Faraday Consultancy Limited
Faraday Consultancy Limited (FCL) is a private limited company registered in England and Wales, Company Registration Number 2938426, and is owned and managed by Chris Angove.

This area will be familiar with surfing enthusiasts, the beach at Newquay on the north coast of Cornwall. The house and island are privately owned but may be available for holiday rental. Newquay also has a station, a 10 minute walk from here, and you can get a train there from Paddington, usually changing at Par, taking about 5 hours.

High Speed Digital Hardware

The following sections describe some of the areas of digital and communications hardware in which FCL has built up significant skills, experience and knowledge. Click on the topic of interest for more information.

Digital Hardware and Communications

Analog to Digital Converters (ADCs) and Digital to Analog Converters (DACs)

Digital Cellular Communications

Mixed Signal PCB Design

High Speed Digital Interfaces

Digital Modulation

Shannon-Hartley Capacity Theorem

Differential (Balanced) and Single Ended (Unbalanced) Transmission Lines

Digital Hardware and Communications

Today, multi-gigabit per second (Gbit/s) digital signaling speeds are deployed even in many of the most basic consumer products. The digital information transmission is normally at 'baseband'. As its name implies, the baseband frequency spectrum extends from at or near zero (DC) to some upper limit, influenced by (1):

Uncoded data comprises strings of bits, conventionally called 1s and 0s, which have not yet been coded. The probability of an uncoded 1 occurring is identical to that of an uncoded 0 occurring so is exactly 0.5. Streams of uncoded data are typically represented by text strings comprising 0s and 1s, such as '01101001' or '11001010. Although those examples have equal numbers of 1s and 0s, that is not a requirement. Equally valid are '11111111' and '00000000' because the probability of a given bit occurring is independent of the state of any earlier bit or bits. The bit states must therefore be truly random, so to generate a stream of uncoded bits we need an accurately random bit generator, a surprisingly difficult challenge. We usually have to use a pseudo random bit sequence (PRBS) generator instead. A PRBS generator per se is actually a form of coding itself but, by definition, it is considered a close approximation to uncoded (random) data. An example of a simple PRBS generator using a linear shift register with exclusive OR (XOR) feedback is shown in Figure 1-1.

Figure 1-1 Generation of a simple PRBS using a clocked 4-bit linear shift register tapped at positions 2 and 3 with feedback using one exclusive OR (XOR) gate.

The taps at positions 2 and 3 are carefully chosen to allow the maximum possible number of pseudo-random bits before repetition. These are generated using the shift register of order (or length) k = 4 and the correctly chosen XOR feedback function. The PRBS block length output before repetition 'N' is given by N = 2k - 1. The recursive equations for this configuration are shown in (1-1), where the symbol '⊕' is for the operator 'exclusive OR'.

Q 0 (n+1)= Q 2 (n) Q 3 (n) Q 1 (n+1)= Q 0 (n) Q 2 (n+1)= Q 1 (n) Q 3 (n+1)= Q 2 (n)
(1-1)The recursive equations for the linear feedback shift register with exclusive OR (XOR) feedback shown in TBA

This may be easily simulated using an application with math logic functions such as Excel® or Matlab®. The condition of all zeros (0000) is omitted, but all other possible 4 bit groups will be generated, their order determined by the design of the XOR feedback. To describe the PRBS, the order is appended to the letters 'PRBS' so this case is PRBS4 with a repeated block length (N) of 15 BITS. The omission of the 0000 word makes, overall, the probability of a 1 very slightly greater than 0.5, so the probability of a 0 is reduced accordingly. That is because the XOR output for inputs of 0 only is also 0, so if this condition could occur, it would 'lock up'.

You might very reasonably ask the question "If there is nothing stopping the '0000' condition occurring, why doesn't it lock up?". Good question, I don't know but it may be something to do with additive white Gaussian noise (AWGN), similarly to the question "What makes an oscillator start oscillating?". The clock input is external so the 'all zeros' condition has to occur at exactly the correct (wrong) time. For example '10000000' would be a risky time should a noise spike occur. I will reserve this for another blog elsewhere.

With faster data signaling speeds, in most real world PRBS circuits we need rather larger 'N' values. Unfortunately, determining the XOR tapping points for these is not trivial. An 8-bit example with tapping points at 1, 2, 3 and 7 is shown in Figure 1-2. In this case the unrepeated block length (N) is 255.

Figure 1-2 Generation of a PRBS using a 8 bit shift register and XOR feedback with taps at 1, 2, 3 and 7.

