HFC-network simulation and analysis redefined

Almost nobody reads clunky guides or manuals, so COPERNICAT does not have one. Instead all information is online on this page. Because working with COPERNICAT is quite intuitive, the intention of this online guide is not to cover every detail or button but only to discuss in reasonable detail the important subjects that you might have a question on. That should be enough to get you started and keep you going.

The best approach for learning to work with COPERNICAT is as follows:

- read the
and**'Miscellaneous'**sections below**'Graphical User Interface'** - start making small projects, explore COPERNICAT and push all buttons
- come back to this page if you have questions

**Note**: This guide contains several references to publications (mainly in the '*Simulation theory*' section). If you find the PDF of such a publication using your favorite search engine then you're done, but if the publication's PDF originates from a paid source (like IEEE Explore) then I am not allowed to make a copy available due to copyright restrictions.

Miscellaneous

How does the licensing system work?

COPERNICAT supports trial licenses and paid licenses:

- When you install COPERNICAT without a license, it will act as a 30-day trial version. The licensing system allows for only one trial every 6 months on each PC, so plan your trial carefully because a thorough evaluation of COPERNICAT may take a few days.
- Datafiles generated with one trial version cannot be processed by another trial version. This means that these datafiles become unusable after the 30-day trial, unless you buy a license.
- When you buy a license, you receive a USB-dongle which controls the license authorization for COPERNICAT. A licensed version of COPERNICAT can read, edit and process all datafiles: from any trial version, and from any other licensed version.

What accuracy can I expect?

It is impossible to give an absolute value in dB, the best we can do is to give an understanding of the aspects that play a role. First of all the specifications of signals and network components you use in COPERNICAT, as well as the built-in theoretical models, only approach the real world. This means that simulations can be used to design a network, but should always be verified by lab measurements before actually implementing the simulation results in a real network. That may be different for relatively small changes you make in an existing network: the changes predicted by COPERNICAT can also be expected in the real network. Some rules of thumb:

- Optical and electrical signal levels as well as electrical (thermal) noise are always spot on.
- Nonlinearities are specified by measurements (CSO/CTB or CINR-curves) which give an estimate of the amplifier's typical behavior. The simulation is based on theoretical models that approach the real behavior.
- Optical distortions are based on theoretical models which may have simplifications in order to be practically useful.
- COPERNICAT uses rules of thumb for the summation of distortions from different origins (electrical or optical) or from different network components.
- For capacity estimates of signals with a digital load, the relation between throughput and CNR is very well known. COPERNICAT must however use rules of thumb when nonlinearities are involved.
- Uncertainties in the simulation results generally increase with the complexity of the simulated network.

What should I do when COPERNICAT crashes?

Despite extensive testing and code-checking it is possible that an unhandled error occurs and COPERNICAT crashes. When that happens, an error-file is generated and you are asked to email that file to COPERNICAT (info@copernicat.net) so we can trace back the problem and debug the source-code. This error-file contains the complete project you were working on because the specific data in your project like signals and/or network components may have led to the error. Please note that your data is safe with COPERNICAT: we will never have a detailed look at the specific simulation you were working on. The error-file is only used as a means to reproduce the error and to solve the problem in the source-code of COPERNICAT. When that is done, the error-file is deleted immediately.

Graphical User Interface

Project tab

This tab covers the standard project options: creating/opening/closing/saving projects, specify a description, etc. The '*Guide'* button refers to this online guide.

Project settings tab

The project settings tab mainly specifies parameters which control the simulation, for example:

- required CNR and CIR for legacy analog signals like analog television and FM
- relative level and required CNR and CIR for sound-carriers of analog television
- effect on legacy analog signals of non-linear distortions that originate from signals with a digital payload
- distortions to be included in the simulations
- etc.

For analog legacy signals like broadcast FM and analog television, the required CNR and CIR values can be entered. Several sound systems for analog television are supported and their level, relative to the ATV carrier, can also be specified. The A2 Stereo or '*Zweikanalton*' consists of two separate mono FM carriers and is treated as such when calculating CNR and CIR. Stereo sound systems like BTSC (USA) and EIAJ (Japan) have similarities with the PSD of broadcast stereo FM and therefore follow the same calculation scheme for CNR and CIR. The following holds for calculation of CNR and CIR:

- The CNR of ATV is calculated in a bandwidth which depends on the used CCIR- and color-system (the bandwidth of PAL-B/G is much higher than the NTSC-M bandwidth).
- The CIR of ATV is calculated using protection curves which depend on the used CCIR- and color-system.
- The CNR of FM-signals, both mono and stereo including variants for ATV sound systems, is calculated in a bandwidth of 200 kHz.
- The CIR of FM-signals is calculated using protection curves according to ITU-R BS.412-9 or NVN-5175 (a Dutch variation).

Intermodulation (IM) due to nonlinearities and originating from broadband signals like DOCSIS have an impact on analog legacy signals quite simuilar to noise. You can therefore choose to incorporate this IM in the calculation of the CNR instead of calculation of the CIR.

The impact of IM on analog legacy signals strongly depends on the frequency offset of that IM relative to the victim signal carrier. In order to obtain an unambiguous CIR value, protection curves are used which weight the IM depending on their offset frequency before the IM-level is used for calculation of the CIR. Only out of curiosity one should disable these protection curves.

For analog FM you can choose for protection curves according to the Dutch NVN-5175 calculation standard, or the ITU-R BS.412-9 protection curves. Details about the difference between these curves are described in the '*Intermodulation and legacy analog signals*' section of the '*Simulation theory*' chapter below. Note that the NVN-5175 method consist of multiple protection curves for different types of nonlinearities. Because these different types of nonlinearities must be saved throughout the network simulation, choosing the NVN-5175 method may have a significant impact on memory usage.

ATV-signals are specified by their top-sync level which is almost 5 dB higher than the rms signal power. Because the top-sync levels of different IM-components in a cluster on the same frequency are not synchronized but all have their own random offset in time, the composite rms power of the IM-cluster is lower than predicted from the top-sync levels only. Depending on the statistics of ATV-signals in the raster and the statistics of the IM-cluster itself, a reduction of the predicted IM-cluster level may be applied. For more theoretical background see the '*Intermodulation and legacy analog signals*' section of the '*Simulation theory*' chapter below.

