:  Sub-Nyquist Sampling for TDR Sensors:

Finite Rate of Innovation with Dithering

Marc Ihle, Hochschule Karlsruhe, Germany

Who We are

Bashar Ahmad

Thomas Weber

Marc Ihle

: Marc Ihle (17.09.2013) 2

 Presentation Outline

:  Introduction – TDR Sensor

:  Problem Formulation

:  FRI and the Proposed Approach

:  Description of the System

:  Simulations

:  Conclusion

: Marc Ihle (17.09.2013) 3

Time Domain Reflectometry Sensor
(Guided Wave Radar Level Sensor)

Aim: to measure liquid level in an industrial container by measuring ToF.

Reflection coefficient:
Z1 ! Z 0 a
R=
Z1 + Z 0

The processed signal (K pulses): b

K!1
x(t) = " ai p(t ! ti )
i=0 c

ai: amplitude of the reflected pulse.

ti: location of the reflected pulse.

Gaussian pulses are typically used

with given σ values.

: Marc Ihle (17.09.2013) 4

TDR: An Example

Sensing requirements:
TDR-Level Sensor
LFP Cubic; SICK AG
Requirement Value

Measuring Range 5 cm ... 10 m

Inaccuracy < 5mm
Resolution < 0.5 mm
Response Time < 100 ms

è Maximum tolerated relative ToF measurement error is:

s
terror = = 33 ps
c

: Marc Ihle (17.09.2013) 5

Problem Formulation and Proposed Approach
1
Original signal without noise and estimated pulse positions

Estimated pulse positions with Cadzow algorithm

0.5

0
Classical Nyquist sampling demands collecting several Giga samples per second.
ï0.5
‒  Infeasible due to practical
5 10 15 20
SWPaC limitations
25 30 35 40 45 50
t [ns] of miniature TDR sensors.
Signal x(t) with noise and interferences, SNR = Inf dB
1

0.5

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2
Alternative sub-Nyquist
0.04 techniques:
y(t) Output Sampling kernel

−  Equivalent Time
Sampling points
0.02
Sampling: Bulky sensitive circuits (PLL) and long signal acquisition times.
0
−  Compressed Sensing: Infinite time resolution and high SWPaC implementation.
ï0.02

−  Finite Rate of Innovation: Can be easily integrated

5 10 15 20 25 30 35 40 45 50
t [ns] into existing TDR sensor architecture.

FRI is an effective solution to the data acquisition problem in TDR sensors.

FRI Limitation: Very sensitive to quantisation noise and high resolution ADCs cannot be used,
e.g. due to TDR sensor practical limitations.
Proposed Approach: FRI with dithering and averaging to combat quantisation noise.
: Marc Ihle (17.09.2013) 6
 System Description: Proposed Approach

Implementation using FRI with Dithering and Averaging

:  Ensemble averaging of consecutive sequences shall improve the ADC resolution.

:  Averaging may lead to a slightly increased response time.

: Marc Ihle (17.09.2013) 7

 Description of the System

Signals along the path

: Marc Ihle (17.09.2013) 8

 Description of the System
ï0.5
5 10 15 20 25
t [ns]
30 35 40 45 50

Signal x(t) with noise and interferences, SNR = Inf dB

1

0.5
Signals along the path
0

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2
0.04
y(t) Output Sampling kernel
0.02 Sampling points

ï0.02
5 10 15 20 25 30 35 40 45 50
t [ns]

: Marc Ihle (17.09.2013) 9

 Description of the System

Signals along the path

: Marc Ihle (17.09.2013) 10

 Description of the System

Signals along the path

: Marc Ihle (17.09.2013) 11

 Description of the System

Signals along the path

Original signal without noise and estimated pulse positions
1
Estimated pulse positions with Cadzow algorithm
0.5

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Signal x(t) with noise and interferences, SNR = Inf dB
1

0.5

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2
0.04
y(t) Output Sampling kernel
0.02 Sampling points

