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In this case, the maximum level of the amplitude Amax =120. Then,A10 = 120*10% = 12A90 = 120*90% = 108Evaluate the sequence number of samples corresponding to these levels:n10 = 163n90 = 317This waveform was recorded at a sample rate = 25 GS/s, then the time between adjacent samples equals: =1= 0.04 25 ∗ 10963Calculate the time of the rising edge, through multiplying by the sampling interval: = (317 − 163) ∗ 0,04 ∗ 10−9 = 6,16 According to [6, 7], when using precision pulses, the rise time should not exceed2 ns for GDS-2202.

However, G5-60 generator’s rise time is finite and specified asless than 10 ns. When measured with oscilloscope LeCroy WaveSurfer 104MXs witha bandwidth of 1 GHz and its own rise time of less than 350 ps, a more accurate valueof the time pulse rise time = 6 ns with the same generator was measured. Inthis case, it is impossible to determine the rise time of GDS-2202 and compare it tothe specifications (Appendix A), as it is 3.43 time less than the pulse rise time. To obtain an accurate step response, a rectangular pulse, close to the ideal, is needed whichis difficult to obtain in practice.

However, in theory, such pulse is easily definedmathematically. For this one we need to use MatLab computing environment.Knowing that the transient response of the oscilloscope is inextricably relatedwith the bandwidth, we can set the GDS-2202 frequency response measured in theprevious chapter and, by approximating it using Yule-Walker equations, design a discrete filter. Yule-Walker equations design recursive infinite impulse response (IIR)digital filters using a least-squares fit to a specified frequency response. Yulewalkfunction performs a least-squares fit in the time domain.

It computes the denominatorcoefficients using modified Yule-Walker equations, with correlation coefficientscomputed by inverse Fourier transformation of the specified frequency response. Tocompute the numerator, yulewalk takes the following steps:1. Computes a numerator polynomial corresponding to an additive decomposition ofthe power frequency response.2. Evaluates the complete frequency response corresponding to the numerator anddenominator polynomials.3.

Uses a spectral factorization technique to obtain the impulse response of the filter.4. Obtains the numerator polynomial by a least-squares fit to this impulse response.Before approximation, reduce the amplitude scale to the reference value of 1, andreduce the frequency scale to a normalized scale where sample rate equals 2.64The result of 11th order approximation is shown in Figure 2.9.Figure 2.9 – Original and approximated frequency response comparisonThen we transmit the ideal pulse through a designed digital filter.

The input andoutput pulses are shown in Figure 2.10.65Figure 2.10 - Comparison of ideal pulse and the pulse passed through the filterAs a result of filtration we have got a step response - the dynamic response of thesystem to input in the form of the Heaviside function:() = {0, < 0,1, ≥ 0(2.3)where A – amplitude of the Heaviside function;t – time variable.Finally, measure the rise time of the output pulse, with the same criterion of 1090%:10 = 0,258671 90 = 1,985251 = 90 − 10 = 1,72658 66This value is close to the rise time specified by the manufacturer (≥1.75 ns),which means that the estimated constant (Formula 2.2) equals: = 200 ∗ 1.72658 ns = 0.345316 ,that is approximately equal to the declared value above 0.35.Rise time of the measured pulse is expressed by formula:22 , = √+ (2.4)where – pulse rise time; – the rise time of the oscilloscope.Checking our case with formula 2.4: = √62 + 1,726582 = 6,24348 ,which is roughly equal to practically measured value meas = 6.16 ns above.

Thus, wecan claim that the rise time specified in Appendix A is confirmed. However, whenmeasuring the signal’s rise time equal or less than the rise time of the oscilloscope,there will be a relative error, as the oscilloscope is limited by its characteristic. Formeasuring time parameters the following rule is valid: the greater the ratio of signalrise time and oscilloscope front, the smaller the measurement error (see Table 2.4).The larger the bandwidth of oscilloscope is, the shorter the rise time and the more accurate the measurements are. This rule is easy to check using the formula 2.4.Table 2.4 - The relative error in the measurement of the rise time to ratioRelative error1:141.4%3:15.4%5:12.0%10:10.5%67.

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