Many distortions play a role in an HFC-network. By enabling//disabling individual distortion contributions one can get a better feeling which type of distortion may play a dominant role in the calculation of CNR and CIR.

Optical WDM distortions have a random and therefore unpredictable component, please see the '*Optical distortions*' section of the '*Simulation theory*' chapter below for a note on this subject. There are two options how to deal with this random component:

- '
*average distortion*': this option calculates the average WDM distortion level on the HFC-network - '
*worst-case distortion*': this option calculates the worst-case WDM distortion level which may ever be present on the HFC-network

Protection curves for FM-signals, used when calculating their CIR, have quite a narrow '*top*' of only a few tenths of kHz around the signal's center frequency. This means that for sufficient accuracy the simulation resolution should be in the order of 5 kHz. However, such a resolution can slow down the simulation significantly and you can therefore choose a compromise between simulation speed and the accuracy of the calculated CIR for FM-signals. If no FM-signals are present, a simulation resolution of 50 kHz is used.

Application settings tab

Only three settings reside here for your convinience:

- the dimension of the signal levels (dBµV, dBmV or dBm)
- the fontsize in the GUI
- the plotting colors of the various simulated spectra

Signal raster tab

On this tab you can specify all signals present in your simulation in terms of frequency, signal level, etc. Signal types include:

- (Euro)DOCSIS 1.0 - 3.0
- DOCSIS 3.1
- legacy analog television (PAL/NTSC/SECAM, B/D/G/K/M/N/etc., various sound systems)
- legacy analog FM (mono/stereo)
- etc.

A subraster number must be assigned to each individual signal. This creates different groups of signals, each identified by their own subraster number. In the network design signals are injected by signal sources, and each signal source can select one or multiple subrasters. This makes the subraster numbering a simple but powerful feature which gives you control on how, and where, groups of signals are injected into the network.

Adding and deleting signals is done using the ribbon buttons. Clicking the 'Add' button adds a single new signal to the raster, while clicking the 'Delete' button deletes all selected signals in the table. The 'Undelete' button provides a list of previous delete-actions which can individually be selected and reverted.

With the 'Import' button one can import a complete raster from a file which was previously saved using the 'Export' button. The 'Import' button gives various options how to process the loaded raster, for example if it must be added to or overwrite the current raster. It also gives several options how to deal with the subraster numbers of each signal.

The level of each signal can be edited independently from all other signals. However, in some cases during the design or evaluation process it can be very efficient to select a group of signals in the table and be able to give all selected signals the same absolute level ('Set' button) or to change the level of all selected signals at once with the same amount of dBs ('Change' button).

For various reasons the specified bandwidth for DOCSIS 3.1 must be the active bandwidth, meaning that guard bands are excluded. So for the largest supported downstream OFDM block of 192 MHz, one must specify 190 MHz when the default minimum guard band of 1 MHz on each side is used.

Network tab

Creating and editing a network is quite simple, the following is important to know:

: Click on the network component from the toolbox on the left side of the screen, drag the component to the design area and release it.**adding network components**: Hover the mouse cursor in the neighbourhood of a network component's connection and another mouse cursor appears. Then click, move the mouse cursor to the intended endpoint of the connection and release. If the connection between two network components contains horizontal and vertical sections, then these sections must be drawn seperately.**making connections**: First click on the network component in order to select it (it becomes gray), then click and drag the network component to its new position. Existing connections will be lost, and must be deleted and created again.**moving a network component**: Double-click on the center of the network component (for a coaxial cable or optical fiber the center is at the bottom of the 'circle') which opens the edit tab containing all specifications for that network component.**edit a network component**: Click on the network component in order to select it (it becomes gray), then press the 'Delete' button (keyboard or GUI ribbon).**delete a network component**

Network components can have several colors:

: The network component is not correctly connected.**red**: The network component is selected and can be deleted or moved.**gray**: The default color for a correctly connected network component in the electrical domain.**lightbrown**: The default color for a correctly connected network component in the optical domain.**lightblue**

With the 'Import' button one can import a complete network from a file which was previously saved using the 'Export' button. The imported network overwrites the current network.

Several additional graphical features are available like saving the network as a drawing to file, zooming in and out, panning, selecting multiple network components and connections, etc. Their behavior is as you are already used to from other graphical design software.

With the 'Delete' button one can delete the selected group of network components, while the 'Undelete' button provides a list of previous delete-actions which can individually be selected and reverted.

When the mouse cursor is hovered (don't click...) over the location where an output leaves the network component body, a signal preview is shown if the network has no errors. There are two different previews:

: This preview shows a plot of the total cumulative gain for the active signals from the signal source to the current output, and it shows a plot of the active signal spectrum.**electrical domain preview**: The optical domain preview shows a text overview of the optical signals active at the current output (optical power, wavelength and optical transmitter which generates the signal).**optical domain preview**

Simulate tab

Several specifications of network components also have an additional 'tolerance' specification. Using these tolerances, COPERNICAT can execute 3 different simulations which give you an insight in the performance sensitivity of all signals for small changes in the network:

: This simulation uses the nominal specifications without applying tolerances.**nominal simulation**: This simulation applies the specified tolerances in such way that signal levels become lower, and as a result CNR decreases and CIR increases.**noise sensitivity simulation**: This simulation applies the specified tolerances in such way that signal levels become higher, and as a result CNR increases and CIR decreases.**IM sensitivity simulation**

After choosing the simulation types to be executed, the simulation(s) can be started using the 'Start' button. The 'Stop' button stops the running simulation(s) as soon as possible.

After simulations finished, several choices can be made in order to display simulation results:

**tolerance**: Selects one of 'nominal simulation', 'noise sensitivity simulation' or 'IM sensitivity simulation'.**output**: Selects the network component for which the simulation results are shown.**result**: This selects the type of simulation result ('spectra', 'signal performance' or 'spectral efficiency'), which is discussed further below.