Although this form of PRBS is not perfectly random, it is predictable. The ends of a real digital channel could be thousands of miles apart, so the bit error rate (BER) send and receive equipment can be set up for the same PRBS test pattern independently. The receive end will therefore know exactly what was sent from the transmit end and its signaling speed. It will receive a 'noisy' version of the transmitted data at the receive end, but slightly later than it was sent due to the propagation delay along the line. Therefore, once it has synchronised, it can compare the actual received bits with what would have been received without any errors and hence determine the BER. Clearly there must be sufficient error-free bits to enable synchronisation in the first place. The choice of PRBS must be appropriate to both the data signaling rate and the approximate BER expected.

Over the years, numerous types of coding have been developed and found to be highly effective for many different applications. For example, in error control including radio paths such as digital cellular, errors in the form of 'bursts' utilise targeted coding. Many reasons for using coding are described by Sklar (1) and Wikipedia:

One reason that we require coding is because, when designing a digital channel, we often require to couple together digital transmission lines using (coupling) capacitors. Often there are (DC) bias voltages present on ICs which could cause problems if the DC was allowed to couple through the line to adjacent components so the coupling capacitors block the DC. Also there are many occasions that transmission lines have to feed media through paths that are non-galvanic and do not (directly) carry DC such as fiber optic cables and wireless paths.

Streams of high-speed coded data propagate across physical interfaces which are hopefully well designed transmission lines. Today, differential (or balanced) transmission lines are more popular for high speed data than single ended types, some of which are still adequate for lower speeds (3). That is because the noise issues related to electromagnetic compatibility (EMC) are very challenging and good balanced transmission line design with accurate differential excitation reduces any common mode (CM) components significantly. This reduces both the susceptibility and emissions aspects of EMC performance.

The large bandwidths necessary mean that the electrical transmission properties of these are critical to achieving good signal integrity, which ultimately means: reliable, low-error performance. Important PCB and layout properties which must be sufficiently controlled in order to achieve this include: controlled impedance architecture, connector mismatches, discontinuities and differential line lengths which cause differential delays, often called 'skew'. Electrical properties include: compliance voltages, common mode rejection, component self resonant frequencies (SRFs), Q factors, loss tangent, skin depth effects, screening and filtering.

We know from Fourier transforms applied to streams of digital data that, noise-free, uncoded digital signals occupy an infinite bandwidth (4). That is not possible of course, so we normally have to settle for the maximum possible available bandwidth that the regulatory authorities will allow us and the minimum occupied bandwidth that will enable adequate transmission of the data channel we have in mind.

To set up a radio link we require to modulate a carrier in some way using the baseband, before radiation. Bandwidth is scarce, expensive and possibly not even available and noise in one or more of its many forms is always present. Additive white Gaussian noise (AWGN) is the systemic type of finite noise which influences the integrity of digital data (5) (6). We know from Nyquist that this always causes transmission errors to some degree: the laws of physics do not allow error-free transmission of data (2). These errors, normally expressed in the form of the bit error rate (BER) parameter will always exist, and BER performance figures for digital communications equipment can range across many orders of magnitude. The spectra of high speed data can extend to very high frequencies, perhaps well into the microwave region. We have to design the associated transmission lines to the same standards we would use for microwave transmission. Many of the performance parameters and the equipment required to measure them are identical to those used for microwaves.

Analog to Digital Converters (ADCs) and Digital to Analog Converters (DACs)

Although we live in a digital world, most natural physical phenomena can generally be described as analog. For example, radio wave propagation and any form of modulation, even if it is described as a 'digital'; may actually be analog. For example, one form of digital modulation is quadrature amplitude modulation (QAM) which relies on creating changes in both the amplitude and phase of the carrier being modulated. One type of QAM is 256QAM which is designed to distinguish amongst 256 different states of amplitude and phase of the carrier, each with a unique 'symbol'. I suppose you could argue that this was digital on the basis that each of those 256 amplitude/phase combinations was discrete but it would certainly not be binary (except for BPSK or 2QAM which does have 2 discrete states). Where do you stop? Electrons have discrete charges and photons have discrete energies.

The modulated carrier may eventually be propagated by an electromagnetic wave from an antenna. It may even be carried by a cable, for example to meet one of the 'DOCSIS' specifications. Even with the propagation path (wave or cable) being analog, with today's digital technology, we will probably need to process digital signals at each end of the communications channel and possibly at intermediate points for functions such as coding, filtering and compression. Processing can of course be analog or digital but today certainly most of it is digital. In many of these types of channel there needs to be at least one digital to analog converter (DAC) at the transmit end and one analog to digital converter (ADC) at the receive end. These are just two examples of applications of ADCs and DACs. There are many others.

