[ieee 2013 2nd international symposium on instrumentation & measurement, sensor network and...

A high speed random equivalent sampling method based on time-stretch

Kuojun Yang College of automation engineering

UESTC Chengdu, China

[email protected]

Shulin Tian College of automation engineering


[email protected]

Jinpeng Song College of automation engineering


[email protected]

Abstract—Random equivalent sampling is realized based on time stretch in this paper. Firstly, the measuring error of time stretch factor K is analyzed. Formulas are derived to describe the effect of measuring error of K and jitter on the waveform reconstruction. According to the formulas, multiple measurements are used to reduce the measuring error of K,and average filter is utilized to reduce the distortion introduced by jitter. The effectiveness of average filter is verified through simulation. And the proposed methods are implemented in digital oscilloscope, the experiment results prove that our method is useful and the equivalent sampling rate reaches 100GSPS.

Keywords-equivalent sampling; time stretch; average filter; time measurement;

I. INTRODUCTION

Equivalent sampling is a useful method to implement high speed sampling for periodic analog signals by using low speed ADC [1]-[3]. It is widely used in digital sampling oscilloscope, achieving the functions such as inhibition of ground penetrating radar (GPR) radio frequency interference (RFI) [4] and measuring the sampling clock jitter [5].

Random sampling is a frequently-used technique in equivalent sampling. Its key technique is high-accuracy time interval measurement [6], which are discussed by numerous literature [7]-[10], such as the stepping delay method, which employs the dedicated carry chains of FPGA for time interpolation purposes inside a clock cycle [7] [8], the pulse shrinking method, exploiting the different fast carry chains delay times for both pulse edges in FPGA [9] and the time stretch method [10], etc. Since the precision of time interval measurement with FPGAs are more sensitive to changes of ambient temperature and supply voltage, the time stretch method is suitable for the proposed implementation.

To achieve higher equivalent sampling rate, two problems need to be solved. One is the accurate evaluation of stretch factor K in time stretch circuits. If there is an offset in the measurement of stretch factor, distortion will be introduced in the reconstructed waveform. The other one is that the measurement precision is greatly reduced because of the jitter which cannot be completely eliminated in the system. In this paper, the effect of stretch factor measurement error and jitter on waveform reconstruction is analyzed, statistics method is adopted to decrease the measuring error, the method is easy to realize. No additional calibration circuit is needed while the measurement precision

can be guaranteed. Furthermore, a method of reducing jitter by average filter is presented. The precision improvement of time measurement enhances the equivalent sampling rate. Eventually, the validity of the proposed method is verified through simulation and experiments. The experiment shows that a 100GSPS equivalent sampling rate is achieved in digital oscilloscope.

II. RANDOM EQUIVALENT SAMPLING FUNDAMENTAL

The essence of random equivalent sampling is reconstructing an integrated waveform via multiple sampling. The sampling rate will be increased by a factor of N when the system acquires N times to reconstruct a waveform. Assuming that N =4, zero crossing is regarded as trigger condition, shown in Fig.1. Time interval t between trigger signal and the first sampling clock after triggering is utilized to determine the position of every sampling point. It can be seen from Fig.1, 4 1 3 2 ,t t t t< < < so the sequence of sampling points is 4,1,3,2. Assuming that N times of sampling are done to create one waveform, the sampling period is T, the position of every sampling point can be expressed as

tPos N

T= . (1)

1t2t 3t 4t

Figure 1. Random equivalent sampling fundamental.

As in [11], the fundamental of measuring t by time stretch method was described. To diminish measuring error caused by the temperature drift of K and threshold level of comparison, “three measurements method” is commonly used. The expression of t is

2013 2nd International Symposium on Instrumentation and Measurement, Sensor Network and Automation (IMSNA)

159978-1-4799-2716-6/13/$31.00 ©2013 IEEE

10

2 1

CP CPt T

CP CP−

=−

. (2)

The position of each sampling point can be derived by substituting (1) into (2)

1

2 1

CP CPPos N

CP CP−

=−

. (3)

CP in (3) is measured in every sampling, so that the position of each sampling point can be calculated. After Ntimes of sampling, an integrated waveform can be reconstructed.