**spectra**: This shows several PSD plots for signals, intermodulation due to nonlinearities, optical distortion, noise, etc.**signal performance**: A table is generated which shows frequency, level, CNR, CIR and CINR of all signals. For signals with a digital payload also the expected data throughput in [Mbps] is calculated. For signals with a digital payload, DOCSIS 3.1 for example, the reported CNR margin is the margin to the first lower available value of QAM-complexity. That is: if 1024-QAM is reported for DOCSIS 3.1 with a margin of 2.6 dB, then the CNR can drop by 2.6 dB before one must switch to 512-QAM.**spectral efficiency**: For signals with a digital payload the expected spectral efficiency in [bps/Hz] can be plotted. With DOCSIS 3.1 the spectrum of the signal is divided into 1 MHz wide bands, and for each band the expected spectral efficiency is calculated and plotted in order to give an indication of the possible benefits of variable bitloading.

DOCSIS 3.1 offers so-called '*mixed modulation*', meaning that different sub-carriers in the OFDM block can be modulated with different QAM-complexities. This means that determining a single QAM-complexity and corresponding data throughput for the simulation results table based on the total channel CINR may be yield values which are far from what is actually possible. A single large interference signal can ruin the overall calculated CINR for the entire channel bandwidth, but in reality only a single OFDM sub-carrier may be affected with negligible loss of data throughput for the entire DOCSIS 3.1 channel. COPERNICAT therefore uses the following approach in order to come up with values which give a good indication of the DOCSIS 3.1 channel's possible performance:

- The DOCSIS 3.1 channel is divided into 1 MHz subbands (we do not use the much smaller carrier spacing because 1 MHz is a reasonable compromise between accuracy and CPU-load for the calculations).
- For each 1 MHz subband the spectral efficiency and data throughput is calculated.
- The total data throughput of the DOCSIS 3.1 channel is the sum of the data throughput of all 1 MHz subbands.
- The reported QAM-complexity is an average of the QAM-complexity of all 1 MHz subbands.

Specific elements of network components

Editing network components

By double-clicking on the center of a network component icon (for a coaxial cable or optical fiber the center is at the bottom of the 'circle'), the edit tab for that network component is opened. The specific elements and characteristics of all network components are described later on, this section only gives the function of the three buttons on the GUI ribbon which are available for all network components:

: This button enables you to overwrite the current specifications with a network component of the same type previously saved using the 'Export' button.**'Import' button**: Save the current network component to a file.**'Export' button**: Finish editing the network component and return to the network tab.**'Ready' button**

Tolerances

Several specifications also have an additional 'tolerance' specification. Using these tolerances, COPERNICAT can execute 3 different simulations which give you an insight in the performance sensitivity of all signals for small changes in the network:

: This simulation uses the nominal specifications without applying tolerances.**nominal simulation**: This simulation applies the specified tolerances in such way that signal levels become lower, and as a result CNR decreases and CIR increases.**noise sensitivity simulation**: This simulation applies the specified tolerances in such way that signal levels become higher, and as a result CNR increases and CIR decreases.**IM sensitivity simulation**

Editable tables

Editable tables are available for several specifications, in particular gain and noise figures which can be specified as function of frequency.

When the specification at hand supports to be edited as a table, a table with a few initial entries is provided in the GUI. If you move the mouse cursor to a certain cell of the table and click right, a context menu appears which gives you several options for the column you selected (add a column, delete or interpolate the current column, etc.).

Nonlinearities

The nonlinearities specification is an important and complex subcomponent present in several network components:

- electrical amplifier
- optical transmitter (specification concerns the optical domain, i.e. DFB-laser or Mach-Zehnder modulator curve)
- optical receiver (specification concerns the final electrical amplifier stage)

COPERNICAT supports 3 different specification types:

: This type of measurement dates from 'the old days' when analog television and FM were the dominant signals in CATV networks. It uses a measurement raster of carriers, and the network component therefore produces clusters of discrete nonlinear spectral components. For second-order intermodulation (IM) these clusters are called 'Composite Second Order' (CSO) and for third-order IM these clusters are called 'Composite Tripple Beat' (CTB). For all clusters the relative level in [dBc] to the nearest measurement carrier can be determined and the specification for single CSO/CTB equals the worst-case dBc value. For more control on how the simulation model behaves, one can define a frequency range in COPERNICAT for which the worst-case dBc value is determined. However, because only the worst-case value is specified, COPERNICAT can only create a frequency-independent model which may not accurately reflect the true behavior of the network component.**single CSO/CTB**: Because the 'single CSO/CTB' specification results in a frequency-independent model with limited accuracy, COPERNICAT also incorporates a specification where multiple CSO/CTB values including their frequency can be specified. This enables COPERNICAT to create a frequency-dependent simulation model which is more accurate. Manufacturers however never provide data for this type of specification, so you have to do the required measurements yourself.**multiple CSO/CTB**: With the introduction of signals with a digital payload like (Euro)DOCSIS, HFC architects would like to see a specification on datasheets which directly reflects the behavior of the network components when using this type of signals. CINR curves specifications give that insight using a measurement raster with broadband signals. When varying the output level of the measurement raster a CINR curve with certain maximum results: the left side of the curve is the safe working area where noise dominates, the right side of the curve is the unsafe area where intermodulation due to nonlinearities dominates and where the BER quickly increases.**CINR curves**

CINR curves were introduced in the IEC-60728-3 standard as a result of the work done by CENELEC TC209-WG7, and give a direct indication of the network component's performance when using a '*full digital load*' with, for example, (Euro)DOCSIS signals. Specifying these curves is quite simple in COPERNICAT: the GUI offers a list of maximally 6 CINR curves which are all disabled by default. When a new CINR curve must be specified one can simply select one of those disabled curves, enable it, specifiy the correct measurement frequency and modify the example values of the curve. As you modify the CINR curve, COPERNICAT shows you a graphical representation of your data (dots) and the result of an optimized model which will be used by COPERNICAT in simulations (solid line). The simulation model generator of COPERNICAT makes sure that the top of the CINR curve is dominated by third-order intermodulation. Only when signal levels are increased beyond the CINR curve maximum, higher order intermodulation may play a role.

In order to obtain a simulation model which correctly incorporates the frequency dependency of the network component, make sure you specifiy at least 3 or 4 CINR curves over the entire frequency range.

