《天然药物化学》课程教学资源（文献资料）Basic Principles of NMR.pdf_P16-P19

16 gna able sign in the frequency domain 1.6.NYQUIST THEOREM AND DIGITAL FILTERS Questions to be addressed in the current section include 1 What is the Nyquist theor 2.Wha dow (SW) sampling rate of 3.What is digital filtering and its application to NMR? with a certain sam theorem (Brac vell 1986)states that for the frea ency to be accurately represented the sampling rate is required to be at least twice the frequency.In other words,the highest fre- quency which can be accurately sampled is half of the sampling rate.This frequency is called the Nyquist frequency,which is defined as the spectral width,or the spectral window (SW).As a result,two time points must be recorded per period of a sinusoidal signal.The time interval of sampling is called the dwell time(DW)and 1/DW is the sampling rate,which based on the Nyquist theorem has the relationship with SW: fa SW=2DW (1.38) va=v-kf 1.39) in which vis the frequency ofthe signal.f is the n auist frea .va is the aliased frequency and k is an integer.For example,if the spectral width is set to 8 kHz,a signal with freq 9kHz will appear at 1 kHz.A signal with a frequency outside the SWcan also be folded into the spectrum,called the folded frequency.For an fa of 8 kHz,for instance,a signal with frequency 9 kHz will be folded at 7 kHz.Usually,a folded peak in an NMR spectrum shows a different phase than the unfolded peaks.In order to remove the aliased or folded signals, analog filters can be utilized before the signal is digitized.However,frequencies beyond the pass band (including noise)can still appear in the spectrum as the folded or aliased frequencies because the transition band between the pass band and stop band of analog filters is rather large (see Figure 1.9 for the definition of the transition band,pass band,and stop band of a filter) The real solution to avoid folding-in of signals and noise is to utilize digital filters combined with oversampling (Winde 1997:Moskau,2001 ng der otes that th gnal is acquired using and a larg spectra s determ ng rate yquist t 10-folary.Beca ersampl e oversampling is to reduce the e produce ed by the ADC in

16 Chapter 1 Therefore, quadrature detection (two detectors aligned perpendicularly) is required to detect NMR signals with a distinguishable sign in the frequency domain. 1.6. NYQUIST THEOREM AND DIGITAL FILTERS Questions to be addressed in the current section include: 1. What is the Nyquist theorem and how does it affect NMR signal detection? 2. What is the relationship between the spectral window (SW) and the sampling rate of detection? 3. What is digital filtering and its application to NMR? In the analog-to-digital conversion of the NMR signal, the analog signal is sampled with a certain sampling rate by the ADC (analog-to-digital converter, Chapter 2). The Nyquist theorem (Bracewell, 1986) states that for the frequency to be accurately represented the sampling rate is required to be at least twice the frequency. In other words, the highest frequency which can be accurately sampled is half of the sampling rate. This frequency is called the Nyquist frequency, which is defined as the spectral width, or the spectral window (SW). As a result, two time points must be recorded per period of a sinusoidal signal. The time interval of sampling is called the dwell time (DW) and 1/DW is the sampling rate, which based on the Nyquist theorem has the relationship with SW: fn = SW = 1 2DW (1.38) If a signal frequency, ν, is higher than the Nyquist frequency, it will appear at a different frequency in the spectrum, called an aliasing frequency: νa = ν − kfn (1.39) in which ν is the frequency of the signal, fn is the Nyquist frequency, νa is the aliased frequency and k is an integer. For example, if the spectral width is set to 8 kHz, a signal with frequency 9 kHz will appear at 1 kHz. A signal with a frequency outside the SW can also be folded into the spectrum, called the folded frequency. For an fn of 8 kHz, for instance, a signal with frequency 9 kHz will be folded at 7 kHz. Usually, a folded peak in an NMR spectrum shows a different phase than the unfolded peaks. In order to remove the aliased or folded signals, analog filters can be utilized before the signal is digitized. However, frequencies beyond the pass band (including noise) can still appear in the spectrum as the folded or aliased frequencies because the transition band between the pass band and stop band of analog filters is rather large (see Figure 1.9 for the definition of the transition band, pass band, and stop band of a filter). The real solution to avoid folding-in of signals and noise is to utilize digital filters combined with oversampling (Winder, 1997; Moskau, 2001). Oversampling denotes that the time domain signal is acquired using a larger spectral width and a larger number of data points than necessary. Because the spectral width is determined by the sampling rate (Nyquist theorem), 10-fold oversampling increases both the spectral width and the number of data points by 10 times (whereas the acquisition time is not changed). The role of the oversampling is to reduce the “quantization noise” produced by the ADC in

