aainstitutetext: State Key Laboratory of Particle Detection and Electronics,
University of Science and Technology of China,
Hefei 230026, Chinabbinstitutetext: Department of Modern Physics, University
of Science and Technology of China,
Hefei 230026, China

# A Low Dead Time, Resource efficient Encoding Method for FPGA based High-
Resolution TDL TDCs

W.Dong a,b,1 C. Feng,111Corresponding author. a,b J. Wang a,b Z. Shen a,b S.
Liu a,b and Q. An<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel encoding method for fine time data of a tapped
delay line (TDL) time-to-digital Converter (TDC). It is based on divide-and-
conquer strategy, and has the advantage of significantly reducing logic
resource utilization while retaining low dead-time performance. Furthermore,
the problem of high bubble depth in advanced devices can be resolved with this
method. Four examples are demonstrated, which were implemented in a Xilinx
Artix-7 Field Programmable Gate Array (FPGA) device, and encoding method
presented in this paper was employed to encode fine time data for normal TDL
TDC, a half-length delay line TDC, and double-edge and four-edge wave union
TDCs. Compared with TDCs from the latest published papers that adopt
traditional encoders, the logic utilization of TDCs in this paper were reduced
by a factor of 45% to 70% in different situations, while the encoding dead
time can be restricted in one clock cycle. Acceptable resolutions of the
demonstrated TDCs were also obtained, proving the functionality of the
encoding method.

###### Keywords:

Instrumentation and methods for time-of-flight (TOF) spectroscopy; Digital
electronic circuits; VLSI circuits

##  1 Introduction

Time-to-digital converter (TDC) plays a key role in many areas, such as
particle physics experiments, medical imaging equipment, and autopilot
systems. For example, a time-of-flight (TOF) detector is often the core part
of large detector systems for particle and astroparticle physics projects
carnesecchi2019performance . Besides, TOF detectors determine the coincidence
time of positron emission tomography imaging, the range of a LIDAR receiver,
and the distance of direct TOF cameras. All of these applications require
high-resolution TDCs brunner2013studies ; keel2019640 ; song2017high .

Time-to-digital converters are mainly implemented in application-specific
integrated circuits (ASICs) or Field Programmable Gate Array (FPGA) in recent
years. Benefiting from their advantages in flexibility and development cycle,
FPGA-based TDCs have been widely applied in many areas where the requirements
of TDCs are non-standard and not massive. In recent years, FPGA-based TDCs
have experienced remarkable advances, and their accuracy has reached the level
of several pico-seconds, which is comparable to the performance of state-of-
the-art ASICs, mainly due to the invention of dedicated carry chain
song2006high ; wu2003firmware  and wave union architectures wu200810 ;
wang201110 .

However, increasing the number of channels in a single chip is still a
challenge. The density of TDC channels has become increasingly important in
various applications as mentioned above. For example, in the ALICE (A Large
Ion Collider Experiment) experiment at the Large Hadron Collider, there were
152,928 readout channels in its TOF detector, and more than 20 thousand high-
resolution TDC ASIC chips were used, with eight channels for each chip
carnesecchi2019performance . In another example of a Compressed Baryonic
Matter spectrometer, the TOF subsystem contained thousands of readout channels
(1,600 of mTOF and 6,912 of eTOF) deppner2020fair . For applications that
contain a large amount of time measurement channels, increasing the number of
channels in a single TDC chip is a direct way to improve the system
integration level. However, the logic utilization becomes a pivotal factor
that restricts the channel density for FPGA-based TDCs. Logic resources are
limited in FPGAs and usually other functions, than the core TDC, are required
in the same chip.

Among all the architectures, a normal tapped delay line (TDL) is a mainstream
type for FPGA-based TDCs. It combines a coarse counter that is driven by the
system clock and a TDL for fine-time interpolation. The latter is usually
implemented by a carry chain in FPGA. In this way, a large dynamic range
together with a high time resolution is achieved. However, to obtain the fine
time which is less than one clock period, the TDL’s raw data must be encoded
into binary code. Since the raw data is usually thermometer code, some
encoders used in flash ADC have been used in this kind of TDC zheng2017low .

For normal FPGA-based TDL TDC, the time resolution is limited by the cell
delay. Some methods have been employed to attain higher precision, among which
multi-chain architecture is an effective way to improve the time resolution
beyond its cell delay. However, this method occupies more logic resources not
only to implement more carry chains but also to encode the fine time data for
all the chains. Therefore, it is not suitable for applications that need many
TDC channels.

Wave Union, proposed by J. Wu and Z. Shi wu200810 , is another effective
architecture to achieve higher precision. It employs two types of ”Wave union
launchers” (named A and B) to make multiple measurements with a single delay
chain structure, which effectively sub-divides the ultra-wide bins in each raw
measurement. For the Wave Union A structure, which is more widely used because
of its low dead time feature, the launcher generates one or several narrow
pulses and all the rising and falling edges are sampled in one clock cycle. In
this case, raw data of the TDL becomes non-thermometer code.

A type of newly presented architecture kong2020resource , referred to as half-
length delay line TDC in this paper, has this type of non-thermometer raw data
as well. For each effective input signal, a square pulse, with the width of
roughly half a clock cycle, is generated, and any one of its two edges is
captured to produce a fine timestamp. In this way, the length of the TDL can
be shorten to slightly longer than half of the clock cycle.

However, for the above two TDC architectures, the nonthermometer-to-binary
(NTH2B) encoding logic is a particular challenge because of the variable and
complex raw data pattern. To deal with this problem, several methods, such as
a stepped-up tree encoder, have been presented hu2013stepped . However, they
are much less resource efficient and less scalable than normal thermometer-to-
binary (TH2B) encoders, which has become restrictions for further increasing
the channel density.

In this paper, a novel resource-saving encoding method is proposed, which can
be used in both thermometer-to-binary and non-thermometer-to-binary encoders.
It is based on the strategy of divide-and-conquer, which is fully parallel in
time and capable of correcting bubble errors. Based on this method, high
channel density TDCs can be realized in a low cost FPGA chip.

##  2 Methodology and design concept

###  2.1 Basic concepts

The architecture of a typical TDL TDC is shown in Figure 1. Besides a coarse
counter that runs at the system clock, it is mainly composed of a TDL, an
array of D flip-flops, and the encoding circuits. A specific waveform is
generated and transmitted on the delay-line when the launcher is triggered by
an input signal (Hit). The outputs of all the delay cells of the TDL are
sampled in the form of raw data by D flip-flops on every rising edge of the
system clock. Finally, the raw data of the TDL is encoded. If the raw data
indicates that a transient waveform appears on the delay line, then the output
of the encoding circuits and the coarse counter is buffered subsequently by
auxiliary circuits.

Figure 1:  Basic components of TDL TDCs.

In this paper, three types of TDCs are discussed, which have the same kind of
delay line, sample D flip-flop, and coarse counter, but the waveforms running
on the TDL are different. The three types of TDCs are 1) step signal for a
normal TDL-TDC, 2) spike pulse for a wave union TDC, and 3) a square wave for
a half-length delay line TDC. The wave union TDC discussed in this paper is
type A, which is measured in one clock cycle wu200810 . Therefore, the
launchers and encoders in the three types of TDCs are also different. The
encoder for a half-length delay line TDC needs to deal with raw data
corresponding to either the rising or falling edge, while the encoder for a
wave union TDC has to deal with more than one edge.

Figure 2 shows the three kinds of snapshots when hit signals propagate on
delay line and their corresponding raw data. The pattern of the raw data of a
normal TDL TDC is in the form of “111…1100…000,” and the pattern of a wave
union TDC is in the form of “00…0011…1100…00.”

As for half-length delay line TDCs, a short delay line is used, which covers
just slightly more than one-half of the clock cycle. There are three kinds of
patterns, “111…1100…000,” “000…0011…1100…000,” and “000…0011…111,” which
correspond to different statuses of signal propagation. Although there are
various types of input data, the output data of the encoder is binary code,
which contains the positions of 1-to-0 or 0-to-1 transitions. These
transitions indicate the edges of the input signal which are captured by the
delay line and DFFs.

Figure 2:  Signal propagations and the delay chains’ data patterns of three
kinds of TDCs: (a) a normal TDL TDC (the rising edge is captured ); (b) a
half-length delay line TDC (either rising or falling edge is used depending on
the status of the propagation in the TDL); and (c) a wave union TDC (both the
rising and falling edges are captured at the same time).

The basic idea of all the encoding methods is to find the edge. A simple
method is to employ the exclusive-OR (XOR) logical operation for every two
adjacent binary codes of the raw data. Then, the edge position is encoded
directly by a priority encoder. However, this method is based on the ideal raw
data which is in form of “111…1100…00”. In advanced devices, researchers have
found that the raw data sometimes have the pattern like “111…1100110100…000”,
in which the zeros between ones are called “bubbles” wu2010several . The
length of the blurring area, such as “001101”, is defined as bubble depth. And
the simple XOR method can’t work in this situation.

The exact source of bubbles is difficult to find out for FPGA users. But
considering the basic principles of digital circuits, the non-ideality of
clock distribution and placement of logic cell may be the main factor. In
addition, the signals propagating in the TDL are not synchronized to the
sampling clock. So metastability may be another root of bubble problem.

In order to solve this problem, the mainstream methods are to count the “1”s
in raw data, among which the Wallace tree encoder is a mature technology that
has been applied widely in normal TDC zheng2017low . Some other encoders that
were originally invented for Flash ADC, such as Ones-Counter, have also been
introduced to normal TDL TDC sall2007thermometer ; wang20173 , but most of
them are not capable of handling non-thermometer-to-binary codes. Another
encoder was proposed to eliminate the bubble problems by using the first 1-0
or 0-1 transition jiao2021resource . However, it is based on the premise that
the maximum bubble depth is four, and the encoder structure is difficult to be
extended for the situations of deeper bubbles.