III. IMPACT OF TIME MEASUREMENT ERROR ON WAVEFORM RECONSTRUCTION

A. Errors Introduced by Stretch Factor From (1) we know the quality of reconstructed waveform

is determined by measurement precision of t . Time measuring error would cause sampling point placed in wrong position, distorting the reconstructed waveform. To reduce the impact of current changes in constant-current source and threshold level of comparison drift caused temperature in time measuring, formula (3) is adopted to determine the position of sampling point. However, it cannot eliminate errors completely. Firstly, large time cost is the major disadvantage of time stretch, and the time cost is directly proportional to K, Frequent measurement of 1CP and 2CPwould waste more time especially with a larger N. 1CP and

2CP is measured after a time period, but they may have been changed in the time period. Secondly, measuring error may be large in every single measurement because of the noise and jitter even “three measurement method” is used. The time measurement error resulted from measuring error of

1CP and 2CP , which is also the measuring error of stretch factor, is derived as follow and the changes of K make the formula for measuring CP turning into

0 0'( 2 )K t T CP T+ = × . (4) In (4), 'K is the changed value of K .

Setting 'K K K= + Δ , then the result of “three measurement method” can be rewritten as

- -=- +

1

2 1

CP CP KPos N

CP CP KΔΔ

. (5)

To deduce the relationship between the final deviation δ and KΔ , which is measurement error of K and time interval t, (5) is equated to (6).

/= ( 1)Kt T K t K tPos N N N

K K T K T− Δ Δ≈ − +

+ Δ (6)

Time measuring error can be obtained from (1) and (6).

= ( +1)K tN

K Tδ Δ (7)

As indicated in (7), δ is proportional to KΔ and t, and inversely proportional to K. To ensure the quality of reconstructed waveform, δ must be no more than one small grid in oscilloscope screen. If five pixels compose one grid, then one can have the following relationship.

( +1) 5K tN

K TΔ ≤ (8)

Assuming that N=320 and K=1000, since the maximum value of t is T, so

5 1000 8640

K ×Δ ≤ ≈ . (9)

B. Errors Introduced By jitter When signals go through the I/O of FPGA, they may be

superimposed by jitter commonly. Stretched pulses need to be counted by standard clock which also has jitter in FPGA. Fig 2 shows the simplified block diagram of time measurement system and the noise model. 1Z is the FPGA internal clock jitter. 2Z is the jitter on the narrow pulse output pin. And 3Z is the jitter on the stretched pulse input pin.

1Z

2Z 3Z

Figure 2. The block diagram and noise model of time measurement system.

Both 1Z and 2Z superimposed on time interval have significant impact on the result, while 3Z superimposed on stretched pulse is almost negligible after divided by stretch factor. In this case, CP can be expressed as

1 2 0( ) /clk trigCP K t t Z Z T= − + + . (10) In(10), clkt is the time of the second rising edge of clock

after trigger signal arriving, and trigt is the trigger time.

CPδ , the error of CP introduced by jitter, in (11), can be inferred from (10).

1 2 0( ) /CP K Z Z Tδ = + (11) Accordingly, Posδ , the error of waveform’s position, is

1 2 0( ) /Pos Z Z N Tδ = + . (12) The system jitter is measurable by histogram. A constant

time interval generated in FPGA is measured 100,000 times. Fig.3 reflects that system jitter is normally distributed. The acquisition system has both jitter in time axis and noise in amplitude which is from the power supply or analog channel. To keep these noises away from waveform reconstruction, points at the same position need to be averaged during the equivalent sampling. Assuming an ideal signal ( )ty f t= ,because of the jitter, ( )tk ky f t τ= − where kτ is the value of jitter. After averaging N times, the mean value

1

1 n

t tkk

Y yn =

= . (13)


160

Figure 3. The statistical histogram of repeated measurement on the same time interval.

The mean approaches to the expectation when n is large. ( ) ( ) ( ) ( )* ( )tE Y f t P d f t P tτ τ τ

∞

−∞= − = (14)

Fourier transform is used for (14), then it can be derived that

( ( )) ( ) ( )tF E Y H j P jω ω= . (15)

IV. METHODS TO REDUCE ERROR

To reduce measuring error of stretch factor from single measure, CP1 and CP2 is measured N times before

calculating the mean value. Hence, the variance is 1n

of that

in single measure. From (9) we get 8KΔ ≤ , then variance of

measuring result after average 42KKδ Δ= ≤ . From Fig.3 we

know that measuring error 15δ ≈ in single measurement, so the measuring error after n times of measurement is

15 4K n nδδ = = ≤ . (16)

Solving inequality(16), we get 14n ≥ . That means 1CP and 2CP need to be measured 14 times to meet the system

requirements. A digital back-ground filter is used to reduce the

distortion caused by jitter. From (15) we know when the response of the filter is the inversion of ( )P jω , the distortion caused by jitter is eliminated. So the problem is equivalent to figure out ( )H jω when both ( ( ))tF E Y and ( )P jω are known. Derived from (15)

( ) ( ( )) / ( )tH j F E Y P jω ω= . (17) Due to the zeros of ( ( ))tF E Y and ( )P jω , if we calculate

( )H jω using (17) directly, many values tend towards infinitude. Besides, probability density function of jitter is hard to obtain, and it will change with the environmental factor such as temperature and voltage.

