2.6 Integration
2.6.1 General constraints
2.6.1.2 Multipole systems
From a circuit design point-of-view, creating a single-pole integrator with a very small pole a is difficult. In version 1 of the RMPI chip, the lowest achievable pole was about 44 MHz. If we allow a multi-pole system, then it is much easier to push the first pole closer to 0. By doing this, the version 2 RMPI chip has its first pole at 302 kHz, and a second pole at about 300 MHz. With such a small first pole, the impulse responseh(t) is very flat over the integration time [0, Tint]. Figure 2.38 shows matched filter tests of an exact 300 kHz integrator, and compares recovery to that of a Bernoulli±1 matrix. The results are averaged over 400 samples, and the matched filter requires no parameters, so this test is quite reliable. The single-pole model without stagger suffers slightly at high noise level, but not by a significant amount, since the time constant 1/(300 kHz) is very long. This test adds noise of the form Φ(x) +z, instead of Φ(x+z), since for the latter model there is no discernible difference in performance. The variance of z was adjusted to account for the gain of the system.
With the former noise model, the additive noisezwill make it slightly more difficult to recoverxat the beginning of time periods. We conclude that the 300 kHz pole has little effect on the system, and is much better than the 44 MHz pole in the version 1 design.
But does the second pole near 300 MHz have an effect? Figure 2.39 shows several tests on different types of integrators. The left column shows a version of Φ generated from a Simulink model that incorporated a realistic version of the integrator, with several zeros and poles. It performs similarly
0 100 200 300 400 500 600 700 800 900 1000 channel 2
channel 1
time, in units of Nyquist time For a short time constant (here, τ = 5 ns), shifting is crucial
Below are two unshifted channels
time, in units of Nyquist time Measurement matrix φ
100 200 300 400 500 600 700 800 900 1000
channel 1
channel 2
channel 3
channel 4
channel 5
channel 6
channel 7
channel 8
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−0.2 0 0.2 0.4 0.6 0.8 1
Figure 2.36: Figure (a) shows copies ofh(nTint−t) forn= 1, . . . ,9 overlaid, forh(t) =e−atwhereais large. The top data (in red) are for channel one, and the bottom data (in blue) are for channel two. We see that without shifts, these align. For timest= [0,20∆T],x(t) is barely measured. This can be fixed by staggering the channels.
Figures (b) shows the matrix Φ of a staggered system.
Figure 2.37: The results of four independent reconstructions using a single-pole model. The four plots show four different values of the channel stagger. In the 0 offset model, all channels aligned; in the 12 offset model, none of the channels aligned; in the 25 and 50 offset models, 2 and 4 channels aligned, respectively. For small values of the time constant, shifting by any amount is helpful.
2 3 4 5 6 7 8 9 10 105
106 107 108 109
Noise level σ
Median error in frequency estimation
± 1
± 1 w/ stagger single pole single pole w/ stagger
Figure 2.38: The±1 model compared to the single-pole model (with 300 KHz pole). This compares performance, not modeling error. The single-pole model suffers a bit, due to the decay of the time domain transfer functionh(t).
Shifting the channels helps this. N = 2048, and simulations used 2 realizations of each Φ matrix and then 200 sample noise vectors and carrier frequencies;xwas a smooth pulse of length 100 ns. The dashed horizontal line shows the frequency resolution of the grid search.
to the other Φ models, so it seems that it has little effect.
Let the second pole be at location a2. The transfer function of a two-pole system is
H(s) = 1
(s+a)(s+a2)
so thatH(s) decays like 1/s2 for frequencies much larger than a2. To reason about the effects of the second pole, we simplify and consider
H(s) =
1
a2(s+a) s < a2
0 s≥a2
.
If a2 is in the range 100 MHz to 1 GHz, this is quite similar to the scaled single-pole integrator H(s) = a 1
2(s+a). The main limit is that if the PRBS has a short repetition period Tchip, then we must have a2 1/Tchip, otherwise signals with frequency that is far from a harmonic of 1/Tchip
will generate very small measurements. For the current chip design, Tchip = 128∆T, so 1/Tchip= 39 MHz, and hence a practical lower-limit ofa2is about 100 MHz, since otherwise all but one or two of the spikes in the spectrum ofx(t)c(t) will extremely attenuated. With only one or two of Fourier series ofc in the passband, it is much more likely for two inputs to generate similar measurements, and hence the system is less robust. The small bandwidth would also exacerbate the power difference between inputs that are exact harmonics of 1/Tchip and those that aren’t.