The encoding method proposed in this paper is based on divide-and-conquer
strategy. It solves the bubble problems by dividing the raw data into pieces
and counting ”1”s in each piece. The number of ”1”s in each piece is called
”local sum” and the MSB of the local sum is the flag of the corresponding
piece. The flags of these pieces are used to find the one or more transitions
in the raw data roughly. For each transition, two adjacent pieces where the
transition locates in are selected . Finally, combined with the local sums of
the two selected pieces, the exact position of the transition is attained.

###  2.2 Architecture of encoding circuits

The encoding method proposed in this paper has several features: resource
saving, eliminating bubbles and an extensible and standard structure for
different types of TDL TDCs. The encoding method has three steps as follows:
The first step is to divide the TDC raw data into small pieces and then the
local sums and flags are attained. By making use of the flags from the first
step, the second step is to find the edges piece by piece. In the final step,
the edge position is calculated from the sequence number of the selected piece
of the second step and the local sums of the first step.

In the first step, the raw bins are divided into m-bit (m equals 24 in Figure
3) wide pieces. The pre-encoded cells get the local sums of these pieces.
After being buffered by D Flip-Flops, the pre-encoded results are sent to
back-end circuits to find the 1-0 transition. As shown in Figure 3, the flag
array consists of the most significant bits (MSB) of each local sum and the
piece select (Sel) array is composed of the XOR results of the adjacent bits
of the flag array.

Figure 3:  The process of first step in the encoding method. It consists of
pre-encoded cells and gets labels that roughly show the edge’s position. a)
The details of pre-encoded cell; b) the process of getting the Sel array.

In the second step, the encoder finds which two adjacent pieces the transition
occurs in. The output of the first step is buffered. The back-end circuit,
which accomplishes the second and final steps, consists of priority encoders,
multiplexers, and arithmetic circuits. The back-end circuit finds pieces that
have a 1-0 or 0-1 transition by the Sel array in the priority encoder.
Finally, the product of the encoding result and the piece width plus the local
are summed in the arithmetic circuits. The arithmetic circuits consist of
multipliers and adders. The binary code of one transition’s location is
attained. The back-end circuit for four-edge wave union TDCs is designed to be
a pipeline structure as shown in Figure 4.

Figure 4:  The back-end encoding circuits consist of priority encoders,
multiplexers, and arithmetic circuits(the M&A block). The diagram shows
encoders for four-edge wave union TDCs.

In further steps, the difference between the encoding method and the Wallace
tree encoder is that the priority encoders and multiplexers in this method
replace the Wallace tree encoder’s adders. The input of the MUXs is not the
most significant bit (MSB) of the sum of these pieces but the XOR of their
adjacent MSB (as the Sel array shows in Figure 3). As a result, compared with
the Wallace tree encoder, the encoding method presented in this paper can deal
with non-thermometer-code. As shown in Figure 4, the encoding method can deal
with multiple transitions by cascading priority encoders.

###  2.3 Three types of encoders

####  2.3.1 Encoder for a normal TDL TDC

Normal TDL TDCs only need to encode one 1-0 transition’s position, which is
illustrated in Figure 2. The encoders of the normal TDL TDCs only have one
stage of back-end circuits to find the 1-0 transition. In the arithmetic
circuits, the adders process the selected local sums and the priority codes to
obtain the final encoded transition position. Meanwhile, the en-flag signal
indicates whether the hit occurs in the corresponding clock cycle. There are
two examples in Figure 5 showing how the encoder attains the transition
positions. The final result has two parts: the base which is the product of
piece width and sequence of the selected pieces, the offset which is the sum
of local sums of the two selected pieces. The equation of encoding result is

$R=Base+Offset=P\cdot{W}+Sum$ (1)

In Equation (1), R represents the result of encoding, P represents the
sequence of the first selected piece, W represents the width of the pieces (24
in Figure 5), and Sum represents the sum of the local sums of the two adjacent
selected pieces. All the calculation is completed by adders because the
weights of the priority codes are fixed.

Figure 5:  Examples of calculating transition position in Normal TDL TDCs.

####  2.3.2 Encoder for a half-length delay line TDC

The waveform and raw data of the half-length delay line TDCs are shown in
Figure 2. To reduce the length of the delay line, half-length delay line TDCs
utilize either rising or falling edge of the pulse signal. The difference
between a half-length delay line TDC and a wave union TDC is that a half-
length delay line TDC only deals with the first edge of the waveform. However,
there are three kinds of situations that the encoder has to process. If
traditional encoders are used for the half-length delay line TDC, the
reduction of logic utilization will not be as large as the reduction of delay
cells kong2020resource .

The above-mentioned encoder used in a normal TDC can be easily adapted to
half-length delay line TDCs. In this case, the flags and the MSB of the local
sums of the selected pieces are different. The first flag is “1” and the
second one is “0” when the 1-0 transition is registered. The flags are inverse
for the 0-1 transition. As a result, the type of the captured edge can be
distinguished in the final arithmetic circuits. Thus, the equation for the 1-0
transition is the same as for normal TDL TDCs, and the equation for the
0-1transition is

$R=P\cdot{W}+({W\cdot{2}-Sum}).$ (2)

The base part in the encoding result changes because the encoder counts ”0”s
in selected pieces in 0-1 transitions.

####  2.3.3 Encoder for wave union TDC

A wave union TDC has to deal with more than one edge in one sampled raw bin
data wu200810 . To solve this problem, more stages in the back-end circuit can
be used to deal with more edges, as shown in Figure 4. If the input data have
more than one 1-0 or 0-1 transition, then there will be more than one ”1” bits
in the Sel array. In the first stage, the priority encoder finds the first
edge and the first ”1” bit in the Sel array is masked. The second stage can
find the second one in the masked output of the first stage. It is the same in
the subsequent back-end encoding stages. Because the type of the transitions
delivered to the arithmetic circuits is clear, the arithmetic circuits handle
the data according to their sequence. For example, the waveform in Figure 2
has two edges, and the first one is a rising edge and the second is a falling
one. The final output is valid only when both edges are sampled and the
encoder can indicate the number of transitions in the raw data.

##  3 Implementation and test results

To verify the newly presented encoding method, three kinds of TDL TDCs were
implemented in a Xilinx low-end FPGA, which was a XC7A100TFGG484 device.
Firstly, the normal TDL TDCs and corresponding encoders were realized. Then,
the encoders were adapted for half-length delay line TDCs and wave union TDCs.
Both double-edge and four-edge wave union TDCs were realized.

A prototype board was designed with a high precision clock generator. The
clock generator can provide different frequency clock signals while being
under a picosecond of jitter. Therefore, the test results were hardly affected
by the system clock jitter. An arbitrary waveform generator generated the
pulse, and the pulse was transmitted to a matching tee joint. Two pulse
waveforms with fixed delay were obtained using a pair of unequal length
cables. As shown in Figure 6, the signal was sent to the TDC board through an
adapter board that was connected to the TDC board via an ERNI 154744
connecter. The digitalized timestamps were transferred to the PC via a USB
port.

Figure 6:  The TDC prototype board.

###  3.1 Test results of the normal TDL TDC

As shown in Figure 2, the normal TDL TDC needs to capture the rising edge of
the step signal in one clock period. This means that the whole delay of the
delay line should be longer than the clock period. As for the FPGA-based TDCs,
the delay line is realized by a carry chain that consists of carry logic in
the FPGA. The carry logic in Xilinx Artix FPGA was CARRY4, which has four
output ports. One CARRY4 corresponds to four delay bins. In the normal TDL
TDCs, we used 54 CARRY4 elements to construct a delay line with 216 delay
taps. The clock of the TDC block ran at 400 MHz. As a result, the input data
of the encoder was 216-bit wide. The output of the 216-bit wide thermometer
code input was an 8-bit wide binary code. One TDC channel consumed 254 LUTs
and 449 registers. Bin-by-bin calibration was employed to get better time
resolution wu2010several .

A statistical code density test was used to find the bin size of the TDL and
the look-up table for bin-by-bin calibration. Figure 7 (a) shows the bin size
test result of a single channel. In the test, 200,000 random hits were applied
to the channel. The test showed that the average bin size was 14.8 ps. After
calibration, the root mean square (RMS) value of the time-intervals evaluated
by the “cable-delay test” of the two channels was 15.0 ps. This result is
comparable to prior works, which means that the encoding method is effective
in normal TDL TDCs.

Figure 7:  Test results of the normal TDL TDC: (a) bin size of a single TDC
channel, and (b) time resolution of two channel time-interval test after bin-
by-bin calibration.

###  3.2 Test results of the half-length delay line TDC

To reduce the length of the carry-chain, the half-length delay line TDCs take
advantage of two edges of a pulse signal. In theory, this method can shorten
the length of the delay-line by half. However, it is difficult to achieve this
target in practice. Due to the uneven pulse width, more redundant delay cells
had to be used to ensure that the pulse was sampled. In this module, the carry
chain had 30 CARRY4 logic elements corresponding to 120 delay taps. One TDC
channel consumed 156 LUTs and 248 registers. The sample region of the half
length TDC is roughly equal to 192-bins in the normal TDCs. Because the width
of the square pulse is slightly shorter than the length of the delay line, the
sample region is slightly shorter than twice the delay line. The frequency of
the system clock was 400 MHz, which was the same as the normal TDL TDC.

Because the encoded fine code of the half-length delay line TDC had an extra
bit to indicate the type of valid transition, the 120-bin carry chain also had
an 8-bit binary fine code to be calibrated. As shown in Figure 8, the encoded
numbers were divided into two parts by the flag bit. The average bin size of
the half-length delay line TDC in this channel was 14.6 ps, and the time
resolution between the two channels was 15.9 ps.

Figure 8:  Test results of the half-length delay line TDC: (a) bin size of a
single TDC channel, and (b) time resolution of two channel time-interval test
after bin by bin calibration.