ï0.02
5 10 15 20 25 30 35 40 45 50
t [ns]

: Marc Ihle (17.09.2013) 12

 Description of the System

Signals along the path

Original signal without noise and estimated pulse positions

1
Estimated pulse positions with Cadzow algorithm
0.5

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Signal x(t) with noise and interferences, SNR = Inf dB
1

0.5
: Marc Ihle (17.09.2013) 13
 Data Acquisition Device

aperture limitation:
ADC selection thermal limitation
equivalent when
using averaging Heisenberg
limit
:  Resolutions > 8bit are expensive
for fs < 1ns.
-0.5bit/oct. -1.0bit
/ oct.

:  High-speed ADCs are mainly

limited by the aperture jitter.
-0.5bit/oct.
:  Averaging adjacent samples is not
efficient; ensemble averaging
however is.

Graph taken from: “Sigma-Delta Modulators: Tutorial Overview, Design Guide, and State-of-the-Art Survey“; IEEE Trans. on Circuits and Systems, Vol. 58, No. 1, Jan. 2011
Limitations according: R. H. Walden: “ADC Survey and Analysis”, IEEE Journal on Selected Areas in Communications, Vol. 17, No. 4, April 1999

: Marc Ihle (17.09.2013) 14

Monte Carlo Simulations – Set Up
Signal Model:
:  K = 5 Gaussian Pulses with σ = 200 ps, each. Pulse sequence with 0 dB dynamic range:
:  Period of the pulse sequence is 50 ns.
:  Two dynamic ranges are examined: 0 dB and 26 dB. 1
Original signal without noise and estimated pulse positions

Estimated pulse positions with Cadzow algorithm

0.5

FRI: ï0.5

:  Sum of Sincs (SoS) sampling kernel is used.

5 10 15 20 25 30 35 40 45 50

Pulse sequence
Signal x(t) with
with noise 26 dB dynamic
t [ns]
and interferences, SNR = Inf dB range:
1

:  Cadzow plus total least squares are applied. 0.5

:  FRI minimum sampling rate is 220 MHz. 0

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2

Dithering: 0.04
y(t) Output Sampling kernel
Sampling points

:  Uniform distributed dither is used.

0.02

:  Maximum dithering amplitude is ±Q/2. ï0.02

5 10 15 20 25
t [ns]
30 35 40 45 50

Assessment:
:  Maximum error and RMS error are used to assess the results accuracy.
:  In practise the maximum error is more important.
: Marc Ihle (17.09.2013) 15
 Simulations
Effect of ADC Resolution 5
10
Max. Error ï without dithering
RMSE Error ï without dithering
Max. Error ï with dithering
RMSE Error ï with dithering
(0 dB dynamic range) random guess
4
10

:  Errors of more than 10 ns

correspond to random
guesses. t (ps)
3
10

:  ADC resolution of at least

10 bits is needed.
2
10

Simulation parameters:
sampling rate: fs = 440 MHz 1
10
oversampling: β = 2 6 7 8 9 10 11 12
ADC Resolution in Bit
13 14 15 16

averaging: 250 times

: Marc Ihle (17.09.2013) 16
 1
Simulations
Original signal without noise and estimated pulse positions

Estimated pulse positions with Cadzow algorithm

0.5

Effect of ADC Resolution

ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Signal x(t) with noise and interferences, SNR = Inf dB 5
1
10
0.5
Max. Error ï without dithering
0
RMSE Error ï without dithering
ï0.5
5 10 15 20 25
t [ns]
30 35 40 45 50
Max. Error ï with dithering
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2
0.04
RMSE Error ï with dithering
(26 dB dynamic range)
0.02
y(t) Output Sampling kernel
Sampling points
random guess
0 4
10
: 
ï0.02
26 dB dynamic range
5 10 15 20 25
t [ns]
30 35 40 45 50

causes the RMSE time

resolution to decrease by
a factor of 2 to 3. t (ps)
3
10
:  random guesses occur 3.0
5.5
with ADC resolutions of
up to 10 bits.
:  maximum error notably 2
10 2.5
increases by 5.5.

