In fact, Eq. (15) shows that Δ T depends on Δ and V which can both be varied to some degree in the fiber design to such an extent that the sum of material and waveguide dispersions can be made zero at wavelengths from 1.3 to 1.55 μm and beyond, simply by control of the core diameter. The available bandwidth of single-mode silica fibers is therefore absolutely enormous. TransAtlantic and TransPacific cables having a bandwidth of 30 Gb/sec (600,000 telephone circuits) are in operation, but this is nowhere near the ultimate capacity of fibers.

Abstract

The most common formulation of the Fermat principle states: the light propagates in such a way that the propagation time is minimal. This chapter outlines the historical background as well as the different formulations of the Fermat principle.

The Fermat principle is valid for any traveling wave, not only for light. In optics, the underlying equation is the wave equation, which describes all details of propagation. Geometrical optics is a limiting case when wavelength tends to zero. The opposite limiting case—when wavelength tends to infinity—is the field of geometrical wave theory. That theory is based on the Fermat principle as well as on the dual concepts of rays and fronts.

Chemical waves can be described in detail by reaction–diffusion equations. These equations may have traveling-wave solutions. The essential features of the evolution of chemical wave fronts can also be derived from the geometric theory of waves.

Some recent formulations of the Fermat principle require only stationarity of the extremals instead of the stricter requirement of minimal propagation time. This problem is discussed, and it is concluded that for chemical waves only the requirement of minimum is relevant: maximum and ‘inflexion-type’ local stationarity does not play any role. The chemical waves and their prairie-fire picture underlines: only the quickest rays have effects. Nevertheless, there are singularities, when there are several paths of rays with the same propagation time. For these cases the related fields of singularities and caustics are relevant.

Special attention is paid to aplanatic surfaces, where all the reflected or refracted paths require the same time, that is stationarity holds globally. Nonaplanatic refraction is discussed, too: the special example of refracting sphere is treated analytically.

Finally, the shape of a chemical lens is derived: this ‘chemical lens’ is able to perfect image formation with chemical waves, that is circular fronts will be also circular after refraction.

Time of Arrival Approach

The time of arrival approach is centered on the ability to time-tag a transmitted signal and measure the exact ToA of that signal at a receiver. The propagation time, at the speed of light assuming LOS propagation, is a direct measure of the propagation distance. This provides a receiver with an iso-range sphere for a given transmitted and received signal. If multiple receivers at known locations receive the same signal, generally at different times, the multiple iso-range spheres intersect at the transmitter's location. It requires four receivers to geolocate one transmitter in three dimensions. A reverse problem may be constructed in which four transmitters provide their location in a time-tagged transmitted signal and the receiver can geolocate itself. More complex situations for which only relative positions can be determined can be constructed. This problem is useful in sensor fields and also for large numbers of cognitive radios. A two-dimensional (2D) depiction of this process is shown in Figure 8.1.

4.17 Propagation time

Suppose that a rectangular voltage pulse is applied to the input of a logic inverter, as shown in Figure 4.26. For any practical logic gate there will be a time delay or propagation time between the change in the input voltage to the corresponding change in the output voltage, and this delay is denoted by tPHL when the output voltage changes from a high to a low level. When the output voltage changes from a low to a high level, the propagation delay time is denoted by tPLH. These two propagation delays may, in principle, have different values. Although described here only in terms of a simple logic inverter, all logic components show propagation time effects to varying degrees, and in the case of complex components there may be differing values of tPHL and tPLH according to which inputs and which outputs are being considered.

The timing diagrams of Figure 4.26(b) are somewhat idealised since they imply that all the voltage transitions take place instantaneously. In practice, the input and output voltages will not change instantaneously, and the propagation times tPHL and tPLH are therefore usually defined as the time delays between the voltages halfway between the steady voltage levels achieved, sometimes called the ‘50% points’, as shown in Figure 4.26(c).

The propagation delays specified by manufacturers usually fall into three categories: minimum, typical and maximum. This is because there is a manufacturing spread for these parameters. In effect, the manufacturer is stating that the maximum delay will never be exceeded, and the wise logic designer will ensure that the design operates correctly if the gates used only meet the maximum quoted values (i.e. ‘worst case design’).