COPERNICAT offers two types of measurement rasters:

: There are several predefined measurement rasters for CSO/CTB specifications (NTSC and CENELEC), and for CINR curves (full digital load from a minimum frequency to a maximum frequency). For all rasters the absolute level can be specified, as a scalar value or a frequency-dependent specification.**predefined rasters**: One can also define a custom raster, assembled using the Raster tab of COPERNICAT and then saved to a file, which then can be loaded manually in the nonlinearities specification. COPERNICAT offers the possibility to modify the levels of the loaded custom measurement raster, not on an individual signal basis, but as a scalar modifier or a frequency-dependent specification.**custom rasters**

Nonlinearity specifications and the corresponding measurement raster levels are interpreted differently depending on the domain in which the nonlinearities are generated:

: This applies to the electrical amplifier and the output stage of the optical receiver. The measurement raster levels are interpreted as**electrical domain****output**levels of those network components.: The measurement raster levels are interpreted as**optical domain****input**levels of the optical transmitter which, using the additional specification which translates input levels to OMI, results in a certain effective OMI which is the primary variable for the optical nonlinear model.

Automatic Gain Control

Two network components have the option to enable an AGC circuit in order to keep the output level constant:

- electrical amplifier
- optical receiver

: This simple type of AGC keeps the level of a single reference signal constant at its nominal value by adjusting the gain of the network component with a frequency independent value.**single frequency reference**: This ideal type of AGC keeps the level of all signals at their nominal value (generally not possible to implement in practice).**full equalization**

Network components

Signal source

The signal source injects signals into the network. Each signal source can select one or more subrasters and their corresponding signals. The signal source icon has 6 small numbers at its output, representing each of 6 possible subrasters:

: The subraster does not exist.**red**: The subraster exists but is not selected.**yellow**: The subraster exists and is selected.**green**

Amplifier

The electrical amplifier consists of 4 parts:

- An input passive attenuation which can have various frequency dependent characteristics.
- A pre-amplifier which determines the amplifier's noise figure (together with the input attenuation).
- A mid-stage passive attenuation which can have various frequency dependent characteristics.
- A power output stage.

Attenuator

An attenuator can have a constant attenuation, or various frequency dependent characteristics.

Coaxial cable

The coaxial cable component allows to choose from several standard coaxial cable types, or a custom specification can be used with the following attenuation in dB: $$A_{dB}(f)=A_{dB}(f_{ref})\sqrt{\frac{f}{f_{ref}}}$$

Combiner

A combiner combines multiple input signals into a single output. The specification sheet of the combiner shows the specified attenuation by the user, as well as the minimum theoretically possible attenuation for the specified number of inputs when using a resistive combiner which terminates all ports with the correct impedance.

Splitter

A splitter splits a single input signal into multiple outputs. The specification sheet of the splitter shows the specified attenuation by the user, as well as the minimum theoretically possible attenuation based on distributing the input power over the specified number of outputs.

Termination

Each cascade of network components must be terminated by the '*termination*' component. Its noise figure can be used to model the in-house attenuation (coaxial cables) plus active equipment noise figure.

Optical transmitter

The optical transmitter can generate multiple optical signals which are combined into a single output using the internal multiplexer. All optical signals of the transmitter are generated using the device type as specified under the '*General*' tab. Two different modulators can be chosen:

- A Distributed Feedback (DFB) laser: this is a diode laser which can be modulated directly. Due to the modulation intensity the optical frequency also changes a little bit which is called '
*laser chirp*' The amount of laser chirp scales several optical distortions which relate to fiber dispersion. - A Mach-Zehnder modulator: this is an external modulator which must be fed by a stable laser source having a certain constant laser linewidth. The intrinsic modulator curve is entirely defined by the architecture of this type of modulator, and can therefore be chosen as the applicable model for IM-calculations.

Specifying optical carriers can be done using the '*Optical carriers*' tab. This tab gives you several additional menu icons on the GUI ribbon to add optical signals, delete them, or change the optical power of selected optical signals in the table:

The following may need a more detailed explanation on the '*Optical carriers*' tab:

- The optical transmitter accepts multiple electrical inputs which can be defined here. Each optical signal must be coupled to one of the electrical inputs using the field '
*Input number*'. - The polarization angle of the optical carriers is generally not controlled ('
*random*') in HFC-networks. This polarization angle affects the amount of optical distortions in WDM-systems. - The laser power uncertainty is used for the sensitivity simulations '
*nominal*', '*noise*' and '*IM*' (see the '*Simulations*' tab for more details).

Optical receiver

The optical receiver is a so-called '*non-coherent direct detection*' receiver which demodulates all optical carriers present at its input. If optical carriers with different modulating electrical spectra are present at its input, then these eletrical spectra are combined in the receiver.

Optical attenuator

An attenuator can have a constant attenuation, or wavelength dependent characteristics.

Optical amplifier

The optical amplifier, usually an '*Erbium Doped Fiber Amplifier*' (EDFA) in HFC-networks, can have a fixed gain or a built-in output level control which enables the amplifier to have a fixed output power.

Optical fiber

The default values for the optical fiber are characteristic and valid for the vast majority of optical fibers used in HFC-networks. Especially for specifications in the '*Raman gain*' tab and '*Other*' tab, these values should only be changed if other values are provided by the fiber manufacturer.

Optical multiplexer

The optical multiplexer can combine multiple optical signals. This may be required when multiple optical transmitter components are used in the network.

Optical demultiplexer

In WDM-systems an optical demultiplexer is required to select an optical signal for each optical receiver. The number of outputs of the demultiplexer can be specified, and if a channel grid type is chosen these outputs can only be editted if the entire network design is valid and at least one optical signal is defined.

Two different types of demultiplexers can be used in COPERNICAT:

- The '
*Frequency range*' or '*Wavelength range*' demultiplexer. - The '
*Channel grid*' demultiplexer.

This type of demultiplexer passes all optical signals which fall within the specified frequency or wavelength range. The isolation characteristics are as follows:

The channel grid demultiplexer selects a single optical signal for each output. COPERNICAT assumes that each optical signal is in the center of its output bandpass function, and uses the specified grid spacing to calculate the isolation values for disturbing channels. The isolation characteristics are as follows:

Simulation theory

Nonlinearities

Nonlinearities are distortions with a very different behavior than thermal AWGN (Additive White Gaussian Noise), and specifications are generally expressed as CSO/CTB values or CINR-curves for optical lasers or electrical amplifiers. This section sketches the basic theoretical background how COPERNICAT models these nonlinearities.