Basic Principles of NMR 17 Stop band Stop band Passband Transition band Frequency → Amplitude V → Figure 1.9. Characterization of a bandpass filter. The signals with frequencies inside the pass band of the filter pass through the filter, while those with frequencies in the stop band are filtered out. The amplitudes of the signals with frequencies inside the transition band are attenuated (Winder, 1997). the case that the receiver gain has to be set to a high value, by spreading the noise over the larger spectral width. As a result, the dynamic range as well as the signal-to-noise ratio can be increased. In addition, application of oversampling causes a flatter baseline of the spectrum. A significantly larger number of data points is generated by the oversampling, which requires much disk storage space. For example, 20-fold oversampling of 32 k data points will generate 640 k data points (k is equivalent to multiplying the number by 1024). To avoid the unnecessary larger data sets a real-time digital filter is used to reduce the spectral width to that of interest by removing the frequency range outside the desired spectral width before the data are stored on disk. The digital filtering is achieved by a digital signal processor integrated with the ADC circuits (real-time) before time averaging, or by software (post-acquisition) after the acquired FID is transferred from the console to the host computer prior to storage on disk. Similar to analog filters, a digital filter is characterized by the pass band (SW of interest) and the shape of the stop band, which describes the steepness of the cutoffs of the filters (the stop band is the region to be filtered out, Figure 1.9). For post-acquisition filters, the steepness of the cutoff is determined by the number of coefficients. A larger number of coefficients defines a filter with sharper cutoffs and flatter pass band (brick-wall type with narrower transition band), whereas a smaller number of coefficients characterizes filters with slower cutoffs (wider transition band). The real-time digital filters can be characterized as two types: brick-wall type with sharpest cutoffs and analog-like with gradual cutoffs. 1.7. CHEMICAL SHIFT From the previous sections, we know how a detectable magnetization is generated and how an FID is acquired with quadrature detection. Now, we would like to know what kind of signals we are going to observe and how the information can be used. In the current section, questions will be addressed such as: 1. What is chemical shift?

18 All nucle would have the same were no other yif there or a give tope,dispersion f th is cau in the e em 0 nd the The di itude onal to Bo.which shields me n of the static field from the nuclei.This electro electrons.The net effect can be described using a quality called the shielding constant by: v=21- (1.40 The shielding constant is always less than i because the induced local magnetic field will not be larger than the applied magnetic field. The absolute zer roofchemical shift is the one obtained for a bare nucleus without electrons Although the absolute value of the chemical shift may be obtained for bare nuclei such as protons,it is convenient to use a specific compound as a reference,whose resonance frequency is set to the chemical shift value of zero.The chemical shifts of other resonances are expressed as the difference in electron shielding to the reference nucleus: 6=y-hef10 1.41) Vref in which v andv are the resonance freq ucleus in units。 nd is the cher nits of per million)of the nucleus with fr ncy v.Chemical shift 8 is independent of ma strength that is the resonances in resent in a spectrum remain the same when obtained at different magnets with different field s en ths. The refer ce comn ound is required to have the follow ties:(a)stability in a variety of solvents,(b)an unchanged chemical shift value over a wide range of ter perature and pH values,and (c)ease of handling.Two compounds are commonly used forH NMR reference:tetramethylsilane(TMS),which is the standard reference adopted by the IUPAC(International Union of Pure and Applied Chemistry) and 2,2-dimethyl-2-silapentane-5-sulfonic acid(DSS),which is a secondary IUPAC reference (Harris et al.,2001).Either of the reference compounds can be added into an NMR sample as an internal reference or used alone as an external reference.For internal referencing,the reference compound is dissolved with the sample,which clearly has limitations such as solu- bility,miscibility,or reaction with the sample.For external referencing,a reference compound issolved alone in a specific solvent and the chemi cal shift is measured for the re erence eithe in Its own tube or in a capillary insert tub inside the s mple tub equen n 1 the sam ope Be of the of t an NMR spe will not change nles s the 2H lock k frequency i adjusted.The gra ual drift of the nagnet

