Imaginary FID

Now we can construct a perfectly balanced real and imaginary FID from these signals by combining all the sine functions into a real FID and all the cosine functions into an imaginary FID ... 

Where X and Y are the real and imaginary FIDs coming out of the ADC. Thus, we can speak of the receiver phase as the point of view from which we view the FID signal in the rotating frame of reference. If the receiver phase is shifted by 90° (from x to y axis), we mean that the real channel has been shifted counterclockwise by 90° from the x axis to the y axis, and the imaginary channel has been shifted counterclockwise by 90° from the y axis to the -x axis. With this 90° shift, for example, a coherence of Iv would be observed as Ix, and in a four-scan phase cycle with Ar = 90°, we would observe a four-scan sequence of coherences lx, y, —Ix, —Iy as 4IX in the sum-to-memory. To keep track of these phase shifts, a shorthand notation is often used where 0 stands for a 0° phase shift (real part of receiver on the x axis), 1 stands for a 90° phase shift (receiver on the y axis), 2 stands for a 180° phase shift (receiver on the —x axis), and 3 stands for a 270° phase shift (receiver on the —y axis). In this notation, the four-scan receiver phase cycle with A
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The chemical-shift evolution during the FID is taken care of by the exponential term in 2b t2, with a positive exponential because it is the 1 that is evolving. In this complex arithmetic, the real part corresponds to the real FID in t2 (Mx component in the rotating frame) and the imaginary part is the imaginary FID in t2 (My component). We can substitute sines and cosines for the imaginary exponentials as... 

Now we have exactly the same kind of data we have with States mode acquisition cosine modulation in t for the real FID and sine modulation in ti for the imaginary FID. In I2 we can equate Mx with the real part and My with the imaginary part, so that in each case we have Mx = —sin(f2bD) and My = cost f2b/2). This represents a vector starting on the y axis at t2 = 0 and rotating counter clockwise (positive offset 2) at the rate of f2b rad s-1. Once these rearrangements have been made in the computer, the data is processed just like States data, with a complex Fourier transform in t. ... 

As was mentioned above, the observed signal is the imaginary part of the sum of and Mg, so equation (B2.4.17)) predicts that the observed signal will be tire sum of two exponentials, evolving at the complex frequencies and X2- This is the free induction decay (FID). In the limit of no exchange, the two frequencies are simply io3 and ici3g, as expected. When Ids non-zero, the situation is more complex. 

The first Fourier transformation of the FID yields a complex function of frequency with real (cosine) and imaginary (sine) coefficients. Each FID therefore has a real half and an imaginary half, and when subjected to the first Fourier transformation the resulting spectrum will also have real and imaginary data points. When these real and imaginary data points are arranged behind one another, vertical columns result. This transposed data... 

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

Real and imaginary parts Two equal blocks of frequency that result from Fourier transformation of the FIDs. 

Adjust zero time until the sum of squares of the imaginary spectrum is zero (i.e., the FID is then centred about zero). 

Real/Imag Toggles between display of the real and imaginary parts of the spectrum or FID. 

Load the ID H FID of peracetylated glucose D NMRDATA GLUCOSE 1D H GH 001001. FID, increase its vertical scale and inspect its last part on the screen. Notice the deviation of the mean FID from the zero horizontal line switch back and forth between the real and the imaginary part of the FID to recognize a DC offset between the two parts. From the Process pull-down menu choose the DC Correction option and apply a DC correction. Inspect the last part of the FID again. Fourier transform the FID with/without a DC... 

The real and imaginary spectra obtained by Fourier transformation of FID signals are usually mixtures of the absorption and dispersion modes as shown in Fig. 2.13 (a). These phase errors mainly arise from frequency-independent maladjustments of the phase sensitive detector and from frequency-dependent factors such as the finite length of rf pulses, delays in the start of data acquisition, and phase shifts induced by filtering frequencies outside the spectral width A. 

Figure 3. The acquired NMR signal (a), Free Induction Decay (FID) in the time domain response.Two transient signals each 90° out of phase comprise the real and imaginary part of the FID (b), its frequency domain response using Fburier transformation.

The actual Fourier transform is a digital calculation, so not all frequencies are tested. In fact, the number of frequencies tested is exactly equal to the number of time values sampled in the FID. If we start with 16,384 complex data points in our FID (16,384 real data points and 16,384 imaginary data points), we will end up with 16,384 data points in the real spectrum (the imaginary spectrum is discarded). Another difference from the above description is that the actual Fourier transform algorithm used by computers is much more efficient than the tedious process of multiplying test functions, one by one, and calculating the area under the curve of the product function. This fast Fourier transform (FFT) algorithm makes the whole process vastly more efficient and in fact makes Fourier transform NMR possible. 

Recall that the raw NMR data (FID) consists of two numbers for each data point one real value and one imaginary value. After the Fourier transform, there are also two numbers for each frequency point one real and one imaginary. In a perfect world, the real spectrum would be in pure absorptive mode (normal peak shape) and the imaginary spectrum would be in pure dispersive (up/down) mode. In reality, each spectrum is a mixture of absorptive and dispersive modes, and the proportions of each can vary with chemical shift (usually in a linear... 

Quadrature detection in t2 gives us two FIDs (real and imaginary) by sampling both the Mx component and the My component of the net magnetization as it precesses. This allows us to put zero audio frequency in the center of the spectral window and defines the left side... 

Both FIDs are acquired with the same t value, and both are encoded with the same frequency 2a in t, but they are 90° out of phase (cosine vs. sine modulation in t ), just as the real and imaginary channels of the receiver (Mx and My) are 90° out of phase. This gives us our quadrature detection in F, allowing us to put zero F audio frequency in the center of the F spectral window. 

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