NUCLEAR INDUCTION / FREQUENCY

NMR Free Induction Spectrometer

Tip an ensemble with a radio-frequency pulse. Retain both quadratures of the induced voltage, transform those records into a spectrum, and estimate the signed frequency offset and effective transverse decay.

Ideal single-resonance ensemble with instantaneous broadband pulses, exponential intrinsic decay and a Lorentzian static field distribution. Calibrated quadrature receiver, white Gaussian voltage noise and complete reset between repetitions. The inferred effective decay cannot separate intrinsic relaxation from static broadening. Synthetic data; no molecule or historical apparatus is fitted.

Vertical static field; horizontal transmit/receive coil. A switch isolates the receiver during the pulse, then routes the induced voltage to a phase-referenced quadrature receiver. Cable motion is schematic. $B_0=0.35\,\mathrm{T},\quad f_{\rm ref}\simeq14.902\,\mathrm{MHz}$

PULSE SEQUENCE / AVERAGED RECEIVER VOLTAGES

After the pulse, listen.

Equilibrium reset

\theta_x

Receiver blankingSigned quadratures

One fully reset acquisition contains 256 complex samples, every 0.4 milliseconds after a 0.4-millisecond dead time. Repeats are averaged sample by sample. The receiver resolves the slow beat against the RF reference, not the fast carrier.

Ready. Predict, then acquire.

RAW VOLTAGE RECORD / SEPARATE SOURCE EXPECTATION

Two quadratures. Keep the sign.

$t/\mathrm{ms},\quad I,Q\;(\mathrm{\mu V})$

Teal: in-phase voltage. Gold: quadrature voltage. Dots: retained observations. Thin lines: source expectations. Purple dashes: fit using records only. Gray: saved observations.

FINITE-WINDOW TRANSFORM OF THE RECORD

This spectrum comes from those voltages.

$f/\mathrm{Hz},\quad |S_k|/\mathrm{\mu V}$

A uniformly spaced record is needed. Rectangular window, no zero filling. Magnitude and finite-window width are not a fitted absorption line or an independent relaxation measurement.

INFERENCE / SOURCE PARAMETERS EXCLUDED

Inferred effective transverse time

\text{—}

Inferred signed frequency offset

\text{—}

Inferred initial voltage magnitude

\text{—}

Retained complex samples

0

Voltage residual RMS

\text{—}

Global bounded search with profiled amplitude. Weak-information and boundary checks are diagnostic heuristics, not parameter confidence intervals. A synthetic recovery does not independently establish the physical law.

Latest raw observations

Index	$t/\mathrm{ms}$	$I/\mathrm{\mu V}$	$Q/\mathrm{\mu V}$

SAVED COMPARISON / MODEL BOUNDARY

Single noninteracting resonance; no molecular chemical shifts, coupling multiplets, diffusion, radiation damping, coil loading or realistic pulse bandwidth. Gain and phase are calibrated; no drift or digitizer quantization. Instantaneous hard-pulse approximation; the Cauchy static-offset model is not a computed field map.

Next: inversion recovery Read the magnetic-resonance evidence TeachSpin original experimental guide Manufacturer instrument manual Bruker transmit/receive probe manual NIST reference gyromagnetic ratio

Start with the essentials

Focus question: Turn a fading voltage into a spectrum.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

A transverse ensemble induces a voltage

\begin{aligned}s(t)&=A e^{-t/T_2^*}e^{i(2\pi\Delta f t+\phi)}\\\frac1{T_2^*}&=\frac1{T_2}+2\pi w\end{aligned}

A Lorentzian distribution of static offsets has an exponential characteristic function. This special distribution gives additive decay rates. The complex amplitude is unknown to the estimator.

Transform the retained voltages

S_k=\frac1N\sum_{j=0}^{N-1}(I_j+iQ_j)e^{-2\pi i k j/N}

The displayed spectrum uses the actual finite, uniformly sampled record with a rectangular window. Its nonnegative magnitude loses phase; the signed frequency axis retains rotation direction. Partial records have coarser bins.

Infer from both voltage channels

\underset{a\in\mathbb C,\,f,\,T}{\operatorname{minimize}}\;\sum_j\frac{|I_j+iQ_j-ae^{-t_j/T}e^{2\pi i f t_j}|^2}{\sigma_j^2}

Profile out the complex amplitude at every candidate frequency and time. The source decay, broadening, phase and flip angle never enter this fit.

Common difficulties

A simulation supplies its law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

Run the experiment

01
Acquire before transforming
Acquire the reference pulse and inspect both raw voltage channels and the finite-window spectrum.
What to observe: The spectrum appears only after retaining a uniform record. It is not a separately drawn ideal line.
02
Distinguish broadening from lifetime
Save the record, then broaden the field without changing the intrinsic source time.
What to observe: The effective time shortens. A single free decay cannot identify which process caused its width.
03
Reverse the beat
Choose the negative-frequency preset, then remove transverse preparation.
What to observe: Complex sampling distinguishes frequency signs. With no coherent signal, the weak-information rule withholds the estimate.