## Analog Noise

### 3.7.3 Dynamic Range

Dynamic range is the range of concentrations or signal intensities over which you can detect samples in a single measurement. If you are trying to find a "needle in a haystack"—for example, observing the XH spectrum of a 1 mM protein sample in 55 M H2O—you need to have dynamic range. Signal to noise is an absolute limitation: It sets a minimum of signal height that can be observed for a "weak" signal, regardless of any strong signals in the sample. Dynamic range is a relative limitation: It determines how small a signal can be detected relative to the largest signal in the sample. The receiver gain is limited by the largest signal in the sample because the digitizer limits will be exceeded first by that large signal as you increase the receiver gain. Thus at this limit, the "top" of the digitizer (largest integer value it can assign to a signal) is set to the signal strength of the largest signal in the sample. We say that the digitizer is "dominated" by this large signal. The small signal that "rides" on top of this large signal FID has to be accurately described by the digitizer. If it is smaller than one integer step in the ADC output, the signal is lost. If it is only a few integer steps, it will be picked up but the peak in the spectrum will be "blocky"—described by large square integer steps in intensity instead of by smooth curves, similar to a bitmap drawing with very low resolution. The ratio between the largest integer and the smallest (one unit) is the dynamic range of the digitizer. A 12-bit ADC uses 12 bits to digitally measure the analog voltage at each sample point, so the dynamic range is 4096 to 1 (212 to 1). More modern instruments use 16-bit digitizers, so they have a dynamic range of 65,536 to 1 (216 to 1). This can be further increased by oversampling (acquiring many samples for each data point and averaging them), since this allows partial integer values when the average is computed. For example, if you digitize four equally spaced data points during an 80 ^s dwell time and average the value to one data point, you can have a result that is, for example, 645.00, 645.25, 645.50, or 645.75. You now have four times as many intensity levels to choose from and you have increased your digitizer resolution by 2 bits. A modern NMR spectrometer can typically oversample by a factor of 32, leading to 5 additional bits or a total of 21 effective bits in the digitizer and a dynamic range of 2,097,152 to 1!

3.8 SPECIAL TOPIC: OVERSAMPLING AND DIGITAL FILTERING

The sampling rate is the rate at which the ADC samples the raw analog FID audio signal and converts the intensities (voltages) into numbers. The delay between samples is called the dwell time (DW) so that the sampling rate can also be expressed as 1/DW in units of hertz. Because we must have at least two samples per cycle of a sinusoidal signal to properly define its frequency without aliasing, the sampling rate is determined by the highest frequency we need to digitize. The user defines the spectral width (SW) and the spectrometer calculates the sampling rate:

For a typical spectral width of 6250 Hz (12.5 ppm for a XH spectrum on a 500 MHz spectrometer), the sampling rate is 12,500 samples per second. For a 13C spectrum the spectral width is larger, so that a13 C spectrum with a 250 ppm spectral width on a 600 MHz spectrometer would require a sampling rate of

Rate = 2 x SW = 2 x 250 ppm x 150 MHz = 75,000 samples per second since the 13C frequency on a 600 MHz spectrometer is 600 x (yC/yH) = 150 MHz. This may seem like an impressive feat, but it is well within the capabilities of even an older generation ADC. Even an inexpensive modern ADC can sample at 400,000 samples per second, so that the capacity is 5.33 times greater than that needed for the 13C spectrum example and 32 times greater than that needed for the XH spectrum example. The question naturally arises: Is there anything useful we can do with the excess sampling capacity?

### 3.8.1 Oversampling

In the XH spectrum example, if we sample at 400,000 samples per second, we will have 32 samples for every data point that we actually need for the FID. The simplest thing to do with all of this extra data is to divide them into groups of 32 consecutive data samples and average each group to give a single data value. Is this any better than sampling 12,500 samples per second? Yes, because any time you repeat a measurement many times and average the results, you get a more accurate measurement. Furthermore, since each measurement is an integer value with a limited dynamic range (e.g., -32,767 to 32,768 for a 16-bit ADC) you would have a much finer range of possible intensities because each bit (0 or 1) can now have 32 possible values after averaging 32 measurements (0, 1/32, 2/32, ..., 31/32). This finer "graininess" of the intensity values might be useful if we are trying to find a very weak signal in the presence of a very strong one (needle in a haystack problem). Essentially we now have more significant figures (precision) in each measurement, increasing the dynamic range (ability to detect small signals in the presence of large ones).

