Hard & soft thresholding

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In usual Fourier transform (FT), the filter is cut-off certain frequency.

This trick is also suitable for wavelet transform (WT). However, there could be some “features” located in high frequency scale (or octave) , a simply cut-off would remove these features.

If the signal to noise level is large, that means the noise has smaller amplitude than that the signal, we can use hard or soft thresholding, which zero any coefficient, which is after the FT or WT,  less then a threshold.

Lets X be the coefficient. The hard thresholding is

Y=\begin{cases} 0, & |X| <\sigma \\ X, & \mbox{else} \end{cases}


The soft thresholding is

Y = \begin{cases} 0, & |X| < \sigma \\ sign(X) f(|X|, \sigma), & \mbox{else} \end{cases}

A popular function

\displaystyle f(x, \sigma) = \frac{x - \sigma}{ X_{max} - \sigma } X_{max}



\displaystyle f(x, \sigma) = x - \sigma



[ Pol. p target ] a short review

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THe system is fairly acceptable now. the signal fluctuation is about ±30 unit. compare with the absolute value of 600 to 1200. it is fair enough.

we have a Hall probe now, but the measured magnetic field is quite different from what we expected before. we expect it should be around 0.300xxx but the measured value, is 0.33xxx that is mean, something is missing in our understanding.

after finishing the optimization, the system is ready for further development.

  1. absolute polarization
  2. spin echo
  3. laser polarization dependency
  4. Fourier analysis
  5. T1 and T2 measurement
  6. cross polarization between H1 and C13
in order to do the absolute polarization measurement, we have to lower the noise level. or, we can increase the magnetic field and reduce it back when measuring it. this requires to measure the T1 relaxation time. another way is spin echo method. since it can avoid the influence of the coil relaxation signal, which cover up the very beginning signal.
For the Fourier analysis, we have to use an external reference frequency for NMR system. currently, we use the same frequency for the pulse and for the reference frequency. Since the pulse frequency must be matching with the Larmor frequency ( more or less), which is the signal frequency. in principle, our signal must be a simple decay curve when exactly matching was archived. in that case, the Fast Fourier Transform will give is same peak at the edge of the spectrum, which is hardly identified. however, if we use an external reference frequency, problem can be solved, and we are able to obtain some peak at the middle of the frequency spectrum. By this, we can understand more about the crystal and the internal field and processes.
and also, when we cross polarize H1 and C13, we can use Fourier analysis to understand the effect much better.


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Q-factor is Not Q-value. in Q-factor, the Q is for Quality.

Q-factor is a dimensionless factor for showing the degree of reasonace of a coil or an Oscillator.

Higer value means a better coil. The definition is the resonance frequency over the Full Width Half Maximum (FWHM).

Q = f/\Delta f

High value means:

  1. high sensitivity
  2. noise reduction due to narrow band of absorption.
  3. over damped with long decay time.
  4. higher energy stored
  5. lower energy loss

For complex electric circuit, the Q factor is:

Q = | im Z/ re Z|

It seem that there is a conflict between impedance matching.

[ Pol. p target ] Trying New Coil

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after getting frustrated with the unstable and non-reliable result of the system ( the stability may result from many things, but we checked all measured parameters, they are all the same ), we decide to change the coil, since it is the only moving part in the system and it may cause the instability.

we make a 4mm gap in a new coil, each side is 15 turns, so the laser can hit the sample without blocked by the coil. However, when we insert the coil inside, the resonance frequency of the µw cavity changed. it becomes borden and more reflection. with all effort, the polarization fail.

However, may be it is due to a poor and naive design of the coil. with better design, may be we can satisfy the µw cavity resonance frequency, which is the ESR frequency and able to make a non-moving coil.

there are several changes made on the system when we return to a moving coil setting.

  1. we checked the turns of the coil, it is 25.
  2. the NMR coil input impedance is 49.7+0.3i Ω
  3. the optics was reset and now it hits the center of the sample with laser spot size about 1mm in diameter. and the incident angle is almost perpendicular.
  4. Reset the µw frequency, now the Gunn Oscillator tuning length is 4.29.
  5. the µw reflected signal has a noise, and the shape is not right
the result is: we get a large NMR signal about 700 unit.
another thing is, according to a supplier manual of the NMR system, the output voltage is proportional to the “level” in the range from 100 to 800.

[Pol. p target] Meeting report (June 8th)

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  1. Calibrating NMR system with water
  • by changing NMR level
  • after found the peak on level = 150, measure the FID area in successively.