###  3.3 Test results of wave union TDCs

To explore the logic utilization of wave union TDCs using the resource-saving
method, double-edge and four-edge wave union TDCs were implemented in FPGA.
The waveform and the raw data are shown in Figure 2. In this case, a double-
edge wave union TDC was realized on a 288-bin carry chain, and a four-edge
wave union TDC was realized on a 360-bin carry chain. The clock frequencies
were the same at 400 MHz. The circuits to obtain the local sums and flags were
the same as in the normal and half-length delay line TDCs. The back-end of the
encoder had more stages of priority encoders, buffers, and multiplexers to
deal with more edges. The arithmetic circuits in each stage calculated the
edge position by the result from the priority encoder and its corresponding
local sums. Finally, the positions of the two or four edges were added
together. The double-edge wave union TDCs consumed 438 LUTs and 761 registers
per channel, and the four-edge wave union TDCs consumed 706 LUTs and 1,263
registers per channel. The pulse generated by the launcher must be longer than
24 bins’ delay because the pieces in the pre-encoded cells were 24-bits wide.
Finally, 9-bit wide binary code results in double-edge wave union TDCs and
11-bit wide results in four-edge TDCs were obtained.

The smallest number of the fine code was not zero because the fine code was
the sum of two transitions’ locations. The bin size test result shown in
Figure 9(a) indicates that the fine code ranged from 66 to 403. Thus, the
average bin size of double-edge wave union TDC was 7.4 ps, and the time
resolution of the two tested channels was 11.5 ps.

Figure 9:  Test results of double-edge wave union TDC: (a) bin size of a
single TDC channel, and(b) time resolution of the two channel time-interval
test after bin-by-bin calibration.

The smallest number of fine codes of the four-edge wave union TDCs was also
not zero. The bin size test result shown in Figure 10(a) indicates that the
fine code ranged from 164 to 825. Thus, the average bin size of four-edge wave
union TDC was 3.8 ps, and the time resolution of the two tested channels was
8.8 ps.

Figure 10:  Test results of four-edge wave union TDC: (a) bin size of a single
TDC channel, and (b) time resolution of two channel time-interval test after
bin-by-bin calibration.

##  4 Discussion

###  4.1 Resource utilization

Table 1 provides comparisons of the resource utilization of three kinds of
TDCs in some recent works. In the table, ‘$\sim$’means that the parameter is
not mentioned in the text of the paper, but deduced from figures with an
approximate value. Most of the TDCs listed in Table 1 are implemented on
Xilinx 7 series FPGAs, which are the same as the ones in this paper. As for
the work in ref xie2020wave , a Xilinx UltraScale Kintex FPGA was used, which
has the same architecture in LUTs and registers as 7 series.

However, it is difficult to compare these encoders directly. The clock
frequency, sensitivity to bubbles and dead time all have impacts on resource
utilization. And all these TDCs have different length of delay line.
Considering that all the works are implemented in similar FPGA devices, the
usages of LUTs and registers divided by the number of total taps in each TDL
are listed in the table, which can be seen as two simplified (though not
precise) indexes for a rough comparison.

As for the normal TDL TDCs, Xie’s encoder, which is the best one among the
previous works, used 83% more LUTs than this paper xie2020multi . It ran at a
clock frequency of about 250M Hz, which is much lower than most of other works
(about 400 MHz or higher). Another TDCs in Zheng’s paper only used half of the
four outputs of the CARRY4 element, and the clock frequency is also 250 MHz
zheng2017low , was lower than other works. It should be pointed out that for a
lower clock frequency, the pipeline stages of the encoder could be reduced,
which results in the usage of fewer registers.

Table 1:  Resource utilization of a single channel. | Taps | LUTs | Registers | LUTs/ Taps | Registers/Taps |  | Device
---
Family
Normal TDL TDCs
TNS’17 zheng2017low  | 140 | 668 | 387 | 4.77 | 2.76 | Artix-7
TNS’17 wang20173  | 200 | 505 | 1109 | 2.525 | 5.545 | Kintex-7
TNS’17 qin20171  |  $\sim$170 | 848 | / | 4.99 | / | Virtex-7
TIE’19 chen2018multichannel  |  $\sim$380 | 1145 | 1916 | 3.01 | 5.04 | Virtex-7
EBCCSP’20 xie2020multi  |  $\sim$380 | 822 | 1219 | 2.16 | 3.21 | Virtex-7
This Work | 216 | 451 | 805 | 1.18 | 2.08 | Artix-7
The half-length delay line TDCs
I2MIC’20 kong2020resource  | 96 | 418 | 852 | 4.35 | 8.88 | Virtex-7
This Work | 120 | 229 | 388 | 1.3 | 3.23 | Artix-7
Double-edge Wave Union TDCs
arXiv’20 xie2020wave  | 480 | 1349 | 1480 | 2.81 | 3.83 |  | UltraScale
---
Kintex
TNS’19 wang20193  | 216 | 1085 | 2164 | 5.02 | 10.02 | Kintex-7
This Work | 288 | 438 | 761 | 1.52 | 2.91 | Artix-7
Four-edge Wave Union TDCs
TNS’19 wang20193  | 288 | 1410 | 2732 | 4.9 | 9.49 | Kintex-7
This Work | 360 | 706 | 1263 | 1.96 | 3.51 | Artix-7

The greatest advantage of the encoding method presented in this paper is that
the logic utilization of NTH2B encoding can be largely reduced. The encoder of
half-length delay line TDC saves half of the resources compared with the
original work kong2020resource . Compared with the presented encoder for wave
union TDC, the advantage of our encoding method is greater than TH2B encoding
chen2018multichannel ; xie2020wave . This encoder uses 83% more LUTs than
encoder in this paper, and the consumption of register is 32% more.
Furthermore, the encoding method proposed in prior work knittel2019novel  only
deals with the double-edge wave union TDCs. The best results in Table 1 used
this method and only realized double-edge wave union TDCs xie2020wave . When
the TDL has more edges, the proposed encoding method will have greater
advantages wang20193 . Because it can locate more transitions by increasing
the stages of back-end circuits without changing the original function blocks,
it is convenient to adapt our encoding method for TDL, which has more edges
with less increased resource usage.

###  4.2 Dead time

The resource-saving encoding method described above is implemented using
pipeline techniques so that the dead time of encoders is one clock period.
However, the flags of the encoder in two adjacent periods are used to generate
the write-enable signal of the FIFO in the TDC sub-system. Therefore, the dead
times of the whole TDC circuits are two clock periods. The dead time is
acceptable in most cases as the clock frequency of the TDC system clock is
very high.

###  4.3 Sensitivity to bubbles

The balance of logic utilization, dead time and sensitivity to bubbles should
be made for different applications. The encoding block in Figure 3 divided the
raw data into 24-bit wide pieces. Assuming that 0 and 1 in the blurring area
of bubbles are roughly uniform, as observed in previous work wang20164 , the
encoder can handle 16-bit wide bubbles in 1-to-0 transition in a worst-case
scenario. The worst case is that the transition locates in the second selected
piece and the first piece has no bubbles. In that case, if the length of
bubbles is to large, two adjacent piece will not cover all blurring area.
Because the threshold is 16 which means the MSB is easier turn to zero in this
example, this encoding circuit is more sensitive to bubbles of 1-to-0
transitions.

The bubble depth is 4 bit in 40nm Xilinx FPGA jiao2021resource . Then it
increases to 5 bit in 28nm device and more than 10 bits in 20nm and 14nm
devices song2018256 ; wang20173 ; wu2021design . The 24bit wide piece in this
paper is capable for the bubble problems in the Artix-7 FPGA which is on the
28nm process. If the encoding method is employed on TDCs with higher bubble
depth, wider pieces can solve the problem.

##  5 Conclusion

In this paper, a novel resource-saving, low dead time encoding method for TDL
TDCs was proposed. This encoding method was inspired by the divide-and-conquer
strategy, and it combines two encoding methods, sum encoding and priority
encoding. This method not only greatly reduces the use of logic resources but
also solves the high depth bubble problem. And three kinds of TDCs have been
realized in the Artix 7 Device. Both acceptable resolutions and improvements
on resource efficiency are attained. With this method, the TDL TDCs will have
greater prospects in emerging applications.

###### Acknowledgements.

This work was supported by the Strategic Priority Research Program of Chinese
Academy of Sciences, Grant No. XDC07020300.

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# Control of multi-modal scattering in a microwave frequency comb

J. C. Rivera Hernández<EMAIL_ADDRESS>Department of Applied Physics, KTH
Royal Institute of Technology, SE-10691 Stockholm, Sweden  Fabio Lingua
Department of Applied Physics, KTH Royal Institute of Technology, SE-10691
Stockholm, Sweden  Shan W. Jolin  IQM Quantum Computers, FI-02150 Espoo,
Finland  David B. Haviland  Department of Applied Physics, KTH Royal Institute
of Technology, SE-10691 Stockholm, Sweden

###### Abstract

Control over the coupling between multiple modes of a frequency comb is an
important step toward measurement-based quantum computation with a continuous-
variable system. We demonstrate the creation of square-ladder correlation
graphs in a microwave comb with 95 modes. The graphs are engineered through
precise control of the relative phase of three pumps applied to a Josephson
parametric oscillator. Experimental measurement of the mode scattering matrix
is in good agreement with theoretical predictions based on a linearized
equation of motion of the parametric oscillator. The digital methods used to
create and measure the correlations are easily scaled to more modes and more
pumps, with the potential to tailor a specific correlation graph topology.