Early ADCs were designed to simply convert an analog input voltage into a digital form using a successive approximation register (SAR). The best conversion times were lengthy by today's standards but SARs are still used today. They are much faster than they used to be and are typically used in accurate but slow responding devices such as digital multimeters and temperature sensors. More recent ADCs are sampling types, found in many different types of communication equipment. In the design of these, sampling theory and the work of Nyquist, Shannon and Hartley feature prominently (7). There are many other ADC architectures including: time interleaved, pipeline, sigma delta, flash/parallel and photonic sampling. The Analog Devices® website for example has excellent resources, datasheets and application notes which describe these.

For digital to analog converters (DACs), we need to carefully select a suitable sampling frequency and make allowances for the shaping of the analog spectrum by an inverse sinc function (7). The quality of the output analog waveform is influenced by the 'order of hold' that is used: what happens to the analog output between samples.

Over the years, as digital signal processing (DSP) overheads and speeds have increased, ADC signal input frequencies have moved from near DC, through audio and are now well into the radio frequency (RF) spectrum. In the typical receiver architecture, it has now become possible to digitally perform many of the functions that were previously analog. The configuration of digital hardware is usually programmable by software, often a form of hardware description language used to generate firmware, specific instructions for configuring the hardware. Such digital implementations have the significant advantage that their configuration may be updated and modified in code, provided that this does not exceed the limitations of the hardware and of course the Nyquist sampling criteria (7). Most types of digitally programmable hardware may be used to operate both across a range of frequencies (digital downconverter) and to demodulate or modulate a variety of modulation types, both digital and analog (DSP).

Software defined radio (SDR) refers to a flexible software-based technique which may be applied to re-configurable radio hardware in order to set up a specific radio architecture. The SDR configuration is defined by how the associated SDR software was used to program the hardware. For example, the SDR software may define the frequency that the target system will operate at, the type of modulation used and various other radio-like parameters. It is often written in a high level programming language like C or C++ and compiled before programming the hardware. Applications like MATLAB® allow the generation of SDR code.

The dynamic range of a SDR radio is improving but is not yet considered equivalent to a good quality superheterodyne radio designed for a similar frequency. The superheterodyne architecture will not however have the re-programmable flexibility of the SDR version. Currently we are at an intermediate stage with SDR. ADC sampling speeds are still insufficient to allow direct (baseband) sampling for some of the higher operating frequencies, such as those used in digital cellular and satellite communications. So it is often necessary to use some analog circuits at the front end of a SDR receiver to convert the frequency band received by the antenna into one that is more manageable by the ADC. Usually a low noise amplifier (LNA), some filtering and an analog mixer are used to change the incoming frequency band to a lower one more compatible with the current technology. The actual sampling may then be performed at the converted baseband, zero-IF, or across a band of frequencies (bandpass or undersampling).

Typical ADC parameters include resolution, integrated non-linearity (INL), differential non-linearity (DNL), kTB (AWGN) and quantisation noise, effective number of bits (ENOB), missing codes, aperture jitter, dithering and multiple carry issues. These have naturally led onto several digital signal processing (DSP) phenomena such as spectral leakage, windowing and the implementation of forward and inverse fast Fourier transforms.

Digital Cellular Communications

The original (analog) form of cellular communications, going back to the 1980s, has become known as 1G, although we may not have appreciated that at the time. The first digital cellular technology came along in the early 1990s which we called '2G' and is now becoming obsolete. This was the Global System for Mobile (GSM) base station equipment and some derivations for different frequencies, but essentially the same technology. The frequency channelisation was frequency division multiplex (FDM) using time division multiple access (TDMA). The 900 MHz bands were rolled out initially followed by the higher frequencies around 1800 MHz and 2000 MHz which. Around the World various other frequency bands also allocated with names such as 'DCS 1800'.

In those days, the relatively fast programming and lock times required for frequency synthesizers were achieved using fractional 'N' (FN) synthesizer ICs. The FN device was an alternative to the other popular type of synthesizer IC at the time, the dual modulus (DM) type. FN ICs had the advantage that the step frequency can actually be a fraction of the phase detector frequency. This is unlike the DM type IC, where the step frequency must be an integer multiple of the phase detector frequency. The FN device is useful for two reasons: it allows a relatively wide bandwidth phase locked loop (PLL) to be used giving a fast lock time and it enables a common issue, the phase detector frequency modulation spuri, to be more easily suppressed.

