This paper presents a frequency domain analysis of spurious tones in frequency dividers. The results of the analysis are used to develop an event-driven model for system simulations which work entirely in the frequency domain. The proposed approach is able to provide a fast and accurate model in a SystemVerilog/C++ environment which takes the frequency conversion effects of the spurious tones into account. A virtual prototype which includes the model was simulated and due to the fast simulation speed it was possible to determine the influence of spurious tones on the bit error rate in a complex receive scenario.

The trend in modern system-on-chips (SoCs) is going towards integration of
many different subsystems on a single die. Especially for integrated
RF-transceivers it brings a challenge due to the complex modulation schemes
needed for high data rates and flexibility of the system. On the one hand, a
high sensitivity for the receivers and a pure spectral mask of the
transmitters are crucial, on the other hand, due to the integration of
multiple transceive paths with control logic and digital signal processing on
one chip, many sources of digital supply noise exist which can couple in
sensitive analog blocks and harm the performance of a transceiver

RF receiver with PLL for LO signal generation.

For frequency planing of the transceivers it is crucial to know where the
unwanted frequency components of the local oscillator are converted to by the
frequency divider. Figure

Circuit simulation results of frequency dividers.

Due to the fact that frequency dividers are strongly nonlinear and have a
memory, the conversion of the spurious tones are not straightforward as one
would expect. Naively one would say that the frequency of every spectral
component just gets divided by the divider factor but frankly this is not the
case. A circuit simulation of a frequency divider by two and a divider by
three showed, that only the frequency of the main (largest) tone is divided
by the divider factor. The spurious component at the offset frequency
(

Sampling of the output phase.

A deeper look in the mathematics of spurs and frequency dividers can explain
this behavior. The input signal with spurious tones of a FD is a
quasiperiodic signal with constant amplitude. Simplified, these class of
signals can be described as follows:

Simulation result of a FD by 8 with far off spurs.

For understanding the modeling approach of the FD, a small detour to the
signal description in the used simulation environment is necessary. The
spectral signal description is similar to the baseband equivalent signal
description, which is a widely used approach to speed up simulations of
event-driven real number models of RF systems. It is based on the fact that
every modulated RF signal can be represented as:

Example of the processing of nonlinearities.

SV/C++ modeling framework.

As mentioned in Sect.

Due to the fact that the input and the output level of the signals might be different, a scaling before and after the amplitude calculation is done.

Frequency divider model.

Virtual prototype of a 2.4 GHz receiver.

The frequency divider model was used in a virtual
prototype of an integrated 2.4 GHz low-if receiver shown in
Fig.

BER simulation results.

In this work a frequency domain analysis of frequency dividers and an event-driven modeling approach which works with a spectral signal description was presented. The analysis focussed on the conversion of spurious tones and showed, that against the intuition the frequency of the spurs are not simply divided by the factor of the frequency divider. It can be shown, that the offset frequency to the main tone stays the same and the spur amplitude gets divided by the divider factor. Furthermore, the sampling of the spurs by the dominant output frequency component can lead to unexpected spurs close to the main tone. On basis of these analysis an event-driven frequency divider model was developed which calculates the output in the spectral domain and allows fast system simulations. The model was used in a virtual prototype of a 2.4 GHz receiver and it was possible to simulate the BER of the system and observe the influence of spurs in the local oscillator signal.

Due to the fact that the modeling and verification code is still used in other projects, the code cannot be published.

The simulation data of the frequency dividers and the sent
and received bitstream for the BER calculation can be found under

The authors declare that they have no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2018”. It is a result of the Kleinheubacher Tagung 2018, Miltenberg, Germany, 24–26 September 2018.

This paper was edited by Jens Anders and reviewed by two anonymous referees.