This simple process of averaging each set of 32 raw measurements to get a single value is called decimation, and we would define the decimation factor as 32 in this case. Thus we see a part of the overall strategy: Oversampling produces many more data points than we need, and decimation averages them to give us the required sampling rate determined by the spectral width. But we can do much more than just increase the accuracy of our measurements: We can use digital methods to construct a filter that rejects signals outside of the spectral window without affecting the desired signals within the spectral window. With digital filtering, we can set a narrow spectral window that covers only part of the spectrum, and none of the other peaks in the spectrum will alias or "fold in" to the narrow window because they are removed by the digital filter. To understand how this works, we need to first understand the "old fashioned" analog filter used in an NMR spectrometer.

### 3.8.2 Analog Filtering

We have already seen that the digitization of the FID signal (sampling at regular intervals) sets a limit on the frequencies that can be detected without aliasing. Any frequency larger than SW/2 (with quadrature detection, the frequency is zero at the center of the spectral window, so the edges are at ±SW/2) will be aliased back into the spectral window. This applies to noise as well as to peaks, so that without some way of rejecting signals outside the spectral window, we would have a huge amount of noise aliased into the spectral window and the signal-to-noise ratio would be abysmally low. To avoid this, an analog audio filter is included before the ADC to remove any frequencies with absolute value greater than SW/2. Analog filters are constructed from capacitors, resistors, and inductors and have switches to match the "bandpass" (region of frequencies passed through) of the filter to the spectral width set by the user. The frequency response curve shows how effectively a filter blocks the signals outside the spectral window and to what extent it affects signals within the spectral window. The ideal filter response would be "flat" throughout the spectral window and would drop instantly to zero outside the spectral window (Fig. 3.20). Unfortunately, real audio filters tend to attenuate signals in the spectral window that are near the edge, and drop off only gradually outside the spectral window. Peaks that are outside the spectral window are aliased with some attenuation into the spectral window, along with the noise. Thus we expect less signal and more noise near the edges of the spectral window, and this will degrade the sensitivity as well as the accuracy of integration. Peaks and noise that are very far outside the spectral window are effectively blocked by the analog filter. The goal of digital filtering is to achieve a nearly ideal filter response curve, matched to the spectral window, with only the computational tools of a computer chip rather than the cumbersome and imperfect electronic components of an analog circuit. The filter will be moved from its place in the analog stream before the ADC and placed in the digital stream after the ADC.

### 3.8.3 Decimation with Digital Filtering

Let us return to our example of a FID oversampled 32 times and decimated by a simple average of each set of 32 data values. It turns out that this method of decimation by simple averaging actually discriminates among frequencies in the FID, so that a high frequency that changes sign many times during the 32 samples will be nearly eliminated (positive swings cancel the negative swings) and a low frequency that is nearly constant for the 32 samples is unaffected. So this simple filter is a crude kind of low-pass filter: It cuts out the high frequencies and passes (leaves unchanged) the signals with low frequencies.

To understand how this works, consider a simpler example: a filter that averages groups of four data points to give a single filtered value. If the data is oversampled with a decimation factor of 4, the sampling rate (1/DW) is eight times the spectral width (8 x SW) instead of twice the spectral width (2 x SW). Consider the effect of this process on a pure sine wave FID with frequency 4 x SW, which is sampled at a rate of 8 x SW (delay between samples is 1/(8 x SW)). The raw data has sampled values of 1, -1, 1, -1, ... and has a frequency of 4 x SW since it goes through a complete sine wave cycle in two data points (period = 1/(4 x SW)). Such a signal would be reduced to zero by the filter, which averages each group of four data points. A zero response would also be obtained for a signal of frequency 2 x SW (0,1,0, -1,...). A signal with frequency SW (data values 0,0.707,1,0.707,0, -0.707, -1, -0.707) would be retained with reduced intensity because decimation would give two data points: 0.604 and -0.604. This pair would repeat leading to a correct frequency measurement of SW. A signal of zero frequency (1,1,1,1) would also be retained with unchanged intensity (1.0 for each averaged value). We can map out the frequency response curve for this digital filter as shown in Figure 3.21: the response is maximum at the left edge of the spectral window, drops to 71% at the right edge, then drops to zero and oscillates and decays for larger frequencies (for simplicity we are not considering real and imaginary data acquired with quadrature detection, so the edge of the spectral window is at frequency SW rather than SW/2). The response is a "sinc" function (sin(x)/x) that effectively passes low frequencies and discriminates against high frequencies, but it has many undesirable characteristics. The response is far from constant in the interval 0-SW, and there is significant nonzero response outside of the spectral window. Furthermore, the filtered data switches sign in some regions, indicating an alteration of the original phase. Clearly we need to design a better filter, and to do so we must examine various digital filters with different sizes and shapes.