The next ('third generation' or 3G) digital cellular equipment used a completely different technology from 2G, spread spectrum code division multiple access (CDMA). This is also known as the Universal Mobile Telephone System (UMTS). In this case, frequency bands of approximately 5 MHz are allocated to a fixed limit of several users simultaneously. The digital baseband is 'spread' meaning that a cleverly designed coding is applied to the digital stream before modulation. This is block based and adds additional bit overheads to the 'payload' data within the same time slot, therefore increasing the occupied spectrum of the original data, causing the spreading. This is applied to all users across the same frequency band but, by careful control of power levels and coding, interference between users is minimised sufficiently to enable reliable communications. The bandwidths are such that data can actually be extracted from signal spectra that are 'below the noise'. However, the noise in this case is from other users of the channel as opposed to additive white Gaussian (AWGN) noise which would apply in the analog case. The noise is therefore correlated, compared to AWGN which is uncorrelated.

The fourth generation of digital cellular communications (4G), also originally known as Long Term Evolution (LTE) started its roll-out in the UK around 2013 and currently has substantial coverage of the advanced version of LTE (LTE-A), the specification for which was released in 2011.

LTE and LTE-A use yet another modulation and access technology compared to their predecessors: coded orthogonal frequency division multiplex (COFDM) for the downlink and single carrier frequency division multiple access (SCFDMA) for the uplink. COFDM allows efficient bandwidth occupancy and SCFDMA has good peak to average power performance, useful for battery powered devices.

LTE-A specified several improvements over LTE including multiple antenna technologies and higher data rates, especially for mobile and nomadic stations. Nomadic stations are those in which the user antenna is fixed, usually to a high point on a building, instead of being mobile as with the earlier generations. Therefore nomadic stations can often serve users in areas with poor or no broadband cable (DOCSIS or ADSL) services. The fixed antennas are usually in better locations than typical hand-held mobile terminals, have better antennas and do not suffer the rapid and deep fades in propagation caused mostly by multipath fading. Therefore the uplink and downlink data signalling speeds can usually be increased compared to the 'mobile equivalent'. Increased data capacity and therefore bandwidth is achieved using carrier aggregation which enables the effective channel capacity to be increased by allowing the combination of contiguous or discontiguous frequency bands, even if they are in different service frequency bands.

RF over optical fiber (ROF) is an interesting technology with one application used to extend digital cellular services into tunnels, buildings or other areas which would otherwise receive poor or no coverage. In this very successful application of ROF, whole uplink and downlink digital cellular frequency bands may be physically moved to a new location such as a tunnel where 'fill-in' coverage is required. In those areas, smaller and usually more directional antennas can be installed to communicate with the local users' terminals. This is much simpler and cheaper than creating and additional service cell, typically used to extend coverage in an open area. Typical optical wavelengths used for the uplink and downlink paths are 1310 nm in one direction and 1550 nm in the other direction. With suitable optical 'add-drop' multiplexing both of these wavelengths must be carried simultaneously in both directions to provide the 'full duplex' operation necessary in digital cellular communications.

A simpler version of ROF may be used to extend the coverage of broadcast frequency bands into tunnels. The principle is exactly the same as described for digital cellular but of course it operates in 'downlink only'. Also some careful design is necessary in order to avoid possible RF leakage from the tunnel ports into the local service area.

Mixed Signal PCB Design

PCB designers normally need to put as much functionality as possible into the often limited PCB board space available to them. The function areas might include: power supplies, clocks and oscillators, SPI, LVDS, CML, PCIe, Ethernet, USB, Data Storage and Audio/DAC/ADC. If this is electrically and physically possible, a well designed multi-function and probably multi-layer PCB will reduce the cost of manufacture of the equipment significantly compared to having a hybrid architecture with separate boards for each of the necessary functions. A good example is the typical personal computer motherboard. Much of the necessary functionality which was previously on expansion cards, is now included on the motherboard itself.

There are stringent PCB electromagnetic compatibility (EMC) standards to meet so features of the PCB design including compartmentalisation, screening, filtering, track routing, controlled impedances, and transmission lines require targeted design. Other design challenges include loss tangent, skin depth, grounding, differential and common mode impedances, coupling and isolation.

Today's PCB designs typically include several sandwiched conductor (track and plane) layers using various technologies. These make connections where necessary through the layers to pads and holes on the surfaces to match the footprints of the components to be fitted.

Without doubt, the most common dielectric material used to electrically separate the layers is the type which was originally known as 'fire resistant grade 4' or now simply 'FR4'. After many years of evolution various manufacturers now produce a range of grades of FR4 which are claimed to perform better at higher frequencies and faster data signalling rates, and are priced accordingly. Although FR4 has been used for many years, it is often still quite adequate on many PCBs with relatively short tracks, even allowing for the rate at which data signalling speeds, and therefore frequency ranges, have increased.

