Hard & soft thresholding

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 be the coefficient. The hard thresholding is

The soft thresholding is

A popular function

or

[ Pol. p target ] a short review

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.

Q-factor

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).

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

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)

Done

1. Calibrating NMR system with water

• by changing NMR level
• after found the peak on level = 150, measure the FID area in successively.

Result:

• the pulse may not be optimized. the 1st pulse gave FID area = 25, and 2nd pulse  gave FID area = 7, 3rd pulse gave 5.

2. 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.

3. 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.

Result:

• the characteristic peak does not change so much. thus, we have confidence that the variation on NMR signal is NOT by the coil.

4. 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.

5. 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

ToDo

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

Discussion

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

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??

ToDo:

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

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.

ToDo:

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.

[Pol. p target] microwave system

the microwave system was fixed and calibrated so that it resonance with the microwave cavity. the step for turn off the trigger signal is, switch off the RF amp then turn off the trigger. and turn on the trigger voltage then turn the RF amp on. if turn the RF amp first, then the microwave has some noise and unstable.

the phase of trigger voltage also important.

the position of the sample and the NMR coil is unstable, cannot have a reliable placing and make many measurements fail.

suggest after every signal recorded, measure the BG.

Todo

1. find the crystal angle to the microwave.
2. polarized the target
3. measure the laser polarization dependency.

[Pol. p target] Found water NMR signal

the DC B-field voltage is 3.100V. i found that i have have to tune the voltage slowly, to let the CRO signal stabilized.

The NMR setting is 12.8MHz, Level 200, pule length is 5uS on +Y axis. FWD is 49W, BWD is 7W. Gain is 35db, LPF is 100kHz and polarization is 90 degree.

the FID signal is about 200mV peak to peak.

Let me briefly explain the NMR circuit, for reader who want to know the theory. HERE.

Black arrows are any cable, Orange arrows are 12.8MHz cables. The direction of the arrow shows the signal flow.

the NMR pulse is a mixture of 2 signal. the RF generator provides the larmor frequency. and the pulse generator provides a square wave, the duration of the square determine the pulse time, it should be matched the requirement for π/2 pulse. and the period of the pulse should be longer than the T2, longitudinal relaxation time. since we don’t want to either distorted the RF or use 1 more RF generator, the splitter simply split the signal, 1 be a main signal and the other one be a reference signal.

The purple mixer or Double Balanced Mixer (DBM). the function of it is multiple the 2 input signal together. the circuit inside, may be i study later.

after the Mixer, the signal will be amplified and pass a λ/4 cable. i am not sure is it necessary to use a λ/4 cable in here. but anyway, we knew that the phase different from the Amplifier to the crossed-diode is λ/4.

the crossed diode has a feature that only allow signal, which larger then 0.6 V to pass. when the pulse come in, it will not see the diode. however, when the in put signal are off, the signal from the NMR coil will be generated and this signal is smaller then 0.6V, thus, the crossed diode act as a switch or duplexer.

the λ/4 cable from the crossed diode and the NMR tuner + Coil is for impedance matching. even there are little bit mismatch of the tuner, the λ/4 cable is still approximately equal to the impedance matching case. that smoothen the signal.

and i don’t know is it necessary to use  a λ/4 cable from the cross diode to the amplifier.

after the amplifier increase the signal and mixed with the reference signal at the differential mixer. The Free induction decay (FID) signal should be at the Larmor frequency. However, if the input frequency is not exactly, with just little bit of mismatch, the spin can still be rotated. by different the NMR signal with the reference signal ( which is the signal we put in the NMR coil) we can tell the different and calculate back the true Larmor frequency by Fourier transform.

and also, by different the NMR signal by the reference signal, we can more easy to tell the different then by just looking at the NMR signal. since the different usually just few kHz, compare with the Larmor frequency 12.8MHz, less then 1 % deferent.

the different mixer also decompose the signal into sine and cosine waveform. this is because, by only look at the cosine wave, we cannot know the different of the signal is positive of negative. the cos wave is natural real part of the complex wave.

the low pass filter (LPF) can be set the maximum frequency pass. this is for seeing the decay envelop of the FID. anther function is reduce the high frequency noise.

[Pol. p target] Matching Impedance of NMR coil

i had played with the impedance of coax cable. connect the signal generator at 50Ω output and a 4meter long coax to a CRO with 50Ω input with a tee. although the web said, the coax impedance is 52, there is no observed different. the peak-to peak signal does not depend on the frequency.

then connect a 12.8MHz coax to the tee and open at the other end. in theory, it should be zero input-impedance and he CRO reading should be

where the CRO reading is the input voltage. when the frequency adjusted to 12.8MHz, which is quata-wavelength, the input voltage is equal to 1.

when connect the end of the coax with a 50Ω resistor, the input voltage does not depends on frequency, as expected.

i haven’t try to short the end of the coax.

after that. i going to matching the impedance of the tuner and coil. we use a short coax to connect the input to the tuner before, i replaced it with a 12.8MHz coax. the reason is, the little bit mismatching of the impedance can be saved by the length of the cable, such that:

and i opened the tuner to see the circuit inside:

The above is the circuit diagram. there is a fixed capacitor with 2.2pF in parallel. i cannot identify the type of the 2 variable capacitor.  the coax cable can be put in port 1 or port 2 and the coil put in the other port. different configuration has different behavior.

i found that, the input is in port 2 and the coil is in port 1, and the box doesn’t ground but just wrapped with metal sticker. i took it out and grounded to NMR system.

For the input at port 2, the total impedance of the coil and tuner is:

The first things to notice is the impedance depends on the frequency. which mean, the impedance matching can only on particular frequency. when the Cs adjusted to matching the frequency. the impedance solely depends on Cp. when the driving voltage gone, the LC circuit will oscillate at natural frequency:

Thus, i tired to measure to inductance of the coil by a parallel resistance. but i cannot find any suitable wire to convert BNC cable to wires. after a long time finding, i gave up and wait unit work with my partner.

the input in port 2 is not a common config, so, i changed it to port 1. and the impedance is :

The impedance also depends on frequency. and the natural frequency when the driving voltage gone is:

there is one way to tune the Cp to match the input frequency and make load impedance solely depends on Cs. by using a pulse signal. and measure the natural frequency of the LC loop, such that the natural frequency is same as input frequency. However, the Low Pass Filter only let frequency less then 1MHz pass and out frequency is 12.8MHz. can i use other pulse source? may be, if i have a mixer.

so, i matching the impedance by very naive way. i use the method on testing the impedance matching. i use a continuous signal source and fixed the frequency at 12.8MHz, then connect it with a coax cable to 50Ω CRO input by a tee, then connect th tee with a 12.8MHz coax cable. the other end of the cable connect to a 50Ω resistor. this setup should be matched impedance. so, i record the input voltage on the CRO. and replace the resistor with the tuner, which port 2 connected to the coil. then adjust the capacitors (both) so that the CRO voltage is same as 50Ω resistor.

by solving the load impedance formula of port 1 configuration, there are mulitple solution for Cp and Cs to give 50Ω. and i think, as long as the impedance is 50Ω at 12.8MHz, any configuration can do the job.

later, i try to find the water NMR signal. although i cannot find any. but the noise level reduced to ±5mV. more or less equal to the background.

i played with the NMR program. the record data is counted by point, so the CRO horizontal setting should set to 5000 points over the screen.

and just before i leave, i don’t know what wrong, the program doesn’t read the CRO signal…