Let \(x(t)\) and \(y(t)\) respectively represent the input and output of a network component with nonlinear behavior. Their relation is given by a polynomial function: $$y(t)=\sum_{i=1}^\infty \alpha_i x^i(t)$$ If we assume that second-order and third-order nonlinearities dominate, we can simplify the above equation as: $$y(t)=\alpha_1x(t)+\alpha_2x^2(t)+\alpha_3x^3(t)$$ where \(\alpha_1\) represents the linear gain of the network component.

COPERNICAT does not simulate with the time representation of signals, instead it uses the signals's characteristic PSD \(|X(\omega)|^2\) in the frequency domain. This makes it, at this point, unclear how we can deal with nonlinearities. The first step to a solution is to realize that the inverse FFT of a signal's PSD is the autocorrelation function \(R_{xx}(\tau)\) which is expressed in the time domain, but independent of the time variable \(t\). Because \(R_{xx}(\tau)\) and \(x(t)\) are not the same quantities we're still not allowed to calculate the nonlinearities using the polynomial function above. However, under the assumption that the signal \(x(t)\) is sufficiently Gaussian, we can apply Price's theorem and conclude that the autocorrelation function \(R_{xx}(\tau)\) may be used as a substitute for the signal \(x(t)\) in a polynomial that describes the nonlinear behavior. This theory is covered in detail in the following article:

IEEE Transactions on Broadcasting, Vol.38, No.3, September 1992

The following notes can be made for the requirement that signal \(x(t)\) must be sufficiently Gaussian:

- Signals with a digital payload already have an amplitude distribution which is close to Gaussian.
- The Gaussian requirement holds for the sum of all signals in the network, and the Central Limit Theorem then comes to our aid. This theorem states, in short, that when many signals are summed together the total sum converges to a Gaussian distribution.

- The autocorrelation function \(R_{xx}(\tau)\) is calculated from the signal PSD \(|X(\omega)|^2\) using an inverse FFT.
- The resulting nonlinear distortion for order \(n\) is calculated as \(y_n(\tau)=R_{xx}^n(\tau)\).
- The corresponding PSD \(|Y_n(\omega)|^2\) is calculated from \(y_n(\tau)\) using an FFT.
- Finally \(|Y_n(\omega)|^2\) is scaled using frequency dependent weighting factors \(A_n(\omega)\), these weighting factors result from the nonlinear specifications using CSO/CTB values or CINR curves.

Calculation of the nonlinearities model

In COPERNICAT the nonlinearities model is used in simulations as shown in the illustration below.

Based on the signal raster on the output of, for example, an amplifier the following calculation steps are applied (described in detail in the previous '*Nonlinearities*' section):

- The time-domain autocorrelation function \(R_{xx}(\tau)\) is obtained using an IFFT.
- \(R_{xx}(\tau)\) is raised to the n-th power for n-th order nonlinearities.
- With an FFT the n-th order raw intermodulation PSD is obtained.

For CSO/CTB specifications, COPERNICAT internally sets up a simulation using the CSO/CTB measurement raster, but in the last step it uses weights which are all one. The output of this simulation is compared with the CSO/CTB specifications, and from there it is directly clear what the weight values must be in order to obtain an intermodulation PSD which exactly fulfills the CSO/CTB specifications. Calculating the nonlinearities model is thus a fairly easy step. For CSO/CTB specifications the following intermodulation orders are used:

- Second-order for CSO
- Third-order for CTB

48.25, 119.25, 175.25, 191.25, 207.25, 223.25, 231.25, 247.25, 263.25, 287.25, 311.25, 327.25, 343.25, 359.25, 375.25, 391.25, 407.25, 423.25, 439.25, 447.25, 463.25, 479.25, 495.25, 511.25, 527.25, 543.25, 567.25, 583.25, 599.25, 663.25, 679.25, 695.25, 711.25, 727.25, 743.25, 759.25, 775.25, 791.25, 807.25, 823.25, 839.25, 855.25The CENELEC rasters are then defined:

- CENELEC-19: first 19 frequencies
- CENELEC-29: first 29 frequencies
- CENELEC-41: all frequencies except 48.25 MHz
- CENELEC-42: all frequencies

55.25, 61.25, 67.25, 77.25, 83.25The NTSC rasters are then defined by the above low frequency block plus additional frequencies on a 6 MHz grid:

- NTSC-60: 55 additional frequencies starting from 121.25 MHz
- NTSC-77: 72 additional frequencies starting from 121.25 MHz
- NTSC-79: 74 additional frequencies starting from 109.25 MHz
- NTSC-96: 91 additional frequencies starting from 109.25 MHz
- NTSC-112: 107 additional frequencies starting from 109.25 MHz
- NTSC-131: 126 additional frequencies starting from 109.25 MHz

When CINR-curves are specified, COPERNICAT also sets up a simulation with intermodulation weights in the model all equal to one. In contrast with CSO/CTB specifications which only define a single CIR point, a CINR curve defines a range of those points and COPERNICAT must find weights for different orders of intermodulation such that the specified CINR curve is optimally reproduced by the nonlinearities model. COPERNICAT uses third-, fifth- and seventh-order nonlinearities in order to achieve an optimal result. The argumentation for choosing these orders is as follows:

- The CINR curve gives no information whatsoever if a second-order nonlinearity might be involved. Because the first goal of the model is to approach the curve's shape around the maximum as best as possible, it is not really important if a simulation uses a second- or third-order nonlinearity for that. Because third-order nonlinearities become more dominant at higher levels, this order is a good choice to use as the dominant nonlinearity order for the first part of the CINR curve.
- For high signal levels a CINR curve may show a very fast deterioration which can not be modeled by third-order nonlinearities alone. COPERNICAT may therefore additionally use fifth- and seventh-order nonlinearities in order to obtain a model which can reproduce the specified CINR curve with minimum error.

Intermodulation and legacy analog signals

For analog signals, generating nonlinear distortion with the correct levels and calculating their impact on the analog signals themselves is not straightforward. This section gives short overview of the aspects that play a role in dealing with nonlinearities and analog signals. Note however that analog signals are phased out in many HFC networks and the contents of this section therefore becomes less relevant over time. Operators who still have a small selection of analog television signals present on their network may even abandon the original high-quality CNR/CIR requirements for those signals and only offer a 'reasonable viewing quality' in order to give more room to digital payload signals like (Euro)DOCSIS.