††preprint: APS/123-QED

##  I Introduction

The storage and manipulation of information encoded in quantum states of the
harmonic oscillator is attracting a lot of interest due to its promise of
measurement-based quantum computation (MBQC). Controlling the infinite-
dimensional Hilbert space of this continuous variable (CV) system is motivated
by the speed of control, resilience against decoherence and the ability to
correct errors grimm_stabilization_2020 ; kudra_robust_2022 ; sivak_real-
time_2023 . A reduction in complexity, of both the quantum hardware and the
classical control and readout hardware, further motivates the development of
controlled entanglement in larger CV systems. Seminal works have paved the
way, providing the building blocks of MBQC braunstein_quantum_2005 ;
weedbrook_gaussian_2012 ; pfister_continuous-variable_2019 , and demonstrating
the ability to engineer the structure of quantum correlations in the form of
graph or cluster states menicucci_universal_2006 ; menicucci_one-way_2008 ;
gu_quantum_2009 ; menicucci_graphical_2011 . Multi-frequency mixing processes
in non-linear optical systems are typically used to create these correlations,
described as combinations of multi-mode squeezing and beam-splitter operations
pfister_multipartite_2004 ; bensemhoun_multipartite_2023 .

Multi-modal squeezing at optical frequencies can now be realized on a chip
yang_squeezed_2021 ; jahanbozorgi_spectroscopic_2023  and through spectral and
temporal multiplexing of ultra-fast pulses kouadou_spectrally_2023 . CV
cluster states have been realized in optical frequency combs
menicucci_arbitrarily_2010 ; pysher_parallel_2011 ; cai_multimode_2017 ;
walschaers_tailoring_2018  and with time-multiplexing techniques
menicucci_temporal-mode_2011 ; chen_experimental_2014 ; asavanant_time-domain-
multiplexed_2021 ; du_generation_2023 . Time multiplexing not only facilitates
the realization of CV quantum gates but also supports MBQC protocols
menicucci_temporal-mode_2011 ; larsen_deterministic_2019 ; asavanant_time-
domain-multiplexed_2021 ; du_generation_2023 . Optical frequencies benefit
from the fact that at room temperature the thermal energy $k_{B}T$ is lower
than the photon energy $hf$, but the analog signal processing techniques used
in optics lack the flexibility and precision of digital signal processing. At
optical frequencies it is difficult to easily and independently control the
relative phase of each individual pump, an effective tool for engineering
correlations jing_experimental_2006 ; hung_quantum_2021 ;
petrovnin_generation_2023 ; pfister_cluster_2023 .

Recent advances in digital signal processing at microwave frequencies,
combined with millikelvin cooling technologies, are enabling new platforms for
CV quantum technologies. Phase control in a multimode superconducting
parametric cavity has been used to simulate ladder-like bosonic lattices
hung_quantum_2021 . Multi-mode squeezing has been demonstrated with surface
acoustic wave resonators andersson_squeezing_2022 , Josephson
naaman_synthesis_2022 ; petrovnin_microwave_2023 , and traveling-wave
parametric amplifiers esposito_observation_2022 . Multi-partite entanglement
jolin_multipartite_2023  and cluster states petrovnin_generation_2023  have
been realized with microwave superconducting circuits.

Here we demonstrate the creation of 2D square-ladder correlation graphs of a
microwave frequency comb, generated by three phase-coherent pumps driving a
single Josephson Parametric Amplifier (JPA). The circuit diagram in Fig. 1a
describes the schematic of the JPA, corresponding to an LC harmonic oscillator
with an effective time-varying inductance controlled by the pump signal. We
reveal the correlations through analysis of the scattering of a weak classical
probe tone. By controlling the relative phase of one of the three pumps, we
selectively modify the output correlations. Verification of the scattering
between input and output modes is an important step towards multi-modal
entanglement in a useful form. Mode scattering theory is an effective tool for
predicting entanglement of Gaussian states weedbrook_gaussian_2012 ;
ranzani_graph-based_2015 ; serafini_quantum_2017 ; peterson_parametric_2020 ;
naaman_synthesis_2022 . We use this theory to predict scattering matrices that
we verify through experiments.

##  II Theoretical Framework

The parametrically driven JPA is described by a time-dependent quadratic
Hamiltonian yamamoto_principles_2016

$H=\frac{\omega_{0}}{2}(A^{\dagger}A+AA^{\dagger})+\frac{\omega_{0}}{2}g_{p}(t)[A+A^{\dagger}]^{2},$
(1)

where $A$ ($A^{\dagger}$) is the annihilation (creation) operator of the time-
independent Hamiltonian of the harmonic oscillator with resonant frequency
$\omega_{0}$, and the arbitrary pump waveform
$g_{p}(t)=\sum_{k}A_{k}\cos{(\Omega_{k}+\phi_{k})}$ is composed of multiple
tones with different frequency, amplitude and phase.

Figure 1:  a) Schematic of the scattering experiment. An external time-
dependent flux $g_{p}(t)$ modulates the JPA’s inductance, enabling frequency
mixing, also called intermodulation. A weak signal $\omega_{s}$ at the input
port mixes with the pumps generating several intermodulation products which
are measured at the output port. b) The measurement basis in the frequency
domain. Fourier leakage is eliminated and mode orthogonality is established by
tuning each mode frequency $\omega_{j}$ to fall on the _sinc_ nodes of the
other modes $\omega_{i}$, $i\neq j$, satisfied when the frequency spacing is
equal to the measurement bandwidth $\Delta=1/T$. c) Illustration of the
pumping scheme. The flux pump with three tones (black) at
$\Omega_{k}=2\omega_{0}+4k\Delta$, with $k=\\{-1,0,1\\}$. A weak single-tone
signal (blue) is stepped through all the modes (grey) generating 2nd and 3rd-
order intermodulation products symmetrically around $\Omega_{k}/2$. For the
sake of clarity we draw only 2nd order products (green).

Both classical and quantum-canonical treatments of a quadratic Hamiltonian
lead to the same equation of motion, a Langevin-like equation describing the
field quadratures inside the JPA yamamoto_principles_2016

$\frac{dA}{dt}=-i\tilde{\omega}_{0}A-i\omega_{0}g_{p}(t)(A+A^{\dagger})+\sqrt{\gamma}\;a_{\text{in}}.$
(2)

Here $a_{\text{in}}$ is an external signal input through a transmission line,
$\tilde{\omega}_{0}=\omega_{0}-i\frac{\gamma}{2}$, and $\gamma$ the effective
coupling between the transmission line and the JPA yamamoto_principles_2016 ;
peterson_parametric_2020 . In this work we analyze the dynamics in the
classical regime where the operators $A$ ($A^{\dagger}$) can be treated as
complex numbers $A$ ($A^{*}$).

We design an orthogonal basis of modes for measurement and control, defined
from the duration $T$ of sequential measurement time windows,
$c_{j}(t)=e^{-i\omega_{j}t}/\sqrt{T}$ with $\omega_{j}=2\pi/T_{j}$, with mode
period $T_{j}$ commensurate to $T$ (see Appendix A for details). In the
frequency domain each mode $j$ is represented by a _sinc function_
$\varphi_{j}\equiv\varphi(\omega-\omega_{j})$, equally spaced by $\Delta$ and
centered at $\omega_{j}$, as shown in Fig. 1b. With this choice of basis, the
maximum of each mode is at the nodes of all the other modes, ensuring
orthogonality and preventing leakage between modes.

Representing the field quadratures of Eq. (2) in this mode basis
$A=\sum_{i}a_{i}(t)=\sum_{i}a_{i}c_{i}(t)$
$\left(A^{\dagger}=\sum_{i}a_{i}^{*}c_{i}^{*}(t)\right)$, one obtains a set of
$n$ coupled, linear harmonic balance equations

$\left\\{\begin{array}[]{rcl}i\omega_{i}a_{i}+i\tilde{\omega}_{0}a_{i}+i\sum_{j}g_{ij}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,i}\\\
-i\omega_{i}a_{i}^{*}-i\tilde{\omega}_{0}^{*}a_{i}^{*}-i\sum_{j}g_{ij}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,i}^{*}\end{array}\right.,\;\;\;\forall
i$ (3)

with the terms $g_{ij}$ that couple mode $i$ to mode $j$ determined by the
amplitude and phase of the pumps (see appendix A for details). The system (3)
can be expressed in matrix form with the usual formalism of mode-scattering
theory peterson_parametric_2020 ; naaman_synthesis_2022 ;
petrovnin_generation_2023 ; jolin_multipartite_2023

$-iM\vec{a}=K\vec{a}_{\text{in}}.$ (4)

The solutions to the multi-modal set of equations of motion (4) are readily
available upon inversion of the matrix $M$. Here the mode-coupling matrix $K$
is a diagonal matrix containing the couplings of each mode to the transmission
line, assumed to be constant $K_{ii}=\sqrt{\gamma}$. The components of the
vectors $\vec{a}$ and $\vec{a}_{\text{in}}$ in (4) contain the complex mode
amplitudes as
$\vec{a}=\left(a_{1},a_{1}^{*},a_{2},a_{2}^{*},\dots,a_{i},a_{i}^{*},\dots\right)^{T}$.
The matching condition between the input/output and internal modes,
$\vec{a}_{\text{in}}+\vec{a}_{\text{out}}=K\vec{a}$, leads to the scattering
matrix,

$S=iKM^{-1}K-I.$ (5)

We numerically compute the scattering matrix through Eq. (5) and compare it to
that measured in the experiment.