There are many PCB design tool applications available ranging from free and open source ones to very expensive and sophisticated types aimed at the more complex designs. Example PCB applications include Altium Designer 16.1®, Cadence Allegro 16.6® and Zuken Cadstar 2019. The CAD part of most PCB designs are broadly achieved in two stages: schematic design followed by PCB layout and routing. A third stage might be some form of simulation. Simulation add-ons for PCB CAD applications are, however, usually expensive additional investments. This has to be balanced against their value: providing an early warning of design and layout errors and inadequacies at an early stage before committing to PCB manufacture.

High Speed Digital Interfaces

High speed interfaces may be PCB based or connectorised in copper or optical fiber.

Using copper conductors, they would normally be at baseband with a frequency spectrum extending from at or near zero (DC) to some upper limit determined by factors including coding, baseband filtering (baseband shaping) and the necessary channel capacity or signalling rate. In some cases a copper transmission line might carry, not baseband, but a spectrum which is the result of some form of modulation, typically quadrature amplitude modulation (QAM). That is not really high speed digital per se but a modulated spectrum occupying a band of frequencies usually with some RF processing such as filtering, amplification and frequency conversion, with a final destination typically at an antenna. There are a few exceptions however. For example DOCSIS, where the modulated spectrum is connected a coaxial cable.

High speed data carried by optical fiber made from high purity glass is not at baseband but across a band of optical frequencies. This is another type of modulation but applied to an optical wavelength instead of a radio frequency. (We tend to refer to wavelengths in the optical spectrum and simply frequencies in the RF spectrum). Typical fiber optic wavelengths are around 1550 nm, 1310 nm and 850 nm. 1550 nm and 1310 nm are used for single-mode or monomode optical fibers and 850 nm is used with multi-mode fibers. Single-mode fibers have a small core diameter of approximately 5 µm allowing very low levels of dispersion due to fewer reflections compared to multi-mode fibers which have a core diameter of approximately 50 µm. In general, single-mode fibers can carry much greater bandwidths of data over greater distances than multi-mode fibers so they are more suited to telecommunications infrastructure. However, single-mode optical components are much more expensive due to the increased precision required in their manufacture.

The optical spectrum differs from the visible spectrum. They were effectively the same in the early days of optical telescopes when the only means of detecting optical wavelengths was with the human eye. Visible wavelengths range from approximately 380 nm to 750 nm, so in fact those typical optical wavelengths used in optical fibers are strictly in the infrared spectrum. However, a range of techniques, originally developed for the visible spectrum, are still effective at longer wavelengths beyond the visible spectrum so they are still referred to as 'optical'. At even longer wavelengths the optical properties of fiber optic cables will breakdown and we have to use other technologies.

Let us consider a few wavelengths and frequencies to demonstrate the sort of bandwidths that are available. Using Recommendation ITU-T G.694 (8), for all frequency and wavelength calculations we must use an accurate value for the speed of electromagnetic radiation in free space (c) of precisely 2.99792458 × 108 m/s. The convention is to always specify the frequencies and wavelengths in free space. Optical fiber is a circular dielectric waveguide so the actual speed within a fiber is always slightly less than c, even for single-mode.

The optical wavelengths of 1550 nm and 1310 nm in free space are frequencies of 193.414 THz and 228.849 THz respectively. Therefore, we have about 35 THz of optical bandwidth available between these two wavelengths with just one fiber. Because optical fibers, especially single-mode, provide huge bandwidths they are especially attractive for telecommunications infrastructure. But that is just one of their applications. Some of the other properties of optical fibers include the following:

Optical fiber technology now allows many optical carriers to be multiplexed onto one fiber, using wavelength division multiplexing (WDM). Although it is theoretically possible to carry huge volumes of data over optical fibers, this must be done with caution as it makes them highly vulnerable to single point failures.

Several forms of WDM have been adopted from Recommendation ITU-T G.694 (8). The reference frequency is taken as 193.100 THz, a free space wavelength of exactly 1552.52 nm. Then the wavelengths are channelised with 100 GHz, 50 GHz or 12.5 GHz spacing. These are broadly called 'coarse', 'normal' and 'dense' WDM respectively. For smaller capacities and shorter terrestrial distances, coarse WDM might be a cost effective solution. For an international submarine optical cable covering thousands of kilometers and costing hundreds of millions of dollars, the highest possible capacity is normally required from the cable, even though it may not be utilised initially. There could be as many as 160 discrete wavelength channels, each separated by 12.5 GHz and each having a capacity of 10 Gbit/s.

In copper transmission lines used in PCB architecture, high speed techniques include low voltage differential signalling (LVDS), current mode logic (CML), positive emitter coupled logic (PECL) and peripheral component interconnect express (PCIe). High and moderate speed transmission standards include Ethernet (Gigabit), SATA, RS-485, SPI, I2C and even RS-232.