Simulation parameters:
2.0
sampling rate: fs = 440 MHz 1
10
oversampling: β = 2 6 7 8 9 10 11 12
ADC Resolution in Bit
13 14 15 16

averaging: 250 times

: Marc Ihle (17.09.2013) 17
 1
Simulations
Original signal without noise and estimated pulse positions

Estimated pulse positions with Cadzow algorithm

0.5

Effect of Averaging
ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Signal x(t) with noise and interferences, SNR = Inf dB
1 5
10
0.5
Max. error: Dithering, no averaging
0
Max. error: Dithering, 125 averages
ï0.5
5 10 15 20 25 30 35 40 45 50 Max. error: Dithering, 250 averages
random guess
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2 Max. error: Dithering, 2000 averages
0.04
y(t) Output Sampling kernel
Sampling points 4 RMS error: Dithering, no averaging
0.02
10 RMS error: Dithering, 125 averages
: 
0

ï0.02 Averaging 125 estimates

5 10 15 20 25 30 35 40 45 50
RMS error: Dithering, 250 averages
RMS error: Dithering, 2000 averages
enhances the RMSE
t [ns]

time resolution by factor 3

10
of 200. t (ps) 200
:  A further increase of the
number of averages to 2
10
200
2000 enhances the 3.0 /1.5
RMSE time resolution
again by at least a factor
of 2.
1
10
3.0
2.0
Simulation parameters:
sampling rate: fs = 440 MHz 0
10
dynamic range: 26 dB 1 2 4 6
Oversampling factor `
8 12

ADC resolution: 6 bits

: Marc Ihle (17.09.2013) 18
 1
Simulations
Original signal without noise and estimated pulse positions

Estimated pulse positions with Cadzow algorithm

0.5

Effect of Oversampling
ï0.5
5 10 15 20 25 30 35 40 45 50
t [ns]
Signal x(t) with noise and interferences, SNR = Inf dB
1 5
10
0.5
Max. error: Dithering, no averaging
0
Max. error: Dithering, 125 averages
ï0.5
5 10 15 20 25 30 35 40 45 50 Max. error: Dithering, 250 averages
random guess
t [ns]
Samples y(t): 4.2e+08 Sps, resolution: 8 Bit, Oversampling: 2 Max. error: Dithering, 2000 averages
0.04
y(t) Output Sampling kernel
4 RMS error: Dithering, no averaging
: 
Sampling points
0.02
10
0 Oversampling by a factor RMS error: Dithering, 125 averages
RMS error: Dithering, 250 averages
ï0.02
of 4 enhances the RMSE
5 10 15 20 25
t [ns]
30 35 40 45 50 RMS error: Dithering, 2000 averages

time resolution by factor

of 300.
3
10

: 
t (ps)
An oversampling factor
exceeding 12 gives a
2
further improvement by a 10

factor of 4.

1
10 Factor 4

Simulation parameters:
sampling rate: fs = 440 MHz 0
10
dynamic range: 26 dB 1 2 4 6
Oversampling factor `
8 12

ADC resolution: 6 bits

: Marc Ihle (17.09.2013) 19
 Conclusion and Outlook

Conclusions:

:  TDR using FRI is a promising method in respect to efficient hardware implementation.

:  However: TDR using FRI is very sensitive to quantisation noise.

:  Dithering and Averaging leads to significant performance improvements.

:  Improvements are not yet sufficient for highly demanding TDR requirements (<33 ps error).

Outlook:

:  Further reduction of the ToF estimation error is needed.

:  Evaluation of the minimum ToF estimation error bound (Cramer-Rao bound) pending.

: Marc Ihle (17.09.2013) 20

Thank you for your attention.