Figure 2:  Magnitude of the second and third-order intermodulation products as
a function of the phase $\phi_{1}$ of the edge pump at $\Omega_{1}$, and phase
$\phi_{0}$ of the central pump at $\Omega_{0}$. The phase dependence of the
other edge pump at $\Omega_{-1}$ (not shown) is identical to that at
$\Omega_{1}$. Second-order intermodulation products at
$\omega_{\mathrm{I}k}=\Omega_{k}-\omega_{s}$, $k=\\{-1,0,1\\}$, show weak
variation with pump phase. Third-order products at
$\omega_{\mathrm{I}k,m}=\Omega_{k}-\Omega_{m}+\omega_{s}$ ($k\neq m$) show a
strong dependence on pump phase, due to interference in the mixing process
involving two pumps. The experiment (dots) is in good agreement with the
theory (solid lines). For clarity, we only plot some intermodulation products.
Identical phase dependence is observed for: $\omega_{\mathrm{I}-1}$ and
$\omega_{\mathrm{I}1}$; $\omega_{\mathrm{I}1,-1}$ and
$\omega_{\mathrm{I}-1,1}$; $\omega_{\mathrm{I}0,-1}$ with
$\omega_{\mathrm{I}-1,0}$. Figure 3:  Theoretical a) and d) and experimental
b) and e) scattering matrices of a tripled-pumped JPA. The upper (lower)
panels correspond to pump phase $\phi_{1}=0$ ($\phi_{1}=\pi$). We measure 95
modes where mode index $0$ is at half of the central pump frequency,
$\omega_{0}=\Omega_{2}/2$. The magnitude of the scattering is expressed in dB,
normalized to the pump-off case. c) and f) show one column of the experimental
scattering matrix, marked by the white dashed line in b) and e). The black-
dashed marks the $-20~{}\text{dB}$ threshold used to generate the correlation
graphs of Fig. 4.

##  III Experiment and Discussion

The experiment measures the scattering of a weak input signal that
intermodulates with a multifrequency pump inside a Josephson Parametric
amplifier, generating multiple intermodulation products, also known as idlers,
which leak out of the JPA and into the transmission line. The JPA acts as a
parametric oscillator, a lumped-element LC circuit with L being the Josephson
inductance of a SQUID (see Fig. 1a). Flux in the SQUID loop, controlled
through a separate pump port, modulates the microwave oscillator’s inductance
or resonant frequency. In the absence of flux the measured resonance frequency
of the JPA is 7.8 GHz, and with $\Phi_{dc}\approx\pm 0.42\Phi_{0}$ we measure
$\omega_{0}/2\pi\approx 4.2$ GHz. At this DC flux bias we apply microwave
pumps at frequencies $\Omega_{k}\sim 2\omega_{0}$, enabling 3 and 4-wave
mixing of signal and pumps. The signal port is over-coupled to the JPA, giving
a parametric oscillator with loaded quality factor $Q=37.5$, corresponding to
a linewidth $\kappa=2\pi\crossproduct 112$ MHz. A circulator separates the
incoming and outgoing modes, with the latter directed to a cryogenic low-noise
amplifier through a double isolator. A complete schematic diagram of the
measurement set-up is given in appendix B.

The pumps are generated and data are collected with a digital multi-frequency
lock-in that coherently modulates and demodulates at up to 192 frequencies
simultaneouslytholen_measurement_2022 . The frequencies are tuned to be
integer multiples of the measurement bandwidth $\Delta=1/T$, fixing the
frequency resolution and defining the orthonormal set of modes $\\{a_{i}\\}$.
In the experiment described here we choose $\Delta=0.1~{}\text{MHz}\ll\kappa$
and we perform IQ demodulation on a basis of 95 tones spanning 0.95 MHz, all
within the bandwidth of the JPA. We begin with the analysis of scattering with
one and two pumps, where we observe that both the phase of the single pump and
the relative phases between pumps, do not influence the measured scattering
matrix (see Appendix C for details). Control of the relative phase of three
pumps is the key to canceling correlations between modes. We set three pumps
of equal strength at frequencies close to twice the resonant frequency of the
JPA, $\Omega_{k}\simeq 2\omega_{0}+4k\Delta$, with $k=\\{-1,0,1\\}$ and
$\Delta<<\kappa$. The central $k=0$ pump and $\Delta$ determine the
enumeration scheme for the frequencies in the scattering matrix,
$\omega_{j}=\Omega_{0}/2+j\Delta$, with $j\in\mathbb{Z}$.

We inject a weak signal tone at $\omega_{s}$, stepping it through the
different $\omega_{j}$ while listening simultaneously at all 95 frequencies to
determine where the signal scatters, as illustrated in Fig. 1c. The signal
undergoes 3-wave mixing with each pump resulting in the generation of multiple
idlers (2nd-order intermodulation products) at frequencies
$\omega_{\mathrm{I}k}=\Omega_{k}-\omega_{s}$. The signal also undergoes 4-wave
mixing with two pumps to generate 3rd-order intermodulation products at
frequencies
$\omega_{\mathrm{I}k,m}=\Omega_{k}-\omega_{\mathrm{I}m}=\Omega_{k}-\Omega_{m}+\omega_{s}$
(for $k\neq m$). Fig. 2 shows the magnitude of 2nd and 3rd-order
intermodulation products as a function of pump phase, or phase of one pump
with respect to that of the other two. In our specific configuration 3rd-order
intermodulation products involving different pumps coincide in frequency,
leading to constructive or destructive interference, depending upon the pump
phase. Consequently, manipulation of the phase of one pump allows us to
selectively cancel specific products, and thereby engineer correlations in the
output modes. Rotation of the phase of $k=\pm 1$ edge pumps generates
$2\pi$-periodic interference (see fig. 4a) whereas rotation of the central
$k=0$ edge pump demonstrates $\pi$-periodic interference (see fig. 4b). The
theoretical model is in good agreement with the measurements.

Figure 3 shows the scattering matrices, normalized to the zero-pump case,
under conditions of constructive and destructive interference. Again, the
theoretical scattering matrices show good agreement with the experiment, with
the latter limited in resolution by the added noise of the first-stage
amplifier. In fitting the theory we minimize the sum of square deviations
between experimental and theoretical scattering matrices adjusting two
parameters: the common pump strength (mode coupling) $g$ and an effective
transmission line coupling $\gamma$, assumed the same for all frequencies (see
Appendix A.2).

Through analysis of the scattering matrix, we can gain insight into the
structure of correlations. Each column of the S matrix gives the strength of
coupling between that mode and all other modes in the matrix (see Fig 3c and
f). To represent the coupling we construct single-mode correlation graphs
wherein each vertex corresponds to a mode, and edges represent any coupling
between modes that is above a certain threshold. These single-mode graphs are
shown in Figs. 4a and 4b for the cases of constructive and destructive
interference shown in Fig. 2a. Connecting these single-mode graphs we can
visualize the coupling topology, as shown in Figs. 4c and 4d. Interestingly, a
$-20~{}\text{dB}$ threshold, corresponding roughly to our setup’s noise level
when vacuum fluctuations are input, creates three independent correlation
graphs isomorphic to a nearest-neighbor square ladder. Lowering the threshold
an additional 6 dB would result in a square-ladder graph with next-nearest-
neighbor couplings, e.g. mode $1$ in Fig. 4b-d being weakly connected also
with mode $9$ and mode $-7$. The creation of a square-lattice topology of Fig.
4d is interesting as it suggests correlations of a CV cluster state at the
quantum level menicucci_graphical_2011 .

Figure 4:  Single mode correlation graphs at threshold $-20~{}\text{dB}$ for
a) pump phase $\phi_{1}=0$ and b) $\phi_{1}=\pi$. c) and d) show the set of
three connected subgraphs obtained by considering correlations among all
modes. Pump phase $\phi_{1}=\pi$ causes destructive interference which removes
selected edges to create a square-ladder topology.

##  IV Conclusion

Digital signal processing at microwave frequencies was used to engineer mode
correlations in a continuous-variable system. The synchronous excitation and
detection of response at multiple frequencies, all derived from one clock and
tuned to one base frequency, established a set of orthogonal modes and a
global phase reference for control and measurement. Controlling the relative
phases of three pumps driving a Josephson parametric amplifier, we generated
the scattering matrix required to create a square-ladder correlation graph.
Creating this graph on a basis of 95 modes is a significant milestone toward
manipulating the quantum correlations of a cluster state sufficient for
measurement-based quantum computation. Future work intends to explore and
verify entanglement on a square-ladder graph.

The three-pump case represents the simplest non-trivial case where
interference can be exploited to affect the topology of the correlation graph.
The addition of many more pumps, with precise control of the amplitude and
phase of each, is a trivial extension with the digital microwave platform.
However, the complexity of the scattering problem grows rapidly with the
number of pumps. In this situation the simulated scattering matrices become a
useful tool for exploring more complicated pumping schemes. With precise
control of both the phase and amplitude of 95 pump tones, digital signal
processing at microwave frequencies provides hope for scaling continuous
variable entanglement to larger systems and enabling measurement-based quantum
computing.

###### Acknowledgements.

We acknowledge NIST and J. Aumentado for providing the JPA used in this
experiment. We thank P. Hakonen, K. Petrovnin and E. Mukhanova for fruitful
discussions. This work was supported by the Knut and Alice Wallenberg
Foundation through the Wallenberg Center for Quantum Technology (WACQT).

## Author declaration

### Conflict of interest statement

D. B. H. is part owner of the company Intermodulation Products AB, which
produces the digital microwave platform used in this experiment.