Digital Modulation

The common adjectives used to describe the process of modulation, analog and digital, refer respectively to either the continuous or discrete changes necessary to the carrier parameters, to carry information. The parameters are amplitude, phase and frequency and all of these are affected by the modulation. Even with amplitude modulation (AM), which does not explicitly modulate frequency or phase, the result of the modulation is to create a finite bandwdith spectrum from the unmodulated (zero-bandwidth) carrier. The greater the bandwidth of the modulated carrier the greater is its information capacity. Also, the relationship between temporal frequency (f), angular frequency (ω), phase (φ) and time (t) (6-1) means that whenever the phase is changed, the frequency is changed and vice versa.

ω=2πf= dϕ dt
(6-1) Temporal frequency (f) and angular frequency (ω) are the rate of change of phase (φ) with respect to time (t).

In digital communications, probably the most common form of digital modulation is quadrature amplitude modulation (QAM) in which both the amplitude and the phase of the carrier are allocated discrete values according to a lookup table of 'symbols'. If k is the number of bits per symbol, the total number of available symbols N is given by N = 2k. The shorthand way of describing this modulation is N-QAM. For example, 2 bits per symbol (k = 2) means that the symbol set would be of size 4 so this would be represented as 4-QAM. This is also known as quadrature phase shift keying (QPSK) since, in this case, the carrier amplitude is equal for all possible symbols.

This form of modulation is digital because there are 'N' discrete QAM values available. A common way of describing these uses 'in-phase, in-quadrature' (IQ) constellation diagrams displayed in two dimensions in which the 'I' and 'Q' axes are the horizontal and vertical axes respectively. These are equivalent to rectangular argand diagrams which represent vectors in magnitude and phase comprising a real component (I) and an imaginary component (Q). Typically the I and Q units are in volts (V) and the angle units are in degrees or radians. Examples of these for 4-QAM (QPSK) and 16-QAM are shown in Figure 6-1 (a) and (b) respectively.

Figure 6-1 IQ diagrams for 4-QAM (or QPSK) (a) and 16-QAM (b)

In each case a dot is located at the position of the end of the carrier phasor to represent one of the available symbols, in polar (magnitude and phase) or rectangular (real and imaginary) form. Therefore there are 4 dots for 4-QAM in (a) and 16 dots for 16-QAM in (b). The symbol dots or positions are shown adjacent to the binary numbers they represent after Gray coding has been applied. Gray coding minimises the number of bit transitions for adjacent symbols. Diagram (b) shows examples of two phasors, OA and OB which represent the symbols or binary groups, with Gray coding, of 1110 and 0111 respectively. For example, if the 16-QAM constellation covers the range -1.0 V to +1.0 V for both I and Q, then OA represents 1/3 + j1 V and OB represents -1/3 + j/3 V.

The only binary (2-state) order of QAM is 2-QAM, more commonly known as binary phase shift keying (BPSK). The greater the order of the QAM, the more sensitive it is to noise because the spacing between adjacent states is normally less than it would be for lower order QAM such as BPSK. The most important form of noise affecting QAM is Additive White Gaussian Noise (AWGN) or 'kTB' noise. This is a function of the absolute temperature and noise bandwidth and includes noise components in both amplitude and phase.

High speed sampling oscilloscopes and bit error rate (BER) test equipment often include modes for displaying constellation diagrams, similar to those shown in Figure 6-1. For each symbol received, a dot is displayed at the measured carrier amplitude and phase on the constellation diagram. An undistorted symbol would appear at precisely the correct position, such as A or B. In practice however, some deviations are usually seen and the same symbol sent repeatedly would form a cluster of dots, hopefully near the correct symbol location. The performance parameter, error vector magnitude (EVM) is a measure of how close the measured symbols are to the intended position. Other quality parameters include: inter-symbol interference (ISI), phase and amplitude imbalance, bit error rate (BER) and Eb/N0. Eb/N0 is the energy per bit (joules, J) divided by the AWGN power density (watts per hertz, W/Hz).

Shannon-Hartley Capacity Theorem

Claude Shannon is considered by many as 'a father of Information Theory'. In 1948, Shannon's article in the Bell System Technical Journal, 'A Mathematical Theory of Communication' (10) is one of his classic works. Shannon's original article can certainly be quite challenging to understand, but plenty of information has been written on the subject over the last 70 years or so to help simplify it. Much of the discussion here is from books by Pierce (10) around 1980 and, more recently, Sklar (9).

Perhaps the most famous equation in information theory comes from the Shannon-Hartley Capacity Theorem (7-1).