The left side below is an illustration of a video signal (not exactly on scale). The video block means '*the signal can have any amplitude that fits within the block*', and the mean amplitude is halfway that block. Most analog TV systems use negative modulation, ITU-system L is the only exception, so the modulated RF signal is illustrated on the right side.

In HFC networks the specified level of an analog TV signal equals the level of the top-sync part of the RF signal, but from the illustration it is clear that the true rms value of the analog TV signal is much lower. As an initial estimate: the mean video level dominates a large part of the total signal, it is halfway the video block and thus a little less than half the top-sync level. We can thus expect that the true rms value of the analog TV signal may be over 6 dB lower than the top-sync level. However, taking the full signal with all top-sync, video and blanking levels correctly into account, it turns out to be approximately 5 dB.

For a single second- or third-order intermodulation spectral component 2 or 3 different '*parent*' analog TV signals are involved. Because the signals are generally not synchronized in time, it is unlikely that their top-syncs (fully) coincide. This means that if one calculates the level of such an intermodulation component based on the top-sync levels of the parent signals, that calculated level is likely to be present much shorter than the top-sync duration or it may not be present at all. This is illustrated below for second-order intermodulation spectral components with 2 parents.

The same mechanism applies when mutliple intermodulation components add up on the same frequency: because each intermodulation component has at least one different parent signal than every other intermodulation component on that frequency, there will never be a full addition of all top-sync levels. In the extreme case where infinitely many intermodulation components add up in a cluster and where we have infinitely many parent analog TV signals, the Central Limit Theorem predicts that the rms power of the signals becomes dominant. Assuming a difference of 5 dB between rms power and top-sync power, this means that the calculations purely based on top-sync levels are 10 dB and 15 dB too high for respectively second and third-order intermodulation.

Based on above argumentation we can thus introduce a reduction factor for intermodulation clusters where multiple analog TV signals are involved. The reduction depends on:

- the number of analog TV parents in an individual intermodulation component
- the number of individual intermodulation components that add up in a cluster
- the number of analog TV signals in the network

Note that the above curves are the results of simulations which can statistically analyze situations that are impossible in practice. For example, with only 5 analog TV signals in the network it is impossible to have 100 intermodulation products in the same cluster on a certain frequency.

For the quality of legacy analog signals like analog TV and FM it matters where the discrete intermodulation components occur in the signal's spectrum. For FM-signals also the number of analog TV parents in the intermodulation components may play a role. In order to take these phenomena into account and to obtain an unambiguous CIR value, weighting curves (also called '*protection curves*') are used although their definition is a little bit different. A protection curve defines for every frequency offset the required CIR in order to achieve a predefined overall CIR, a weighting curve is based on such a protection curve and defines every frequency offset the weight that must be applied to all interferers in order to calculate a valid CIR.

The figure below shows some examples for analog TV.

The protection curves for analog TV are valid for all types of intermodulation. Note that the curves show combinations which are never used in practice (for example B/G-SECAM). ITU-systems like B/G and others have specific characteristics which determine the shape of the protection curve, and the same applies for the color systems PAL/NTSC/SECAM. If a protection curve is not defined in the standards for a specific combination of analog TV (color) standards, a sufficient accurate curve can therefore be calculated based on the existing curves and characteristics like vestigial sideband width, video bandwidth, offset of the color-carrier, etc.

The figure below shows some examples for analog stereo FM with 75 kHz deviation.

For FM multiple protection curves may exist for different types of intermodulation because they have different characteristics. This was done for the Dutch NVN-5175 calculation standard, and in the above plot '*T*' stands for 'Television' and '*C*' stands for 'Carrier' (generally FM). The curve TTC is therefore a third-order intermodulation protection curve for intermodulation components where two parents are analog television and one parent is a carrier-like signal. The ITU-R BS.412-9 standard only defines two protection curves: one for '*steady interference*' (used by COPERNICAT) and one for '*tropospheric interference*'. COPERNICAT implements these curves for 50 kHz (most analog TV sound systems) and 75 kHz deviation (broadcast FM) although their difference is not highly significant.

Nonlinear distortion originating from broadband signals have a much more dominant tail in their amplitude distribution than Gaussian noise. In other words: for broadband intermodulation and Gaussian noise of the same power, the former has a higher content/probability of very high amplitudes than the latter. Signals with a digital payload like (Euro)DOCSIS are relatively sensitive for those high amplitudes which leads to high BER values. Legacy analog signals like analog TV or FM are however less sensitive for those high amplitudes, and the effect of broadband intermodulation is almost equivalent of that of Gaussian noise. COPERNICAT therefore offers the possibility for legacy analog signals to use the broadband intermodulation power in the calculation of the CNR instead of the CIR.

Summation of nonlinear spectra

In a simulation with multiple coax/amplifier sections, each amplifier generates intermodulation components and all intermodulation clusters from all amplifiers must be summed on a frequency basis. The basic rule is as follows:

- second-order intermodulation adds up with 10∙log (power)
- third-order intermodulation adds up with 20∙log (amplitude)

The following analysis contains a lot of formulas, but it is essentially all quite simple and easy to follow so don't let the equations scare you off! We'll start with three signals present at the output of the first amplifier (A), all with their own frequency and phase offset. Notation \(s_{1,A}(t)\) means signal 1 at the output of amplifier A: $$\begin{aligned} s_{1,A}(t)&=\cos(\omega_1t+\phi_1) \\ s_{2,A}(t)&=\cos(\omega_2t+\phi_2) \\ s_{3,A}(t)&=\cos(\omega_3t+\phi_3) \end{aligned}$$ Note that we leave out the amplitudes because they do not play a role in this analysis, and leaving them out simplifies writing a lot. The essence of nonlinearities is that multiple individual signals are multiplied and that cosine functions are generated with sums and differences of the arguments of the signals. For example the multiplication of two cosines: $$\cos(x)\cos(y)=\tfrac{1}{2}\cos(x+y)+\tfrac{1}{2}\cos(x-y)=\tfrac{1}{2}\cos(x\pm y)$$ For second-order intermodulation components we use the notation \(c_{\,IM2,A,A}(t)\), which means: an IM2 component, generated by amplifier A and observed at the output of amplifier A. For third-order intermodulation components the equivalent notation \(c_{\,IM3,A,A}(t)\) is used. We do not care about amplitude scaling constants and we therefore get: $$\begin{aligned} c_{\,IM2,A,A}(t)&=\cos\left((\omega_1t+\phi_1)\pm(\omega_2t+\phi_2)\right)=\cos\left((\omega_1\pm\omega_2)t+(\phi_1\pm\phi_2)\right) \\ c_{\,IM3,A,A}(t)&=\cos\left((\omega_1t+\phi_1)\pm(\omega_2t+\phi_2)\pm(\omega_3t+\phi_3)\right)=\cos\left((\omega_1\pm\omega_2\pm\omega_3)t+(\phi_1\pm\phi_2\pm\phi_3)\right) \end{aligned}$$ Instead of writing out all possible cosine functions using sums and differences of the signals' arguments, we used the \(\pm\) operator which simply represents all possible options.