### Author contributions

| J.C.R.H. | F.L. | S.W.J. | D.B.H.
---|---|---|---|---
conceptualization | equal | supporting | supporting | equal
data-curation | equal | equal |  |
formal-analysis | supporting | lead | supporting |
funding-acquisition |  |  |  | lead
investigation | lead | supporting | supporting | supporting
methodology | equal | equal | equal | equal
project-administration |  |  |  | lead
resources |  |  |  | lead
software | equal | equal | supporting |
supervision |  |  | supporting | lead
validation | equal | equal | supporting | equal
visualization | equal | equal | supporting | supporting
writing-original-draft | equal | equal | supporting | supporting
writing-review-editing | equal | equal | supporting | equal

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Derivation of the theoretical model

The circuit of the JPA shown in Fig. 1a corresponds to a lumped-element LC
circuit with a non-linear Josephson inductance $L(\phi)=\frac{\phi_{0}}{4\pi
I_{c}|\cos(\pi\phi/\phi_{0})|}$. By Taylor-expanding
$\frac{1}{L(\phi)}\simeq\frac{1}{L_{J}}[1+\kappa_{1}\phi^{2}+\kappa_{2}\phi^{4}]$,
and considering a small time perturbation of $1/L_{J}$ by the pumping signal
$g_{p}(t)$, one obtains the Hamiltonian yamamoto_principles_2016

$H=\frac{1}{2C}q^{2}+\frac{1}{2L_{0}}\phi^{2}+\frac{\omega_{0}}{2}g_{p}(t)\phi^{2}+\kappa_{1}\phi^{4}$
(6)

where $\omega_{0}$ is the resonant frequency of the unperturbed oscillator,
$q$ the charge in the capacitor and $\phi$ the magnetic flux in the dc-SQUID.
Introducing the ladder operators, $A=\frac{\phi+iq}{\sqrt{2}}$ and
$A^{\dagger}=\frac{\phi-iq}{\sqrt{2}}$, one derives the quartic Hamiltonian

$H=\frac{\omega_{0}}{2}\left(A^{\dagger}A+AA^{\dagger}\right)+\frac{\omega_{0}}{2}g_{p}(t)\left[A+A^{\dagger}\right]^{2}+\kappa_{1}\left[A+A^{\dagger}\right]^{4}.$
(7)

The quartic term in (7) is responsible for the Kerr non-linearity in the
equation of motion yamamoto_principles_2016 . In the limit of weak pumping,
the quartic term can be neglected yamamoto_principles_2016 ;
petrovnin_generation_2023  leading to simpler linear dynamics. In this limit,
the Hamiltonian of the JPA reads

$H=\frac{\omega_{0}}{2}\left(A^{\dagger}A+AA^{\dagger}\right)+\frac{\omega_{0}}{2}g_{p}(t)\left[A+A^{\dagger}\right]^{2}$
(8)

Eq. (8) shows a quadratic, time-dependent Hamiltonian where the second term
describes the dependence of the system’s dynamics on the pumping signal
$g_{p}(t)$. From (8) the equations of motion can be readily derived using the
standard canonical approach. By adding the input driving contribution coupled
to the oscillator by a factor $\sqrt{\gamma}$ yamamoto_principles_2016 ;
naaman_synthesis_2022  one recovers the usual Quantum Langevin equations of
motion

$\left\\{\begin{array}[]{lcr}\frac{dA}{dt}&=&-i\tilde{\omega}_{0}A-i\omega_{0}g_{p}(t)(A+A^{\dagger})+\sqrt{\gamma}\;a_{\text{in}}\\\
\frac{dA^{\dagger}}{dt}&=&i\tilde{\omega}_{0}^{*}A^{\dagger}+i\omega_{0}g_{p}(t)(A+A^{\dagger})+\sqrt{\gamma}\;a^{\dagger}_{in}\end{array}\right.$
(9)

where $\tilde{\omega}_{0}=\omega_{0}-i\frac{\gamma}{2}$, and $\gamma$
describes the losses due to the coupling to the transmission line. In the
classical limit, $A^{\dagger}\equiv A^{*}$, the equations of motion become a
system of two, coupled, linear equations in the complex-valued functions of
time $A\equiv A(t)$ and $A^{*}\equiv A^{*}(t)$.

Figure 5: Schematic of the measurement setup including room temperature and
cryogenic electronics.

Let us now define an orthonormal set of frequency modes
$c_{i}(t)=e^{-i\omega_{i}t}/\sqrt{T}$, properly defined from the time interval
of the measurement $t\in[0,T]$, together with the scalar product that ensures
the orthogonality condition

$\int_{T}dt\,c_{i}^{*}(t)c_{j}(t)=\delta_{ij}.$ (10)

For the orthogonality condition to be satisfied, modes frequencies
$\omega_{i}$ would need to feature a period of oscillation
$T_{i}=2\pi/\omega_{i}$ commensurate to $T$ (i.e. $T=nT_{i}$ for
$n\in\mathbb{N}$). By expanding $A$ and $A^{*}$ in the frequency-mode basis

$A(t)=\sum_{i}a_{i}(t)=\sum_{i}a_{i}c_{i}(t),\;\;\;\;A^{*}(t)=\sum_{i}a_{i}^{*}c_{i}^{*}(t)$
(11)

where $a_{i}=\int_{T}dt\,c_{i}^{*}(t)A(t)\in\mathbb{C}$ are the complex
amplitudes of mode $i$; the Langevin equations become

$\sum_{i}\Big{[}\frac{da_{i}(t)}{dt}+i\tilde{\omega}_{0}a_{i}(t)+i\omega_{0}g_{p}(t)\left(a_{i}(t)+a^{*}_{i}(t)\right)\\\
-\sqrt{\gamma}\;a_{\text{in},\,i}(t)\Big{]}=0.$ (12)

For the sake of brevity, we omitted the complex conjugate equation in (12),
but it is intended to be part of the system of equations. By Fourier
transforming Eq. (12), the time derivatives $\frac{d}{dt}\rightarrow i\omega$,
and the system of equations of motion becomes

$\sum_{i}\left[i\omega
a_{i}(\omega)+i\tilde{\omega}_{0}a_{i}(\omega)+i\omega_{0}\,g_{p}(\omega)*\left(a_{i}(\omega)+a^{*}_{i}(\omega)\right)\right.\\\
\left.-\sqrt{\gamma}\;a_{\text{in},\,i}(\omega)\right]=0.$ (13)

where $a_{i}(\omega)=a_{i}\varphi_{i}(\omega)$. $\varphi_{i}(\omega)$ is the
frequency domain representation of mode $i$, which is the Fourier transform of
a cosine function of frequency $\omega_{i}$ multiplied by a square window of
length $T$, namely a _sinc_ function centered around $\omega_{i}$. As shown in
Fig. 1b, if the integer proportionality between the mode’s periodicity $T_{i}$
and window’s length $T$ is chosen, the maximum of each mode $j$ sits on the
_sinc_ nodes of the other modes. This choice eliminates spectral leakage
between modes and sets the mode spacing and frequency resolution of the
measurement basis to $\Delta=1/T$. In practice we achieve these conditions by
tuning the mode frequencies in relation to the sampling clock of our digital
microwave platform.

The product in the time domain (12) becomes a convolution in the frequency
domain (13). Let us consider the pumping signal as composed of a sum of pure
frequency tones

$g_{p}(t)=\sum_{k}A_{k}cos\left(\Omega_{k}t+\phi_{k}\right)=\sum_{k}g_{k}e^{i\Omega_{k}t}+g_{k}^{*}e^{-i\Omega_{k}t}$
(14)

where $g_{k}=\frac{1}{2}A_{k}e^{i\phi_{k}}\in\mathbb{C}$ represents the
complex pump strength. The Fourier transform of Eq. (14) is

$g_{p}(\omega)=\sum_{k}g_{k}\delta(\omega-\Omega_{k})+g_{k}^{*}\delta(\omega+\Omega_{k}).$
(15)

Exploiting the convolution properties of the Dirac’s deltas
$a_{i}(\omega)*\delta(\omega\pm\Omega_{k})=a_{i}\varphi_{i}(\omega\pm\Omega_{k})$
we eliminate the convolution in Eq. (13). If $\Omega_{k}$ and $\omega_{i}$ are
commensurate such that $\Omega_{k}+\omega_{i}=\omega_{j}$, that is,
$a_{i}(\omega)*\delta(\omega\pm\Omega_{k})=a_{i}\varphi_{j}(\omega)$, one
obtains mode index exchange petrovnin_generation_2023 . This implies the
appearance of mixed terms $a_{i}\varphi_{j}(\omega)$ with $i\neq j$ in the
equations of motion. However, by factoring out of the sum on $p$ the terms
with the same index $j$, one can rewrite the equations of motion as

$\sum_{i}\left[(i\omega+i\tilde{\omega}_{0})a_{i}\varphi_{i}(\omega)+i\sum_{j}\left(g_{ij}a_{i}+g_{ij}^{\prime}a_{i}^{*}\right)\varphi_{j}(\omega)\right.\\\
\Bigg{.}-\sqrt{\gamma}\;a_{\text{in},\,i}\varphi_{i}(\omega)\Bigg{]}=0.$ (16)

where $g_{ij}$ ($g_{ij}^{\prime}$) are the couplings between $a_{i}$
($a_{i}^{*}$) and $\varphi_{j}$. Depending on the pump frequencies
$\Omega_{k}$ connecting mode $i$ to mode $j$, the coupling takes the form
$g_{ij}=\omega_{0}g_{k}$ ($g_{ij}=\omega_{0}g_{k}^{*}$) if connected by
$+\Omega_{k}$ ($-\Omega_{k}$). Notice that couplings $g_{ij}$ and
$g_{ij}^{\prime}$ may differ, depending on the particular pumping scheme
adopted. Writing explicitly the two sums in Eq. (16) and grouping together
terms sharing the same $\varphi_{j}(\omega)$, one can rewrite Eq. (16) as

$\sum_{m}\left[(i\omega+i\tilde{\omega}_{0})a_{m}+i\sum_{\ell}\left(g_{\ell
m}a_{\ell}+g_{\ell m}^{\prime}a_{\ell}^{*}\right)\right.\\\
\Bigg{.}-\sqrt{\gamma}\;a_{\text{in},\,m}\Bigg{]}\varphi_{m}(\omega)=0.$ (17)

Equation (17) is solved if each term in square brackets is zero. Integrating
over $\omega$ we recover the system of $2N$-coupled linear equations Eq. (2),
where $N$ is the number of measured modes.