C=B log 2 1+ S N
(7-1) The maximum theoretical capacity available from a band limited, digitally encoded communications channel with additive white Gaussian noise (AWGN)

Note that the logarithm is to base 2 because here we are considering binary (two-state) signalling. The following list explains the symbols used in (7-1)

This equation may be extended to derive the Shannon Limit.

One fascinating comment on the Shannon-Hartley Capacity Theorem is that it is independent of any coding or error correction methods that might be applied. With a digital channel there are usually several options for correcting errors and reducing the bit error rate (BER), even by simply repeating the same message. However, all of these add extra overheads and therefore reduce the net (payload) capacity bit rate of the channel. The Shannon Limit is the value of energy per bit (Eb) divided by noise power density (N0), or Eb/N0, below which no data communication is possible, whatever the provision of error correction. Some quick checks on Wikipedia will confirm the SI base units for: energy (kg⋅m2⋅s-2), power (kg⋅m2⋅s-3) and frequency (s-1). Therefore Eb/N0 is a unit-less quantity which, converted to logarithmic form, is approximately -1.6 dB.

In the last 70 years or so, the Shannon Limit has served as a benchmark against which to compare the performances of competing digital communication systems. They should of course be compared also with the achievable payload information rate and BER.

A practical digital communications channel has the following properties:

These will be discussed in the following sections.

The Information is Coded in Discrete Form

Information carried in binary form has two states commonly referred to as zero (0) and one (1). The transitions between these states will be fast changes and will therefore extend the spectrum that is created. There are many ways in which the binary states may be represented as voltage-time characteristics but those are not central to this discussion about channel capacity. It is often appropriate for strings of binary digits (bits) to be grouped into blocks for processing. By definition, the blocks are no longer bits, so they are often called 'symbols'. A symbol contains and integer number of bits. To carry the same information therefore, the bit signalling rate may be greater than the symbol signalling rate.

At Least One Form of Coding Will be Used

As we saw from Section 1, the only truly uncoded form of binary data has the following properties:

However, uncoded data would not be very useful because it would not contain any information. Once the data is processed in some way it becomes coded and then contains information. The process of coding creates bit overheads which reduce the capacity of the raw data, or the data payload, if the bit signalling rate is not changed. Further 'layers' of coding may be applied to the data stream on its route to the receiver, but on each occasion there is similarly an additional 'overhead' and therefore the speed at which the payload data may be carried.

The Channel Transmission Response Will Be Bandwidth-Limited

The path through which the electrical signal and the noise spectra progress on their route from the transmitter to the receiver, may be considered a cascade of two port networks. These will have many different functions, for example amplifiers, filters, frequency translation, modulation and demodulation. Some functions will be extensive, for example if there are stages including wireless or optical fiber propagation. In a well designed digital channel, ultimately these do not matter because the digital and analog processing will allow for degradations such as multi-path fading and cable attenuation. However, these will add both processing delay and propagation delay to the signal.

Figure 7-1 shows a typical plot of transmission magnitude against frequency for a communications channel which is the net effect of all of the cascaded components in the path from the transmitter to the receiver. Note that this does not include delay and the vertical scaling is power density with respect to frequency (W/Hz). This is appropriate for finite bandwidth spectra, in this case those of the AWGN and the signal spectrum.

Figure 7-1 The transmission frequency response of a communications channel and the equivalent noise bandwidth, B

Also, it is assumed that the bandwidth of the response is a small fraction of the center frequency so, since B << f0, therefore the response is approximately symmetric.

Figure 7-1 also shows the equivalent noise bandwidth of the channel (B hertz). This is the bandwidth of the theoretical (rectangular shaped) passband that would transmit the same AWGN power as the actual channel response bandwidth, often colloquially called a 'brick wall' response.

If G(f) is the function of frequency which describes the actual transmission response and N0 is the peak power density of the response at f0, then (7-2) is the relationship between the integrated AWGN (total noise power over the response) which is equivalent to the product of B and N0.

f=0 f= G(f)df=B N 0
(7-2) The integrated noise power over the actual transmission response is equivalent to the area of the equivalent noise bandwidth

The following two criteria must be met:

Electrical Noise Will Be Present.

Assuming that the communications hardware is at a constant temperature, additive white Gaussian noise (AWGN) produces a constant noise power density (PN/B) across a frequency range up to about 100 GHz, given by (7-3).

P N =kTBW P N B =kTW/Hz
(7-3) Additive white Gaussian noise (AWGN, PN) is a function of the product of Boltzmann's constant (k = 1.381×10-23 J/K), absolute temperature (T, kelvin) and noise bandwidth (B, hertz)

Although there will be some variation of the temperatures of the transmission equipment, to a first approximation assuming a temperature of T = 290 K and substituting for k into (7-1), results in a (linear) noise power density of 4 × 10-21&; W/Hz. In logarithmic units that is equivalent to -174 dBm/Hz.