18 Chapter 1 2. Where does chemical shift originate? 3. What are the references and units of chemical shift? All nuclear spins of the same isotope would have the same resonance frequency if there were no other kinds of interaction in addition to the Zeeman interaction. In fact, for a given isotope, dispersion of the NMR signals of nuclei is caused by the difference in the environment surrounding the nuclei. One of the factors causing the difference in frequency is the electronic shielding. The torque generated by the magnetic field also causes a precession of electrons around the magnetic field direction. The directional electronic precession produces a local magnetic field with a magnitude proportional to B0, which shields some portion of the static field from the nuclei. This electronic precession is different from the random motion of electrons. The net effect can be described using a quality called the shielding constant σ by: ν = γ 2π B0(1 − σ ) (1.40) The shielding constant is always less than 1 because the induced local magnetic field will not be larger than the applied magnetic field. The absolute zero of chemical shift is the one obtained for a bare nucleus without electrons. Although the absolute value of the chemical shift may be obtained for bare nuclei such as protons, it is convenient to use a specific compound as a reference, whose resonance frequency is set to the chemical shift value of zero. The chemical shifts of other resonances are expressed as the difference in electron shielding to the reference nucleus: δ = ν − νref νref 106 (1.41) in which ν and νref are the resonance frequencies of the nucleus under study and the reference nucleus in units of megahertz, respectively, and δ is the chemical shift in units of ppm (parts per million) of the nucleus with frequency ν. Chemical shift δ is independent of magnetic field strength, that is, the resonances in ppm present in a spectrum remain the same when obtained at different magnets with different field strengths. The reference compound is required to have the following properties: (a) stability in a variety of solvents, (b) an unchanged chemical shift value over a wide range of temperature and pH values, and (c) ease of handling. Two compounds are commonly used for 1H NMR reference: tetramethylsilane (TMS), which is the standard reference adopted by the IUPAC (International Union of Pure and Applied Chemistry) and 2,2-dimethyl-2-silapentane-5-sulfonic acid (DSS), which is a secondary IUPAC reference (Harris et al., 2001). Either of the reference compounds can be added into an NMR sample as an internal reference or used alone as an external reference. For internal referencing, the reference compound is dissolved with the sample, which clearly has limitations such as solubility, miscibility, or reaction with the sample. For external referencing, a reference compound is dissolved alone in a specific solvent and the chemical shift is measured for the reference either in its own NMR tube or in a capillary insert tube inside the sample NMR tube. The zero frequency is set to the resonance frequency of the reference nucleus, which is used for all other experiments with the same isotope. Because of the high stability and homogeneity of NMR spectrometers, the external reference is of practical use for biological samples. In fact, once the chemical shift reference is calibrated on a spectrometer, the reference frequency will not change unless the 2H lock frequency is adjusted. The gradual drift of the magnetic

Basic Principles of NMR 19 field is corrected by the z0 current of the shimming assembly (see shimming and locking in Chapter 4). TMS is commonly used as the reference in 1H and 13C spectra for samples in organic solvents. Chemical shifts for all isotopes should include two decimal digits. DSS is commonly chosen as 1H and 13C external references for biological samples, which is dissolved in water in a pH range of 2–11 and used at 25◦C. The chemical shift of water or HDO has a temperature dependence that can be expressed as referenced to DSS over a wide temperature range: δH2O (ppm) = 4.76 − (T − 25)0.01 (1.42) For 31P NMR, 85% H3PO4 is the IUPAC standard reference which can be used externally (with a capillary insert). The chemical shift reference for 15N is complicated and sometimes very confusing. Because there is no compound similar to DSS or TMS available for 15N referencing as in the case of 1H or 13C, a variety of reference systems have been used to define 0.00 ppm for 15N. Although CH3NO2 is the IUPAC 15N reference, liquid NH3 is the most popular 15N reference for biological NMR. The disadvantage is the difficult handling of the sample. Indirect reference compounds are usually used such as 15N urea in dimethyl sulfoxide (DMSO) and saturated ammonium chloride in water. 15N urea in DMSO is a convenient sample as an indirect reference and has an 15N chemical shift of 78.98 ppm relative to liquid ammonium. It should be noted that an 15N urea reference sample must be locked at the frequency of 2H2O. A simple method to achieve this is to place a capillary tube with 2H2O inside the NMR tube of the urea sample. An alternative way to obtain the correct reference frequency using the 15N urea sample is to acquire the spectrum without 2H locking. After the lock frequency is set to be on the resonance of 2H2O using a 2H2O sample, the 15N urea sample is placed into the probe without altering the lock frequency. Shimming can be done with 2H gradient shimming. After setting the resonance frequency of 15N urea to 78.98 ppm, the frequency at 0.00 ppm is the reference frequency for 15N experiments in aqueous solutions. Recently a more convenient referencing system has been introduced that uses the 1H reference for heteronuclei through the frequency ratio of the standard reference sample to DSS (or TMS): % = 100 νX νDSS (1.43) in which νDSS and νX are the observed 1H frequency of DSS and the observed frequency of X nucleus of the reference sample. The values of for different isotope reference samples are listed in Table 1.1. The reference frequency for X nuclei can be calculated from the 1H reference frequency on the spectrometer according to: νX ref = ref νDSS = (ref /%)νDSS 100 (1.44) For example, if liquid NH3 is used as the 15N reference sample, the 15N reference frequency is given by: ν 15N ref = 10.1329118νDSS 100 (1.45)