$\left\\{\begin{array}[]{rcl}i\omega_{i}a_{1}+i\tilde{\omega}_{0}a_{1}+i\sum_{j}g_{1j}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,1}\\\
-i\omega_{i}a_{1}^{*}-i\tilde{\omega}_{0}^{*}a_{1}^{*}-i\sum_{j}g_{1j}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,1}^{*}\\\
i\omega_{i}a_{2}+i\tilde{\omega}_{0}a_{2}+i\sum_{j}g_{2j}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,2}\\\
-i\omega_{i}a_{2}^{*}-i\tilde{\omega}_{0}^{*}a_{2}^{*}-i\sum_{j}g_{2j}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,2}^{*}\\\
&\vdots&\\\
i\omega_{i}a_{i}+i\tilde{\omega}_{0}a_{i}+i\sum_{j}g_{ij}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,i}\\\
-i\omega_{i}a_{i}^{*}-i\tilde{\omega}_{0}^{*}a_{i}^{*}-i\sum_{j}g_{ij}(a_{j}+a_{j}^{*})&=&\sqrt{\gamma}\;a_{\text{in},\,i}^{*}\end{array}\right.$
(18)

In Eq. (18) we take $g_{ij}=g_{ij}^{\prime}$ for simplicity, but this is not
the general case. When integrating over $\omega$ we use the fact that in the
limit $\Delta\rightarrow 0$ integrals
$\int\omega\varphi_{j}(\omega)\,d\omega\approx\int\omega\delta(\omega-\omega_{j})\,d\omega=\omega_{j}$
and $\int\varphi_{j}(\omega)\,d\omega=1$.

A standard convention in literature is to derive the above set of equations
within the Rotating Wave Approximation (RWA) peterson_parametric_2020 ;
naaman_synthesis_2022 ; petrovnin_generation_2023 . The time dependence of the
pump is eliminated by moving to a reference frame co-rotating with the pump
(redefining the ladder operators $A\rightarrow
A\,e^{\,i\frac{\Omega_{k}}{2}t}$) yamamoto_principles_2016 . While this is
perfectly valid in the case of a single pump, it becomes more difficult to
interpret when multiple pump frequencies are introduced. We prefer the
approach adopted above relying on the convolution properties of delta
functions, without the need to introduce a change of reference frame.

Figure 6: Theoretical a) and experimental b) scattering matrices for the case
of one parametric pump at $\Omega_{k}\approx 2\omega_{0}$. Panels c) and d)
show the single-mode correlation and total correlation graphs respectively.

###  A.1 Definition of the Scattering Matrix

We write the linear system of equations (18) in the matrix form following the
standard formalism of mode scattering theory peterson_parametric_2020 ;
naaman_synthesis_2022 ; jolin_multipartite_2023 ; petrovnin_generation_2023

$-iM\vec{a}=K\vec{a}_{in}.$ (19)

The solutions are readily available upon inversion of matrix $M$. The mode-
coupling matrix $K$ is a diagonal matrix containing the wave-guide couplings
for each mode (assumed to be constant $K_{ii}=\sqrt{\gamma}$ $\forall i$),
while $\vec{a}$ and $\vec{a}_{in}$ are the vectors containing complex
amplitudes, e.g.
$\vec{a}=(a_{1},a_{1}^{*},a_{2},a_{2}^{*},\dots,a_{i},a_{i}^{*},\dots)^{T}$.

The scattering matrix $S$ is defined as the input-output relationship

$\vec{a}_{out}=S\cdot\vec{a}_{in}$ (20)

By considering the mode matching condition
$\vec{a}_{in}+\vec{a}_{out}=K\vec{a}$ between the mode basis internal to the
JPA and the external input/output modes of the transmission line
peterson_parametric_2020 ; naaman_synthesis_2022 , one retrieves the usual
definition of the scattering matrix:

$S=iKM^{-1}K-I.$ (21)

###  A.2 Fitting experimental and theoretical $S$ matrices

Figure 7: Theoretical a) and experimental b) scattering matrices for the case
of two parametric pumps at $\Omega_{k}\approx 2\omega_{0}\pm 2\Delta$. Panel
c) shows the single-mode correlation graph obtained by setting a threshold at
$-20~{}\text{dB}$. Panels d)-f) show the three connected subgraphs that arise
in the global correlation scheme.

To compare the numerical solution of the theoretical model with the
experimental results, we normalize both theory and experimental data to the
respective zero-pump case. Throughout this work we considered the balanced
case where all the pump signals are tuned to the same strength
$|g_{-1}|=|g_{0}|=|g_{1}|=g$. To compare the theoretical model (18) with our
experiment, it is sufficient to treat $g$ and $\gamma$ as free parameters and
fit their values. The fit is performed by solving Eq. (18) for different
values of $g$ and $\gamma$ to determine a theoretical $S_{T}(g,\gamma)$, which
is compared to the experimental scattering matrix $S_{E}$. By minimizing the
distance function

$d(g,\gamma)=\sqrt{\sum_{ij}|S_{E,\;ij}-S_{T}(g,\gamma)_{ij}|^{2}},$ (22)

we find the best-fit $g$ and $\gamma$. For three parametric pumps, we found
that the minima of (22) lie along a line $\omega_{0}g/\gamma\approx 0.14$. Our
assumption of frequency-independent coupling, a valid approximation when all
modes are well within the bandwidth of the JPA, makes the fitting procedure
sensitive to only the ratio $g/\gamma$.

##  Appendix B Measurement setup

A schematic of the measurement setup is shown in Fig. 5. The JPA is cooled at
$10~{}\text{mk}$ in a Bluefors LD250 dry dilution refrigerator with
appropriate attenuation of the pump and signal at in multiple stages. We
operate the JPA in reflection with a circulator to separate the input and
output fields. The measurement signal-to-noise ratio is set by the first High-
Electron-Mobility Transistor (HEMT) amplifier at $4~{}\text{K}$. A double
isolator reduces the amplifier noise backacting on the sample.

A multi-frequency lock-in amplifier tholen_measurement_2022  provides the
probe signal and the multi-frequency pump through separate output ports which
are synchronous with the input measurement port. It also provides the DC bias
for the JPA which is combined with the pump in a diplexer at base temperature.
The lockin uses direct digital synthesis to create the signal in the band
$3-6~{}\text{GHz}$ and the multi-frequency pump in the band
$5-10~{}\text{GHz}$ with high-speed digital-to-analog converters working in
the 2nd Nyquist zone. Similarly, input signals appearing in a higher Nyquist
zone of the analog-to-digital converter are under-sampled, appearing at
aliased frequencies in the baseband. Analog filters at room temperature select
the appropriate Nyquist zone. In this way we perform digital signal processing
at microwave frequencies without analog IQ mixers. This direct digital
approach is essential for maintaining the orthogonality of the mode basis and
eliminating Fourier leakage between modes.

##  Appendix C One and two parametric pumps results

As a prequel to the more complex three-frequency pump, it is important to
understand and verify the more simple single and double pump case. We use the
same methodology explained in section III and the same experimental setup.
Results shown here for one and two pumps are consistent with the covariance
matrix analysis published in previous work jolin_multipartite_2023 .

A single monochromatic pump applied at roughly twice the resonant frequency of
the JPA $\Omega_{k}\approx 2\omega_{0}$ enables 2nd order intermodulation
between the pump and signal, generating intermodulation products (idlers)
$\omega_{j}=\Omega_{k}/2+j\Delta$, with $j\in\mathbb{Z}$ (three-wave mixing).
The scattering matrix for the single pumped JPA is shown in Fig. 6 where we
see the anti-diagonal mixing products. From the scattering matrix we also
generate the single-mode correlation graph shown in Fig. 6c. There are only
pair-wise connections in the generalized graph Fig. 6d. We verify that the
scattering matrix is independent of the phase of the pump.

The frequencies of the bichromatic pump are roughly centered at twice the JPA
resonance and detuned by twice the spacing of the mode basis,
$\Omega_{k}=2\omega_{0}\pm 2\Delta$. Our mode basis becomes
$\omega_{j}=\frac{1}{4}\left(\Omega_{1}+\Omega_{2}\right)+j\Delta$, with
$j\in\mathbb{Z}$. In Fig. 7 we show the scattering matrix for the double-
pumped JPA. This pumping scheme creates single-mode correlations shown in Fig.
7c), where each mode is connected to two mixing products. Generalizing the
single-mode graph over all modes in the scattering matrix, we find three
independent linear subgraphs shown in Fig. 7d-e. We verify that the scattering
matrix is independent of the relative phase of the two pumps. Experimental
results, in both cases, show good agreement with our theoretical predictions.

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# Using Harmonics for Low-Cost Jamming

Vasilis Ieropoulos  Vasilis Ieropoulos 1,*,Eirini Anthi2 1Cardiff University,
School of Computer Science & Informatics, Cardiff, UK *Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

The digitalisation of the modern schooling system has led to multiple schools
and organisations buying similar hardware. Electronic equipment like wireless
microphones, projectors, touchscreen displays etc., have been almost
standardised with a few well-known brands leading the market. This has led to
the adoption of common frequency ranges between brands with many sticking
between 600-670 MHz. The popularity of low-cost computing devices like the
Raspberry Pi which has been used in a plethora of applications has also taken
the path of being used as low-cost transmitters. There have been many
implementations where the Raspberry Pi has been used as the target device but
few cases where the PI is the actual threat. In this paper, we explore the use
of the Raspberry Pi as a stealth radio frequency jamming device to disable
wireless conference microphones. Harmonics were used to achieve frequencies
outside the Pi’s transmission frequency by taking advantage of its unfiltered
transmission.