So, all still quite simple. The next step is to analyse how the signals and intermodulation components look like when they are transfered through the coaxial cable and amplifier B. We can then determine how the intermodulation components generated at the output of amplifier B look like, and compare them with the intermodulation components which were generated by amplifier A. Again, amplitude is not important and we'll focus on the phase:

- The coaxial cable has a constant delay \(\tau\), and the corresponding phase-shift is proportional with that delay:\(\theta_{coax}(\omega)=\omega\tau\)
- An amplifier is assumed to have broadband characteristics with a near-zero phase shift for its active components, but the diplexer filter which isolates upstream from downstream introduces a constant phase-shift \(\theta_{amp}\) for the downstream band.

Now that IM2 is analyzed in detail, we follow the same procedure for IM3 and quickly find: $$ \varphi_{\,IM3}=\mp\theta_{amp}\mp\theta_{amp} $$ We now have two \(\mp\) operators here, and the value of \(\varphi_{\,IM3}\) and summation method depends on them as follows:

first \(\mp\) | second \(\mp\) | \(\varphi_{\,IM3}\) | summation |
---|---|---|---|

\(-\) | \(-\) | \(-2\theta_{amp}\) | power |

\(-\) | \(+\) | \(0\) | amplitude |

\(+\) | \(-\) | \(0\) | amplitude |

\(+\) | \(+\) | \(2\theta_{amp}\) | power |

For IM3 it is worthwhile to analyse the error that is made by summing all IM3 components on an amplitude-basis, knowing that some IM3 components should be summed on a power-basis. Assume we have two cosines with amplitude equal to one, and summing both on a power- respectively amplitude-basis therefore gives: $$\begin{aligned} P_{power} = 1 \\ P_{amplitude} = 2 \end{aligned}$$ Now assume that amplifier A and B both generate 3 different IM3 clusters of equal power on 3 different frequencies, according to the last 3 rows of the above table. This means that one IM3 cluster should be summed on a power-basis and two IM3 clusters should be summed on an amplitude-basis. We thus get for the total combined power of this correct but mixed approach: $$ P_{total,mixed}=P_{power}+2P_{amplitude}=5 $$ If we follow the basic rule, which is an approximation, we sum all IM3 clusters on an amplitude-basis and we get: $$ P_{total,approx}=3P_{amplitude}=6 $$ The difference between the two methods is only 0.8 dB which is small enough to justify the approximation that all IM3 clusters are summed on an amplitude-basis.

As a last note: the question whether nonlinear spectra should be summed on a power- or amplitude-basis is especially relevant when the spectra to be summed have comparable power. When one spectrum has a dominant power compared with the other spectra, that spectrum will also be dominant after the summation and the summation method is less relevant.

Optical distortions

Many types of optical distortion are covered by COPERNICAT, but the underlying theory is too extensive and complex to cover it here in detail (apart from the fact that this '*Simulation theory*' section is meant to give some understanding and insight of the main principles, it's not a thorough lecture). In this section we'll give some equations '*as is*' and additionally explain something about the magnitude of WDM distortions.

COPERNICAT supports the following optical distortions where only a single optical carrier plays a role:

- laser RIN (Relative Intensity Noise)
- EDFA noise
- fiber IIN (Interferometric Intensity Noise)
- fiber SPM (Self Phase Modulation)
- fiber CSO (Composite Second Order, due to laser-chirp acting with fiber dispersion)
- photodiode shot-noise
- receiver post-amplifier noise, equivalent input current
- receiver post-amplifier noise, TIA (Trans Impedance Amplifier) with noise figure and feedback impedance

The parameters are as follows:

- \(e\) is Euler's number (2.71828)
- \(B\) is the signal bandwidth in [Hz]
- \(m_{pk}\) is the signal optical modulation index
- \(f\) is the signal RF-frequency in [Hz]
- \(\lambda\) is the optical wavelength in [nm]
- \(P_o\) is the optical input power in [dBm]
- \(NF_{EDFA}\) is the EDFA Noise Figure in [dB]
- \(NF_{TIA}\) is the TIA Noise Figure in [dB]
- \(\alpha_0\) is the fiber attenuation in [dB/km]
- \(L\) is the fiber length in [km]
- \(D\) is the fiber dispersion in [ps/(nm·km)]
- \(R_{Rb}\) is the Rayleigh backscattering reflectance in [dB]
- \(\sigma_f\) is the total laser-linewidth and -chirp in [Hz]
- \(\sigma_{f,ch}\) is the laser-linewidth and -chirp per channel in [Hz]
- \(N_{CSO}\) is the number of CSO components
- \(R\) is the photodiode responsivity in [A/W]
- \(I_{eq}\) is the equivalent noise current in [pA/sqrt(Hz)]
- \(R_f\) is the feedback impedance in [Ohm]
- \(T_0\) is the temperature in [K]

The following WDM distortions are supported by COPERNICAT:

- demultiplexer adjacent channel isolation
- SRS (Stimulated Raman Scattering)
- XPM-FD (Cross-Phase Modulation acting with Fiber Dispersion)
- XPM-TS (Cross-Phase Modulation acting with demultiplexer Transmission Slope)
- OKE-PDL (Optical Kerr Effect acting with demultiplexer Polarization Dependent Loss)

WDM distortions are mainly characterized by nonlinear effects where the modulating RF-signal of one optical carrier is induced on another optical carrier which results in a decreased CINR. The magnitude of the WDM distortions depends on the polarization angle between the two optical carriers or '*polarization overlap*'. Because this polarization overlap is generally not controlled in HFC systems it is therefore an unknown and random variable. Note that the polarization overlap can vary over time, and the WDM effects therefore may also vary over time.