AWGN is present in any hardware which is operating at a temperature above absolute zero. It cannot be separated from the signal so it is not possible to have a signal without AWGN. Therefore, if we try to measure the signal power we actually measure the signal power plus the AWGN noise power (S + N). We can however measure the AWGN power alone (N).

Figure 7-1 shows the spectral power densities for a typical signal only (S) in (a) and AWGN only (N) in (b), both across identical noise bandwidths B. Both plots are in linear units (W/Hz) and not to scale. Figure 7-2 shows the result when these are added, S + N.

Figure 7-2 Power density spectra for a signal spectrum with no noise (a) and AWGN only (b), in linear values but not to scale.
Figure 7-3 Signal plus noise power density spectrum for the spectra in Figure 7-2 added together.

Errors Will Always be Present Due to Electrical Noise

For an information channel subject to additive white Gaussian noise (AWGN), the ratio Eb/N0 describes the ratio of the energy per bit (Eb) in joules (J) to the AWGN noise power density (N0) in watts per hertz (W/Hz). This is a measure of the quality of signal plus noise combination, like signal to noise ratio (SNR). This theorem provides a means of calculating the maximum possible theoretical capacity of a given channel provided sufficient (noise) bandwidth and signal power are available. For a given Eb/N0, the result is a value in bit/s/Hz. That is a measure of channel capacity against the available bandwidth. For short range communications, for example over a local line-of-sight path, or even a cable path, a relatively high Eb/N0 value would be possible which would allow a high capacity channel to be set up, typically using a fairly high order quadrature amplitude modulation (QAM). However, as Eb/N0 reduces at greater distances, the capacity is reduced and lower order QAM is required to tolerate the higher levels of AWGN. Ultimately the order may be reduced to the minimum of 2QAM or BPSK to allow some information flow, albeit at a rather slow rate.

The Nyquist Sampling Criterion Must be Applied

The Nyquist Sampling Theorem (7) contributes a factor of 2 to the Shannon-Hartley Capacity Theorem. When a sampling component such as an analog to digital converter (ADC) samples a voltage against time waveform, the sample spacing has to be chosen appropriately. For example, to resolve a complex pulse shape of width td with 'fast edges', the sample spacing may require to be one tenth of this (td/10) or even less. However, with a signal spectrum, we need to consider the highest sinusoidal frequency component, that with the shortest period, say T. If we know that the voltage time function is sinusoidal, some fairly basic trigonometry will tell us that we need to know a minimum of the positions of two points on the waveform, in instantaneous voltage and time, in order to determine its amplitude and frequency. For a spectrum the sample spacing would be chosen on the basis of the highest frequency component expected in the spectrum.

Differential (Balanced) and Single Ended (Unbalanced) Transmission Lines

In the early days of RF and wireless engineering, the coaxial cable transmission line became popular because it provided a grounded screen surrounding a 'live' center conductor. The screen was often referred to as a type of 'Faraday cage' architecture and was intended to screen external interference from penetrating the cable and coupling to the center conductor. The same theory is true in reverse: the screen would block potential interference radiating from a similar cable. We know that the effectiveness of the screen depends on several factors including the type of foil and/or braiding, the metal used, its thickness and the measurement frequency. We also assume that the whole of the screen is at the same (ground) potential for the full length of the cable. As data signalling speeds and frequencies increase however, and we have to meet more stringent EMC requirements, we find that these imperfect screens and grounding are inadequate and a better form of transmission line is required.

With just about any transmission line, external radiated interference is more heavily coupled in the common mode than the differential mode. That is because the spacing of the conductors normal to the axis of the line is usually much smaller than the axial length of cable exposed to the interference. By using the signal to excite the cable in as close to perfect differential mode as possible, the common mode interference should be heavily reduced.

Unfortunately, the requirement for differential transmission lines does complicate PCB design. We may still need some form of screening. How close should the individual lines be? How long should they be and of what dimensions? How much can we tolerate them being of different lengths?

References

  1. Sklar, Bernard
  2. Sklar (op. cit.)
  3. Hall, Stephen H. and Heck, Howard L
  4. Sklar (op. cit.)
  5. Sklar (op. cit.)
  6. Pozar, David M.
  7. Sklar (op. cit.)
  8. International Telecommunications Union; Recommendation ITU-T G.694.1 (10/2020)
  9. Sklar (op. cit.)
  10. Shannon, C. E.
  11. Pierce, John R.