  • the pulse may not be optimized. the 1st pulse gave FID area = 25, and 2nd pulse  gave FID area = 7, 3rd pulse gave 5.
  1. Observed the Coil Relaxation signal depends on the coil impedance. there is a characteristic peak to indicate the change in impedance. it can be used to measure the impedance.
  1. the Coil was wrapped by Teflon tape and fastened the copper wire. Adhesive was used to fix the join of the coil and cable. the insertion mechanism was fixed by optics mounting.


  • the characteristic peak does not change so much. thus, we have confidence that the variation on NMR signal is NOT by the coil.
  1. Optimization
  • the microwave delay time was measured. since it is not easy to have trigger on -10 us after the laser pulse end. we use 0us instead.
  • the microwave power is optimized at 1.0W
  • Laser pulse duty and chopper frequency.
  1. Laser polarization angle
  • the change is smaller then signal fluctuation.

Wakui San comments

  1. Laser mode
  1. the laser is running at multiline mode, but the power detector is at 514.5nm
  • Crystal expired
  1. the crystal we use is about 5 years ago


  1. have to measure the statistics of the data, due to a improvement of the coil.
  2. Crystal orientation
  3. Optimization
  1. laser pulse duty
  • NMR pulse calibration by water
  • T1 and T2 measurement
  • Q-factor of the coil
  • more detail measurement on each parameters
  • thermal polarization


  1. the coil is being fixed, we are able to have a more reliable data. we have to find out the statistics. we can compare with a previous data @ Puw = 1.0 on June 3rd, we had collected 15 data for same setting and the s.d. is 30 unit.
  2. To have the absolute polarization measurement
  1. we have to lower the variation of signal
  2. we have to lower the noise level
  3. we have to get same setting for the NMR system
  1. impedance
  2. level
  3. pulse time
  4. gain
  5. Forward and Backward power
  • the change of FID area due to change of external H-field
  1. the data shown the angular frequency is pi per 30us, about 0.1 rad per 1us.
  2. the angular frequency for proton is 267.5 rad per Tesla
  3. if the field change for 1%
  4. the change of the angular frequency will be 2.67 rad per us
  5. or to say, the fluctuation of the field should be less then 0.05%
  • In order to preform Fourier transform, or the Finite time Fourier Transform, we can use wavelet analysis.
  • Before polarization transfer to 13C, we have to optimize the system.
  • the sample NMR signal is not flowing sine and cosine function
  1. it it due to crystal field

[Pol. p target] getting polarization signal

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after my colleague’s work on yesterday on fighting the noise, the noise level of the NMR system reduced to 25mV. and but water NMR signal also same magnitude, due to impedance mismatch. thus. i tuned the impedance and observed water NMR signal to be 330 mV. signal to noise ratio is around 10.

then i insert the sample and start polarization. i pick the data from 15 us to 25 us after the pulse, since there has a peak, which gives the FID amplitude different from BG significantly. i.e. BG FID amplitude is around 6 unit and polarized sample has FID amplitude around 20.

however, i cannot obtains a consistent data. say, if i set the crystal angle be 90 degrees, then on 95 degree. then i do again 90 degree, the 2nd time, the FID amplitude may change alot. even though i took 3 times on each angle.

the sample space has fluctuation due to the noise. thus, we need to take several data on each setting to get a  reliable result. but each data need 300 seconds, thus the data taking is quite frustrated, due to inconsistent data.

the possible reasons for inconsistent data is due to NMR coil position and connection. since it is the only thing changing on experiment. each time, we have to unplug and plug in the coil to get the NMR date. even a tiny change of the coil position will change the NMR signal strength ( this was tested on  water sample).

also, the reflected NMR pulse is about half of the input pulse. even though we matched impedance by smith chart. How is this possible??


  1. fix the NMR coil position and get consistent and reproducible data.
  2. find the maximum Crystal orientation
  3. Optimization
  4. find the laser polarization dependency

[Pol. p target] try to polarize sample

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the microwave system was tuned again. the minimum resonance deep of the reflected microwave is 60mV at cw mode. when it is pulse mode, during the triggered microwave, the reflected wave is still large (8.4V), and we can see a clear charge and discharge signal. we also correct the microwave trigger.

the NMR coil is 60 turns now, compare to the old one is 10 turns. thus it is more sensitive. we cannot find a polarized signal. the NMR signal either has large noise or large reflected wave. may be the impedance is not matching or the resonance deep is not enough.


  1. minimize the NMR noise
  2. reduce the reflected NMR pulse.
  3. polarize target
  4. optimization
  5. laser polarization measurement
  6. spin echo
  7. polarization transfer.

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