### 3.8.4 Digital Filtering and the Convolution Theorem

Consider a filter that averages in groups of three data points without any decimation. Every group of three consecutive raw data points is summed together, and the sum is applied to the center value (second of three data points) in a new data set—the digitally filtered FID. We can think of the filter as a rectangle-shaped window, three data points wide, which moves through the FID data, stopping to add together three points and deposit the sum in the new data set and then moving one data point to the right-hand side and repeating the process. What effect will this have on a simple sine function FID? The math is shown in Figure 3.22(a). Each data value on the bottom is the sum of the three data values above it: one above and to the left-hand side, one directly above, and one above and to the right-hand side. Note that the filtered data has one extra data point at each end since the filter is three data points wide and begins to encounter data when the first raw data point is reached. After two anomalous points at the beginning (the "group delay" of the filter) the filtered data are identical to the raw data. Thus this frequency (2 x SW sampled at a rate of 8 x SW) is passed without any change by the filter. With a raw FID of frequency 4 x SW we see that the data are passed by the filter but with inverted phase (Fig. 3.22(b)). A frequency of zero (all data points equal to 1) is passed with high efficiency (all data points equal to three). Thus the frequency response for this digital filter is a sinc function with a maximum response at zero frequency, a smaller positive response at 2 x SW, and a negative response at 4 x SW.

A wider filter function gives a narrower frequency response. Consider a digital filter that is four points wide, with all four values given equal weight. A frequency of 4 x SW is

Raw: 0-1010-1010-1010-1

Filtered: 1 01 00000000000 -1 0 -1

Raw: 0-1010-1010-1010-1 blocked by the filter after a brief transient response (Fig. 3.22(c)). Likewise a frequency of 2 x SW is also blocked (Fig. 3.22(d)). As before, a zero frequency (all data points equal to 1) is passed with maximum response. Similar arguments show that odd multiples of SW (SW, 3 x SW, etc.) are passed by this filter. The filter response curve for the four-point filter is a sinc function with null points at 2 x SW and 4 x SW, so it is narrower than the response curve for the three-point filter (Fig. 3.23). Note that the shape of the frequency response curve (frequency domain) is the Fourier transform of the shape of the filter function (time domain). A filter function that is rectangular in shape (three or four equally weighted points with all other points weighted zero) leads to a frequency response curve that is a sinc (sin(x)/x) function, and the sinc function is narrower in frequency as the filter function is made wider in time. This is just what we expect for the Fourier transform of a rectangular shape in time domain. This principle can be stated more generally if we consider that the filter function need not be rectangular, that is, the points in the filter do not have to be weighted equally. A general digital filter has N coefficients or weighting factors ci, c2, c3,____, cN, and it is passed through the raw data stopping at each new position where the weighted average is calculated (Fig. 3.24, top), where r1, r2, r3,... represent the raw (unfiltered) FID data and the filtered data value for point 7 is dj = Ci r3 + C2 r4 + C3 r5 + C4 re + C5 r7

The filter is then moved to the next position and the weighted average is again calculated (Fig. 3.24, bottom). The value for point 8 is dg = ci r4 + C2 r5 + C3 r6 + C4 rj + C5 r8

This process of moving the filter function through the raw data and calculating weighted averages is called convolution, and the digitally filtered data d1, d2, d3, ... are called the n r2 r3 r4 r5 r6 r7 r8 r9

c2 c3 c4 c5

Raw FID filter

Raw FID

convolution of the raw data (r1, r2, r3, ...) with the filter function (ci, c2, c3, ...). In mathematical terms, d(t) = c(t) ® r(t)

where d(t) is the digitally filtered FID, c(t) is the filter function, and r(t) is the raw FID, all of them digital time-domain functions. The process of digital filtering is the same as the mathematical operation of convolution, represented by the symbol

The convolution theorem states that the Fourier transform of the convolution (d) is simply the product of the Fourier transforms of the two functions (c and r) that are combined by convolution to make d. Thus convolution in time domain is equivalent to simple multiplication in frequency domain:

where D(f) is the spectrum obtained by Fourier transformation of the digitally filtered FID d(t), R(f) is the spectrum obtained by Fourier transform (FT) of the raw FID r(t), and C(f) is the frequency response curve obtained by FT of the filter function c(t). To determine the effect of any digital filter on the spectrum, we simply look at the Fourier transform of the digital filter's shape (its coefficients).