Trying to choose a value for the polarization overlap which fits a worst-case analysis for all combined WDM distortions is quite complex because the WDM distortions are RF-frequency dependent and it requires a separate definition for determining the worst-case CINR result. Specifically, for this worst-case CINR we could take the single worst-case RF-signal in the victim optical carrier or the overall worst-case mean CINR of all RF-signals. Either choice seems arbitrary and may have its pros and cons.

Bassed on the above argumentation, COPERNICAT uses another approach and gives you two choices:

- COPERNICAT maximizes the individual types of WDM distortions, which means that for each type of distortion a different polarization overlap is chosen in order to achieve the corresponding worst-case result. Note that this is not possible in practice because at any moment in time there can only be one polarization overlap value beteen two signals. The calculated CNR result is therefore generally worse than the actual CNR in the system.
- COPERNICAT chooses a value for the polarization overlap which results in an average level of all WDM distortions for a single as well as multiple interfering optical carriers. In more detail: COPERNICAT predicts the magnitude of the WDM distortion level for all possible values of the random polarization overlap angle, and then takes the rms-value of them in order to obtain the '
*statistical average*' distortion level.

An overview of noise and distortions in the optical domain is described in:

Morgan Kaufmann Publishers in an Imprint of Elsevier

More detailed information on specific optical distortions can be found in the following articles, several of these publications form the basis of the equations in the book '*Modern cable television technology*':

IEEE Transactions on Electron Devices, Vol. ED.13, No. 1, January 1966

Journal of Lightwave Technology, Vol. 14, No. 3, March 1996

Journal of Lightwave Technology, Vol. 17, No. 10, October 1999

IEEE Photonics Technology Letters, Vol. 12, No. 8, August 2000

Electronics Letters, Vol. 35, No. 20, September 1999

Scientific-Atlanta, Inc.

Ortel Corporation

Scientific-Atlanta, Inc.

Spectral efficiency estimates

Especially for downstream which has a fixed FEC, the spectral efficiency estimate is quite straightforward. For example:

- With an SNR of 25.5 dB you can use 64-QAM EuroDOCSIS and get a spectral efficiency of 4.8 bps/Hz.
- With an SNR of 31.5 dB you can use 256-QAM EuroDOCSIS and get a spectral efficiency of 6.4 bps/Hz.

For upstream things are different. Not only the range of possible modulations is much wider, from QPSK to 128-QAM, but several parameters of the FEC can also be adjusted and this offers a much finer granularity for the spectral efficiency. Because the goal of COPERNICAT is to give you a reasonable estimate of the spectral efficiency, the upstream values were fitted to one continuous CNR versus spectral efficiency curve and all estimates calculated by COPERNICAT are derived from that curve. This continuous curve is based on the analysis described in:

Juniper Networks

In DOCSIS 3.1 several mechanisms are used which provide a much more flexible range of spectral efficiency than its preceeding DOCSIS versions:

- The upstream supports three different LDPC code rates (8/9, 28/33 and 3/4), and codeword shortening can be applied in order to offer more FEC protection of the payload data.
- The downstream supports only a single LDPC code rate (8/9), but codeword shortening is also supported. Additionaly, mixed modulation offers a 1.5 dB granularity on itself.

IEEE International Conference on Consumer Electronics, 2015

Because of the fine granularity that both upstream and downstream provide using the above described mechanisms, the DOCSIS 3.1 performance was fitted to one continuous CNR versus spectral efficiency curve and all estimates calculated by COPERNICAT are derived from that curve. The estimates depend on the DOCSIS 3.1 signal's settings for carrier spacing (25 kHz or 50 kHz) and cyclic prefix length. The figure below gives the spectral efficiency curves of DOCSIS 3.1 for a cyclic prefix of 2.5 µs.

With 25 kHz carrier spacing (40 µs symbols) the cyclic prefix imposes a relatively smaller performance penalty than with 50 kHz carrier spacing (20 µs symbols) and therefore has a bit higher spectral efficiency. Because upstream only uses LDPC for error correction while downstream combines LDPC with BCH, upstream requires an approximately 2 dB higher CNR than downstream in order to enable a certain M-QAM. This 2 dB difference is only approximately visible in the above plot because the default FEC effective code rate for downstream (0.8785) is a little lower than the default FEC effective code rate for upstream (0.8889). An example verification calculation for downstream, 40 µs symbols with 2.5 µs cyclic prefix (blue curve):

- The FEC effective code rate is 0.8785.
- The efficiency factor due to the cyclic prefix is 40.0 / (40.0 + 2.5) = 0.9412.
- Carriers that do not transfer user data (pilots, etc.) result in an efficiency factor of 0.9721. Note that, for various reasons, COPERNICAT defines the DOCSIS 3.1 bandwidth as the effective/active bandwidth without guard bands which gives a little different definition of the spectral efficiency than commonly used. This does however not affect correct calculation of the resulting total throughput of the DOCSIS 3.1 channel.
- For 256-QAM (requires 27.0 dB according to the standard): 8 bits per symbol, the spectral efficiency is therefore 8×0.8785×0.9412×0.9721=6.43 bps/Hz.
- For 4096-QAM (requires 41.0 dB according to the standard): 12 bits per symbol, the spectral efficiency is therefore 12×0.8785×0.9412×0.9721=9.65 bps/Hz.

Arris

Most standards, including DOCSIS, generally base the correct functioning of the signals on CNR only. This introduces an uncertainty in the calculation of spectral efficiencies because the HFC-network not only suffers from noise but also from nonlinearities. Measurements and theory show that, in general, a certain amount of imtermodulation power has a stronger impact on the BER of a signal than noise with the same power. It is difficult, or even inpossible, to correctly quantify this effect and a rule of thumb must therefore be used: COPERNICAT assumes that intermodulation has a 6 dB stronger impact on the signal's BER than noise. A weighted \(CNR_w\) based on the actual noise \(N\) and intermodulation \(I\) power is therefore calculated: $$ CNR_w = \frac{C}{N+4\,I} $$ and this weighted \(CNR_w\) is used as input for calculation of the spectral efficiency.