In fact, sample point in waveform will deviate from their theoretical position because time measurement is inaccuracy. Assuming that ideal position is n, the practical one may be at any point of [n-m, n+m]. Hence, taking the average of these points can reduce the waveform distortion. The function of average filter is

1 0,1,...,( )

0N n N

h nelse

== . (18)

The corresponding time domain difference equation is 1

0

1( ) ( )

1 [ ( ) ( 1) ... ( 1)].

N

k

y n x n kN

x n x n x n NN

−

=

= −

= + − + + − + (19)

And frequency response is ( 1)/21 1 1 sin( / 2)( )

sin( / 2)1

j Nj j N

j

e NH e eN Ne

ωω ω

ωωω

− −−= =−

. (20)

The average filter is a low pass filter, and its cutoff frequency is related to N. However, N cannot be too large, otherwise the main lobe will be too narrow, and some useful signals will be filtered. Moreover, the larger N, the lower speed of data processing becomes, and the practicability of acquisition system is degraded.

V. SIMULATION AND EXPERIMENTAL VERIFICATION

A simulation for average filter is done in Matlab first. In the simulation, the interpolation factor of equivalent sampling is chosen to 25, and the sampling rate of ADC is supposed to be 1GSPS, so the equivalent sampling rate is 25GSPS. A 500MHz sine wave is sampled. The variance of time measuring error introduced by jitter is supposed to be 4ps. In order to simulate the case in real system, Gaussian noise is added in the sine wave, the sine wave amplitude is set to 1, and the Gaussian noise amplitude is set to 0.01. N of average filter is set to 6.

Figure 4. The waveform before and after average filer.

Fig.4 is the waveform before and after average filter. Dot-and-dash line is the waveform after average filter, and solid line is the waveform before average filter. Obviously, after the filter, the waveform distortion is less. Fig.5 shows the signal spectrum before filtering, and Fig.6 illustrates the


161

signal spectrum after filter. We can see that before filter, the noise floor is about -70dB, and some distortion spectrum is about -50dB. After filter, the noise floor is about -80dB when frequency is greater than 3GHz and the distortion is about -70dB. Therefore, the average filter suppressed the distortion.

Figure 5. The signal spectrum before average filer.

Figure 6. The signal spectrum after average filer.

The proposed method is implemented in a digital oscilloscope. A 100GSPS equivalent sampling rate is achieved. The oscilloscope is mainly composed by ADC, FPGA, DSP and time stretch circuit. The model of ADC is EV8AQ160, its sample rate is 5Gsps. FPGA is used to implement the function such as storage, trigger, generating pulse and counting the stretched pulse, etc, its model is XC5VLX30. DSP is adopted to process data and display waveform, its model is ADSP-BF531. The average filter is built in DSP. Fig.7 is the 1GHz sine waveform acquired utilizing the proposed method. In Fig.7, each time-base grid

of the oscilloscope in composed by 50 sample points, and the time-base is 500ps, so the time interval between two sample points is 10ps, which means the equivalent sample rate of our oscilloscope is 100GSPS.

VI. CONCLUSION

In this paper, the fundamental theory of random sampling is briefly introduced, and the impact of time measurement error on waveform reconstruction is analyzed. By using multiple measurements, a more precision time stretch factor can be obtained. And average filter is used to reduce the distortion caused by jitter. The proposed methods can significantly facilitate high speed equivalent sampling. The experiment shows that a 100GSPS equivalent sampling can be achieved.

REFERENCES

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[2] F. M. C. Clemencio, C. F. M. Loureiro, and C. M. B. Correia, “An easy procedure for calibrating data acquisition systems using interleaving,” IEEE Trans. Nucl. Sci., vol. 54, no. 4, pp. 1227–1231, Aug. 2007.

[3] M. O. Sonnaillon, R. Urteaga, and F. J. Bonetto, “High-frequency digital lock-in amplifier using random sampling,” IEEE Trans. Instrum. Meas.,vol. 57, no. 3, pp. 616–621, Mar. 2008.

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[7] Eugen Bayer and Michael Traxler, “A High-Resolution ( <10ps RMS) 48-Channel Time-to-Digital Converter (TDC) Implemented in a Field Programmable Gate Array (FPGA)”.IEEE TRANSACTIONS ON NUCLEAR SCIENCE Mar. 28, 2011: P1-6.

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Figure7. 1GHz sine wave acquired by proposed random equivalent sampling process methods.


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