Now that we understand the exact relationship between the shape of the weighting factors (coefficients) used in the digital filter and the frequency response curve it produces in the spectrum, we can begin to design a digital filter with ideal properties. The ideal frequency response curve is flat throughout the spectral window and falls off instantly to zero outside the window. In mathematical terms, this is a rectangular shape in frequency domain. The design question then boils down to this: What time-domain function (digital filter shape) will give, after FT, a rectangular function (frequency response curve)? The answer is simple: A sinc (sin(t)/t) time-domain function gives a rectangular frequency-domain function upon FT. The narrower we make the sinc function in time domain, the wider will be the rectangular frequency response curve. So this is our goal: to construct a set of digital filter coefficients that correspond to a sinc function in time domain.

### 3.8.5 Optimizing the Digital Filter

The digital filter cannot be infinitely long, so we will have to cut off (truncate) the sinc function at some point. This will affect the frequency response curve, so it will not be a perfect rectangle, and some of the proprietary (trade secret) information guarded by NMR instrument makers (Bruker and Varian) has to do with the optimization of finite-sized filter functions to give optimal frequency response. We can start with a fairly simple filter: a 15-point sinc function with "sinc" coefficients (Fig. 3.25). Note that this is a symmetrical sinc shape with a maximum at the center (c8) and two null points (c2 and c5, c11 and c14) on each side. The effect of this filter was tested on a "fake" FID that gives, after FT, a spectrum with 41 equally spaced peaks of equal height and width. This raw FID was digitally filtered by sliding the 15-point "sinc" filter (Fig. 3.25) through it, calculating the sum of 15 products (ci x rj) at each stop. The Fourier transform of the digitally filtered FID is shown in Fig. 3.26. Clearly the effect of truncating the sinc function (using only 15 points) is dramatic: the response sags in the center, and the cutoff is not very sharp at the edges of the spectral window. In addition, there is significant intensity outside the spectral window with alternating phase. But this frequency response curve is much better than the simple Figure 3.25

sine functions obtained with "flat" (rectangular) digital filters (Fig. 3.23). The filters in use on modern spectrometers use many more coefficients and are optimized to compensate for truncation effects, so that rejection of signals outside the spectral window is excellent and the cutoff is very sharp.

### 3.8.6 Combining Decimation with Digital Filtering

Our discussion of digital filtering was inspired by a need to reduce the sampling rate from the maximum possible permitted by the ADC to the rate desired for the spectral window of interest (2 x SW). We found that a simple average is not a good way to decimate the oversampled data because it introduces a sinc-shaped frequency response curve into the spectrum. After a detailed examination of the effect of filter weighting on frequency response, we found that this seeming disadvantage can be used to construct a frequency filter that is far better than any analog audio filter. Thus we can get the advantages of oversampling (greater accuracy and dynamic range) as well as the advantages of digital filtering (very sharp or "brick wall" audio filters) by using a carefully planned shaped digital filter to average the oversampled data and reduce (decimate) it to the desired sampling rate. The only difference in our convolution process is that the filter function does not stop at every point in the raw FID; instead, it jumps ahead by many points each time. For example, if the raw FID is oversampled by a factor of 24 (sampling rate 48 x SW), the filter function will jump 24 points forward each time and calculate a weighted average. The filter function can contain many points: for example, 3000 points for a decimation factor of 24. The sinc-shaped "footprint" of the filter function moves forward through the raw FID jumping 24 points forward each time and calculating the weighted average over the whole filter function width (3000 points) at each stop. This weighted average becomes the data value for the digitally filtered FID at each stop, so that the new FID has only 1/24 the number of points as the raw FID (decimation factor = 24).

### 3.8.7 Practical Considerations and Applications

Digital filtering is more or less invisible to the routine user. You will notice that the filtered FID has a "dead time" at the beginning during which intensities are very low, and then the normal FID "blossoms out" after this group delay. The group delay is the time necessary for the digital filter function to "walk into" the raw FID and start generating significant intensity. For a sinc function, most of the intensity of the function is at the center, so the digitally filtered FID does not start to show intensity until this part of the filter function reaches the beginning of the raw FID. This may be as far as 64 points into the digitally filtered FID. The effect of this delay is the same as the effect of a delay in the start of acquisition after a pulse: It introduces a very large first-order (chemical shift dependent) phase error into the spectrum. This will appear as a lot of "squiggles" in the baseline of the spectrum in a shape similar to a sinc function centered at the center of the spectral window. First-order phase errors in the order of 30,000 are typical, so that it is nearly impossible to correct them manually. On the spectrometer, the NMR software calculates this phase correction from the decimation factor and automatically applies it, so the spectrum never shows any unusual phase errors. When using a "third-party" software package (e.g., Felix), the decimation factor must be supplied so that the software can calculate these phase corrections. The only other noticeable difference between digitally filtered data and analog filtered data is that the Bruker "brick wall" filter function produces a slight downturn in the baseline at the extreme edges of the spectral window. This "Bruker frown" is more preferable to the old "Bruker smile" baseline distortion because the baseline is extremely flat through nearly all the spectral window.

