Читать книгу Essential Concepts in MRI - Yang Xia - Страница 30

The frequency ω₀ in Eq. (2.29) is usually too high for the voltage signal to be observed directly after amplification (a good linear amplifier at radio frequency is also more expensive). An electronic process named heterodyning is commonly used for signal detection in NMR. This process employs a number of phase-sensitive detectors to reduce the carrier frequency but retain the individual amplitude and phase information. This approach is identical to how we listen to a radio program – we do not really listen to our favorite broadcast program at hundreds of megahertz frequency (radio frequency); we listen to the audio frequency modulation of the radio broadcasting.

When Δω is used as the offset of the heterodyning signal, the NMR signal in the time domain, previously expressed in Eq. (2.30) in ω₀, becomes

(2.32)

where S₀ is proportional to M₀, and ϕ is a spectrometer parameter called the receiver phase.

With the use of Fourier transformation, we can derive the signal in the frequency domain in both real (Re) and imaginary (Im) parts [2], as

(2.33a)

(2.33b)

When ϕ = 0, the above equations become

(2.34a)

(2.34b)

These two equations can be plotted as in Figure 2.14c–d, where the real part is called the absorption signal, while the imaginary part is called the dispersion signal. By comparing Eq. (2.34) with Eq. (2.23) in the CW NMR experiment, we find that the results are of the same form as the absorption and dispersion components. However, the results in Eq. (2.34) are not centered at the Larmor frequency ω0, but at the difference Δω, the offset frequency.

When ϕ ≠ 0, which is common in practice and means that M is not along any axis in the transverse plane, the real and imaginary parts of the signal contain a mixture of absorption and dispersion components. We call the spectrum “out of phase.” We can correct this phase by multiplying the signal by exp(–iϕ); that is, we apply a 2D rotation matrix to the signal, as we did in Eq. (2.22). This process is termed to “phase” the spectrum in NMR experiments (cf. Chapter 6.10), which illustrates that in actual NMR experiments, the phase of the signal detector can be adjusted continuously.