To summarize the findings, we can view the integrator as a low-pass filter in the sense that we do not need frequencies above about 100 MHz, but the passband behavior of the filter must not be
Model Zero First pole Second pole Third pole
NG InP NA 12 MHz NA NA
Caltech ver. 1 NA 44 MHz NA NA
Caltech ver. 2 360 MHz 302 KHz 361 MHz 1.696 GHz
Table 2.4: Poles and zeros of the integrating filter for various design. Poles are calculated at 3 dB cutoff from magnitude response. See also Figure 2.40.
flat! The exact poles and zeros of the RMPI are shown in Table 2.4, and the extracted frequency response of the integrator, taken from SPICE simulations, is shown in Figure 2.40.
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.010 New Design
original recovered
0 20406080100120140160180200
−0.01
−0.005 0 0.005 0.01 0.015
time (ns) Time domain.
rel l2 error: 7.6⋅10−2; l∞ error: 1.8⋅10−1 l∞ freq error: 5.4⋅10−2
demodulation of exact signal demodulation of recovered signal error
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.010 Ideal integrator
0 20406080100120140160180200
−0.01
−0.005 0 0.005 0.01 0.015
time (ns) Time domain. Ideal integrator rel l2 error: 1.1⋅10−1; l∞ error: 2.1⋅10−1
l∞ freq error: 9.9⋅10−2
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.010 Gaussian matrix
0 20406080100120140160180200
−0.01
−0.005 0 0.005 0.01 0.015
time (ns) Time domain. Gaussian matrix rel l2 error: 5.5⋅10−2; l∞ error: 1.1⋅10−1
l∞ freq error: 2.7⋅10−2
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.010 Nyquist−rate samples
0 20406080100120140160180200
−0.01
−0.005 0 0.005 0.01 0.015
time (ns) Time domain. Nyquist−rate samples rel l2 error: 5.6⋅10−2; l∞ error: 9.7⋅10−2
l∞ freq error: 9.1⋅10−3
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.001 New design
original recovered
0 20406080100120140160180200
−1
−0.5 0 0.5 1 1.5x 10−3
time (ns) Time domain.
rel l2 error: 4.5⋅10−1; l∞ error: 1.1⋅100 l∞ freq error: 3.1⋅10−1
demodulation of exact signal demodulation of recovered signal error
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.001 Ideal integrator
0 20406080100120140160180200
−1
−0.5 0 0.5 1 1.5x 10−3
time (ns) Time domain. Ideal integrator rel l2 error: 3.7⋅10−1; l∞ error: 6.7⋅10−1
l∞ freq error: 2.9⋅10−1
0 1 2
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−170
−160
−150
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−130
Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.001 Gaussian matrix
0 20406080100120140160180200
−1
−0.5 0 0.5 1 1.5x 10−3
time (ns) Time domain. Gaussian matrix rel l2 error: 3.3⋅10−1; l∞ error: 7.1⋅10−1
l∞ freq error: 2.2⋅10−1
0 1 2
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Frequency (GHz)
Power/frequency (dB/Hz)
normal pulse, 2.29 GHz carrier CWAMP: 0.001 Nyquist−rate samples
0 20406080100120140160180200
−1
−0.5 0 0.5 1 1.5x 10−3
time (ns) Time domain. Nyquist−rate samples rel l2 error: 5.9⋅10−1; l∞ error: 1.3⋅100
l∞ freq error: 9.2⋅10−2
Figure 2.39: Does the new integrator suffer compared to, say, a Gaussian matrix? Not much. This test used a pessimistic noise level, and then ran three different amplitudes. For each test, 4 types of Φ matrix were used: a realistic version from the calibration, a single-pole model, a matrix with iidN(0,1) entries, and a Nyquist sampled version (Φ = I). There is no significant difference. Only the two smallest signals are shown, since the large amplitude signal had visually perfect reconstruction.
Figure 2.40: Top: integrator frequency response. Bottom: LNA frequency response. Blue curves are gain (scale on right axis), red curves are frequency response (scale on left axis). Results are from SPICE-extracted simulations.
See Table 2.4 for values of poles and zeros. Other major blocks were modeled as well, and the resulting design parameters used to inform the Simulink model.