For the 74TTL logic family, typical values of propagation delay lie in the range 2 to 33 ns depending upon the particular type of technology being employed (the most common of which are currently LS or ALS, and high-speed CMOS gates (HCT) that are designed to be compatible and interchangeable with TTL gates). Reduction in the propagation delay using bipolar technology can be achieved by employing emitter coupled logic (ECL) where propagation delays as low as 1 ns can be achieved. However, CMOS circuits are widely used in a great number of present system designs. They have the advantages of cheapness, low power consumption per gate and considerably higher packing densities (the number of gates manufactured per chip).

Electronic engineers are also interested in the rise and fall times of the voltage waveforms. The rise time is defined as the time taken for the voltage to change from 10% to 90% of its final value, while the fall time is defined as the time taken to change from 90% to 10% of its initial value. This parameter is also frequently referred to as the transition time.

Charts

Charts are best used when presenting numerical information, or system metrics. Metrics can help shed light on the health and status of the network and supporting equipment before, during, and after an incident. This is a place where a witness can really get into the weeds with lots of deeply technical information during testimony, but it will be important for this type of case. The jury may not fully understand the information being presented, but if the chart is done right it can show peaks and valleys and what normal resource utilization versus abnormal resource utilization looks like. The main types of metrics for the purposes of router and switch technology are technology and network-level metrics.

Monitoring and controlling individual network elements such as routers and switches is dependent on performance monitoring through existing management information bases (MIBs) and network probing of measurements to gather metrics. A few examples of relevant metrics include:

Propagation time The time required for a signal or wave to travel from one point of a transmission medium to another.

Transmission time Also considered the propagation delay, or the amount of time it takes one bit to go from the start of the link to its destination.

Effective bandwidth The actual bandwidth of a particular device (e.g., a router or switch) or the total effective network bandwidth.

Buffer size Also referred to as the cache, normally refers to the size of the data store where the memory elements are stored.

Queue size of a router interface Usually represented in megabytes, the number of packets that can be held before being processed.

Congestion level Similar to a traffic light. When the congestion level exceeds a configured value (maximum number of packets), the sending rate of all packets is reduced to a minimum committed information rate (minCIR). As soon as the rate drops back below the queue size the system increases it back to the maximum queue size allowed, or the committed information rate (CIR).

Route processing delay The time it takes to put the bits on the wire. Regardless of the queue size, this never changes.

You can find more information regarding MIBs in RFC 1212—Concise MIB definitions, at www.faqs.org/rfcs/rfc1212.html.

13.5 Photonic Implementation of Quantum Relay

In this section we first describe the implementation of a Bell-state preparation circuit in integrated optics, required in quantum teleportation systems, which is shown in Fig. 13.17. Among many possible versions of Hadamard and CNOT gates we have chosen two with similar propagation times. The upper circuit operates as a Hadamard gate, while the rest of the circuit operates as a CNOT gate as explained in the previous section. It can be shown that output quantum state |βij〉 is related to the input state |ψin〉 by:

(13.134)|βij〉=12[10000 ][1010010110−10010−1]|ψin〉=12[cHtH+cVtVcHtV+cVtV cVtH−cVtVcHt H−cVtH].

For example, by setting cH = tH = 1 and cV = tV = 0 we obtain the Bell state |β00〉=[1001 ]T/2=(|00〉+|11〉)/2. In Fig. 13.18 we describe how to implement the quantum relay based on the Bell-state preparation circuit (shown in Fig. 13.2B), Hadamard, controlled-X, and controlled-Z gates described earlier. We employ the principle of differed measurement and perform corresponding measurements only in the last intermediate node. The measurement circuits in Fig. 13.3 represent the APDs, which are used to detect the presence of cV photons in corresponding control qubits. The detection of cV photons triggers the application of required control voltages on phase trimmers to perform controlled-X and controlled-Z operation at the destination node.

Calculating the Timing Parameters

Setting the timing parameters of the nodes are very important for the reliable operation of the bus. Given the microcontroller clock frequency and the required CAN bus bit rate, we need to calculate the values of the following timing parameters:

Baud rate prescaler value,

Prop_Seg value,

Phase_Seg1 value,

Phase_Seg2 value,

SJW value.

Correct timing requires that

Prop_Seg + Phase_Seg1 ≥ Phase_Seg2,

Phase_Seg2 ≥ SJW.

An example is given below to illustrate how the various timing parameters can be calculated.

Example 7.2

Assuming that the microcontroller oscillator clock rate is 20 MHz, and the required CAN bit rate is 125 kHz, calculate the timing parameters.