###### Index Terms:

Internet of Things (IoT), Open-Source, Radio Frequency, Software Defined Radio

##  I Introduction

The uses of wireless technologies has risen substantially in the past few
years, and the adoption of wireless devices with integrated sensors has become
ever more prevalent in the modern age[1]. Many industries have become
dependent on wireless devices for monitoring buildings, infrastructure,
health, organizations etc[2]. A large portion of these devices are part of an
interconnected network, with many of them being portable or located in places
where hard-wiring is not an option. Therefore, it is unavoidable that these
devices are dependent on radio communication for the exchange of data. The
knowledge on how to perform radiofrequency attacks has usually been confined
to a small group of people with very specific knowledge on the subject, as
well as the use of specialized hardware[3]. This makes it difficult to assess
security vulnerabilities that are present in RF technology, as the time and
financial constraints of third-party contractors hired to assess these are
limited[3]. This paper focuses on the uses of a low-cost single-board computer
like the Raspberry Pi to perform a stealthy jamming attack on a wireless
conference microphone. The paper emphasizes the ease of use of open-source
software to achieve this, as well as the low cost of the overall platform.
Furthermore, the use of Harmonics is used to overcome the physical limitations
of the device itself by using basic physics to exploit the otherwise useless
harmonic emissions of unfiltered signal transmissions. Using a combination of
low-cost hardware and the exploitation of the unfiltered signals produced by
the device allows for simple, yet effective RF penetration.

This paper is structured as follows: Section II discusses related work.
Section III describes the methodology and testing of the experiment. Section
IV covers mitigation strategies, while Section VI outlines plans for future
work and improvements.

##  II Related Work

The field of Radio Frequency penetration testing is complex but interesting
and is usually done using very expensive equipment. In the case of Jean-Michel
Picod et al.,[4] an Ettus USRP B210 board was used to perform an RF
vulnerability assessment of different wireless devices like Zigbee sensors.
The use of such a device, although it is effective, requires plenty of
training to use, as well as the high price tag compared to more low-cost
solutions makes it less appealing to the average hobbyist. The Raspberry Pi
has been used in penetration testing mostly for network-related attacks as
demonstrated by Maryna Yevdokymenko et al.[5] which used the Pi along with
Kali Linux to perform network penetration tests. The size of the Pi makes it
perfect for stealth attacks as its low profile makes it hard to notice, and
its low power consumption allows it to be run using only a power bank. The Pi
is no stranger to being used as a wireless network tool as well as
demonstrated by Dr Chafic BouSaba et al.[6] who also attached a 16×2 Character
LCD and Keypad Kit. This allowed information to be displayed to the user
directly from the Pi without the need for a remote connection. Hence, this
opens up the Pi for automated scripts that can be executed without physically
accessing it. Therefore, there exists the possibility of using the Raspberry
Pi as a portable, standalone and stealthy RF penetration tool.

Paper | Equipment | Use of external equipment |  |
---|---|---|---|---
[4] | USRP | Yes |  |
[5] | Rpi, Usb dongle | Yes |  |
[6] | Pi, External Screen | Yes |  |
This Paper | Pi | No |  |

###  II-A Harmonics

Harmonics are produced by every oscillating wave, with each level of the
harmonic being a different frequency. Harmonics have a reduction of bandwidth
with every iteration, while also reducing their power[7]. They can be
calculated using the following formula:

$\mathbf{Harmonic=2\cdot\pi\cdot\textit{f}\cdot n}$ (1)

where f is the frequency of the carrier and n is the number of the harmonic.

The advantage of using harmonics, in this case, is that they can be used to
send signals outside the 500MHz range, assuming no filter is attached.
Furthermore, they can cause the capacitance level of power factor capacitors
to fluctuate and the inductance of the power supply to become unstable,
causing the device or neighbouring devices to behave irregularly[8]. Spurious
emissions are also one of the main reasons for using harmonics, as they can be
used to create a wide range of frequencies close to the fundamental frequency
of oscillation, which can cause interference. Many of these spurious emissions
are generated by Low Noise Amplifier embedded in the devices, which are used
to amplify the low powered signal transmitted from the device.

###  II-B rpitx

rpitx is a general radio frequency transmitter created by Evariste
Courjaud(F5OEO)[9] for the Raspberry Pi which doesn’t require any other
hardware to operate, but a filter is recommended to avoid unwanted
interference. It can handle frequencies from 5 kHz up to 1500 MHz. It uses the
GPIO pins of the Raspberry Pi to control the frequency and the power of the
transmitter. Specifically, it uses GPIO4 or pin 7 of the GPIO header. The
signal being transmitted is unfiltered which means that a bandpass filter is
required for it to be used without causing interference. It allows the Pi to
transmit using multiple modulations types such as AM, FM, and SSB. Observing
figure 1 it demonstrates where the short piece of wire can be installed on the
GPIO header.

Figure 1: Raspberry PI schematic

The early versions of the software only allowed for transmissions up to
500MHz, which made testing out the hypothesis difficult. This meant a
different approach had to be taken for a more out-of-the-box solution. Hence,
using harmonics made sense as it would allow the transmission of signals
outside the 500Mhz range along with the Raspberry Pi’s unfiltered signal
transition. Even though sending unfiltered RF is dangerous, the low power of
the Raspberry Pi allows us to harness the harmonic transmissions in a
meaningful way in a controlled environment.

##  III Testing and Methodology

###  III-A Methodology

The model number and the frequency needed to be identified. Photographic
material was taken of the device, identifying it as a ”SHURE QLXD1” with a
frequency range between 606-670 MHz. This was clearly labelled on the back of
the device, as shown in figure2.

Figure 2: Microphone Backplate

The next step was to identify which was the frequency of operation of the
device by scanning the frequency range using GQRX and the SDR. The frequency
was identified as 614 MHz. We continued monitoring the frequency before and
after the device was turned off and on during use. It was identified that
there seemed to be an initial handshake packet that was used to establish a
connection between the microphone and the wireless receiver. This was also
used for establishing frequency hopping between the two devices. As
demonstrated in figure 3 white noise was created using audacity as a sample
file. This allowed us to create the widest possible signal possible, which
would allow the transmission to cover a large portion of the band equally.

Figure 3: White Noise Generation

This potentially could give us the best results. While monitoring the signal,
it was observed that the transmission was using FM modulation, hence the
sample file had to be converted to accommodate for this. This was done using
the following command:

⬇

./pifm sample.wav fm.ft

This converted the WAV file into an FM-modulated sound file, which allows
rpitx to transmit it in this fashion.

###  III-B Testing

Since the microphone was operating at 614Mhz, which was outside the operating
frequencies of the software. An out-of-the-box solution had to be used. Hence,
the Raspberry Pi was set up to operate at 304 MHz which would create a second
harmonic at 614 MHz, allowing us to attack the frequency of operation. To test
our hypothesis, we started transmitting the sound file while the microphone
was in use. This did not affect the microphone, as the handshake signal which
was initiated in the beginning also initiated some form of frequency hopping,
making it difficult for the Raspberry Pi to stop or interfere with the
transmission. We ran the test a second time but in this case, started
transmitting before the microphone was connected to the receiver. This caused
the microphone to not be able to connect to the receiver as long as we had the
Raspberry Pi transmitting. The following was also attempted but resulted in
the microphone never connecting:

  1. 1.

Turning both devices on and off again

  2. 2.

Hard resetting the devices by removing the batteries

To further expand the testing, the transmission was also done on different
frequencies which produced harmonics at different increments but still on the
frequency of operation of the microphone. As demonstrated in Table I the
experiment was successful until the fifth harmonic, after that, the reduction
in power and bandwidth did not allow for any further results.

Frequency | Harmonic | Result
---|---|---
304 MHz | 2nd | ✓
204.7 MHz | 3rd | ✓
153.5 MHz | 4th | ✓
102.4 MHz | 5th | ✗
TABLE I: Harmonics Tested

##  IV Mitigation Strategies

Although this is quite a complex topic with a simple implementation, the
following are some of the mitigation strategies that can be used to mitigate
the effects of jamming attacks. One of the main mitigation strategies is to
use a filter to avoid interference[10]. This can be done by using a bandpass
filter to filter out the unwanted frequencies, as well as using an attenuator
to attenuate the signal[10]. Even though in this case the Raspberry Pi managed
to stop the microphone from initiating the handshake signal, thus stopping it
from working, there are solutions to try to reduce the chance of this
happening. For example, the charging dock could use the physical pins on the
back of the microphone transmitter to initialise the handshake signal
directly, instead of relying on the wireless capabilities of the
transmitter[11]. This would eliminate the chance of an outside source
interfering with the transmission.

##  V Conclusion

The substantial increase in wireless devices over the past decade has made
them an attractive target for malicious threat actors. The use of wireless
devices has also made it easier for malicious actors to target them, as they
can be easily located by using a wireless scanner such as an SDR. This makes
it easier for hackers to target the devices and then exploit them for their
own purposes. The intimidating part about pure radio frequency attacks is the
difficulty of detecting that an attack is happening, especially in the form of
sniffing. The Raspberry Pi has proved itself to be a very reliable device for
wireless attacks, as well as being used for more sophisticated attacks such as
replay attacks and packet manipulation. Its low cost makes it an excellent
solution for hobbyists and tinkerers alike. In our scenario, the Raspberry Pi
was used to jam the communications of a wireless conference microphone, which
could disrupt the talk or conference in motion. This could be expanded into
eavesdropping on confidential or private meetings which use such devices,
which as demonstrated are transmitting unencrypted voice transmissions. The Pi
can be concealed easily due to its small size, making it a potential threat in
the next RF attack.

##  VI Future Work

The experiment does not come without its limitations. The newer versions of
the RPTIX software allow for transmissions of up to 1.5 GHz, which gives the
user more freedom to explore the vulnerabilities in the spectrum and to use
the device for more than just jamming. More sophisticated attacks using a
cheap software-defined radio could be used for replaying packets back to
devices, as well as opening up the possibility of sniffing and packet
manipulation. At the time of writing, the Raspberry Pi 4 is available and
might be more effective in these types of attacks. Furthermore, some boards
take advantage of the GPIO pin, giving the user the ability to connect an
antenna using an SMA connector with embedded filtering on the board. One such
example is the CaribouLite RPi HAT[12] which can tune up to 6GHz, this would
allow the Pi to be used in more high-frequency scenarios like Wi-Fi[13],
Bluetooth[14], Zigbee[15] etc.

## References

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