Because digital filtering can produce a "brick wall" frequency response, any peak that falls outside the spectral window is removed completely and will not alias. This can be a problem if you set the spectral window too narrow: You will never be aware of the peaks you miss. If you accidentally set the spectral window to include nothing but noise, you will get just that in the spectrum: nothing but noise! The good news is that if we are only interested in a small part of the 1D spectrum, we can "cut out" the rest of the spectrum using the digital filter. For example, in a 2D 15N-1H HSQC spectrum of a protein, we are only interested in the HN (H of the peptide NHCO linkage) region of the spectrum (7-11 ppm). The rest of the spectrum, including the intense water resonance at 4.7 ppm, can be cut out by setting our spectral window to include only the HN region. That does not allow you to turn up the receiver gain, however, since it is the raw, unfiltered analog FID that is being digitized by the ADC. In 2D NMR the digital filter only applies to the directly detected dimension (F2). Any excitation that occurs outside the F1 (vertical) spectral window will alias into the spectral window.

3.9 NMR DATA PROCESSING—OVERVIEW

When you have finished acquiring your NMR data, you will need to process the data into a spectrum and plot that spectrum on paper with a ppm axis, and possibly with integrals, peak lists, and other features. You may want to expand interesting or complex regions of the spectrum as insets or on a separate plot so that the fine structure of peaks (splitting patterns, J values, etc.) can be analyzed. Each NMR instrument has its own software for data processing, and it can be daunting to try to learn all of the different commands and operations. The actual data processing task, however, is the same in all cases and the learning curve will be more efficient if we first deal with these tasks in general without discussing individual NMR programs. Starting with an FID (raw time-domain data), we need to carry out the following operations:

(a) Multiply the FID by a multiplier or window function.

(b) Fourier transform the time domain data to obtain a frequency domain spectrum.

(c) Correct for phase errors by adjusting the phase.

(d) Find a reference standard peak and set its chemical shift to the reference value in parts per million.

(e) Expand the desired region of the full spectral window to be plotted.

(f) Plot the spectrum.

In addition, there are several optional operations we might want to perform:

(g) Add zeroes to the end of the FID to increase digital resolution ("zero fill").

(h) Flatten the spectrum baseline (average of noise regions where there are no peaks).

(i) Measure the area under individual peaks by integration.

(j) Plot the chemical shift values of peaks on the spectrum, or print a separate list. (k) List the acquisition and processing parameters on the spectrum or in a printout. (l) Expand and plot smaller regions of the spectral window.

To gain a better understanding of what is involved in these steps, we will start with a look at the raw time-domain data (FID).

Raw time-domain NMR data (the FID) consists of a list of numbers, usually negative and positive integers, as a function of time in equal time increments. The list is usually quite long, with as many as 16,000 or 32,000 entries. There are two types of data values (reflecting the two channels of the NMR receiver): real and imaginary. Data are arranged in the order: real, imaginary, real, imaginary, , regardless of the acquisition mode (alternating or simultaneous) of the pairs. This is the "raw" data of the NMR experiment. The data are contained in a computer disk file (if you saved it!) as a binary file containing a header (with some information about the spectrometer settings—not used on Bruker AMX and DRX instruments) and a list of numbers without the time values. For newer instruments, the NMR data are saved as a directory that contains the binary FID file (fid), and a number of text files containing parameters (Bruker acqu, Varian procpar) and other information relevant to the experiment. On the older instruments (Bruker AM and Varian Gemini) you simply save the FID as a single binary file.

Some experiments involve more than one FID: For example, DEPT analysis (Chapter 7) performs a 13C experiment four times with different parameter settings; 2D experiments involve collections of up to 750 similar FIDs. These can be combined in a single binary file. The FIDs are just listed one after the other in a single continuous list of data that Bruker calls a serial file. Varian treats these multiple FID files in the same way as single-FID files, so they can be used for "arrayed" experiments (a set of 1D experiments acquired by varying some parameter such as pulse width) or for 2D experiments. The file name is the same in either case that is, "fid" Bruker uses the filename "fid" only for single FIDs, and instead uses "ser" for the binary data of all serial files.

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### Responses

• Jean
How varian nmr deal with group delay?
26 days ago