Project B: A Program to Demonstrate the Fourier Transform Technique in NMR Spectroscopy.

Objectives.

• You have to write a computer program that creates a set of superimposed cosine waves of different frequencies, multiplies them by a decaying exponential function, Fourier transforms the data and displays the result on a terminal.
• You have to transfer the data to a Macintosh computer and plot it using the Cricket Graph program.

Background.

Procedure to calculate a Fourier Transform

• Write code to dimension (maximum 1024) the following one-dimensional arrays;

  BROAD, FID, XREAL, XIMAG. The array BROAD will contain the decaying exponential function, FID the superimposed cosine waves before weighting with BROAD, and XREAL and XIMAG are the so called 'real' and 'imaginary' parts of the of Fourier Transformation.

• Write code to define a character array TITLE to contain column headings and open a new file 'cricket.dat' which will contain data to be used by the Macintosh program Cricket Graph.
• Define a character variable TAB*1, and set it equal to;

  TAB = CHAR(9). Here, 9 represents the 'ascii' code for the TAB character, which will be needed later for writing out the data file 'cricket.dat'

• Write code to input the total number of spectral points N (Maximum 1024), to check that this number is exactly a power of 2 (ie 2**n) and to zero all the above arrays prior to calculation. Read in also T1, the initial time delay before measurement of the signal commences. The default value of T1 should be 0.
• Write code to input a frequency K and its intensity AI of a cosine curve. K is simply how many 2 ¼ cycles will occur within the space of the total data points N. Note that you need at least two data points to define a cycle of the cosine, and hence K < N/2 (the Nyquist Theorem). The fraction of a cosine wave between any two adjacent data points (in radians) is defined by;

  STEP = ¼ * K / (N/2) = ¼ * K * 2 / N

  Make sure that STEP is a floating point number by correctly converting any integers (ie N or K) using the intrinsic function FLOAT.

• Write code to evaluate the value of the cosine curve at each of the total number of points N using the equation for the nth point (1 <= n <= N);

  FID(n) = FID(n) + AI* COS ( (n-1+T1)*STEP)

  Here n (for NMR data at least) has the units of time, and the n-1 arises because you are assumed to measure the value of the FID at time = 0 (ie 1-1) rather than at time=1. T1 corresponds to any time delay before acquisition of the signal. This delay has important consequences regarding the phase of the signal (see below).

• Read in the frequency K and the intensity AI of the next cosine curve (if any) and repeat the above steps up to a maximum of five. After the last desired frequency, you should have an array FID cotaining N points each value of which corresponds to the sum of a number of superimposed cosine waves.
• You will now need to evaluate BROAD(n) for each of the N points. BROAD is defined by;

  BROAD(n) = EXP (-BR * ((n-1)/N) )

  where BR is a broadening factor. Note that in the above expression, real and integer numbers are (incorrectly) mixed and your own code should correct this using the FLOAT function. If BR = 0, BROAD(n) = 1; if BR > 0, BROAD contains a decaying exponential function. It turns out the the value of BR controls the WIDTH of the lines contained in XREAL after Fourier transformation. If BR = 0, the lines are infinitely narrow. This might cause arithmetic problems, and you should ensure in your code that BR > 0 under all conditions.

• Write code to multiply arrays BROAD and FID together and put the product in array XREAL.
• Call the subroutine FASTF(XREAL,XIMAG,N). This subroutine is 'linked' in using ugfor. Click here to see the source code.

  * After calling the subroutine, XREAL and XIMAG contain the real and imaginary parts of the spectrum. Actually, only the first N/2 data locations of each array contain the spectrum (see the Nyquist Theorem above). Write code to normalise each array such that the maximum value of any element in each array is 1 (ie the absolute intensity of the cosine waves is arbitrary).

• Write out to the terminal screen the following five variables per line for n=1 to N;

  n-1, BROAD(n), FID(n), XREAL(n), XIMAG(n)

  n-1 corresponds to the frequency of the cosine curves present in the FID, and has the value n-1 because the contents of XREAL(1) actually correspond to a zero cosine frequency.

• What should have happened by now is the following. If you defined N=128 data points, and requested cosine frequencies of say

  K=4, 17, 24 and 52 with intensities of say AI=1.0, 2.0, 2.0 and 1.0,

  the values of the variables XREAL(5), XREAL(18), XREAL(25) and XREAL(53) should be 1.0, 2.0, 2.0 and 1.0. The other locations should contain smaller number, but will not be zero if you used BR != 0, ie the lines are not infinitely narrow. This means the 'spectrum' contains lines at frequencies of 4, 17, 24 and 52 with intensities in the ratio 1,2,2,1.

• Check you program is working by trying some of the following;
• Alter the intensity of each cosine curve and verify the absolute value of XREAL(n) changes accordingly.
• Play around with the broadening factor and verify that larger values of BR lead to wider peaks.
• Try entering a value of the cosine frequency that is larger than N/2 and see what happens. The phenomenon is known as 'folding' and is a cause of many 'operator' errors during 'real' FT NMR spectroscopy.
• Try entering non-zero values of T1. This quantity is called the 'first order phase correction'. What effect does T1 have on the contents of XREAL and XIMAG? Actually, on a real spectrometer, 0<T1<1, and corresponds to the finite time taken for the electronics to respond before acquisition of the signal can commence.
• For the more adventurous, try changing the algorithmic form of BROAD. Many different weighting functions are used in NMR spectroscopy, and each is used to 'massage' the data in different ways. You could try two successive functions, ie an exponential function followed by a Gaussian function centred in the middle of the Free induction decay.

Procedure to Display a Fourier Transform

N N N N/2 N/2

The first row after the * contains the five titles of each column (the above are only suggestions, you can try your own). These will appear eventually to identify the data in the plotting program. TAB means write out character variable TAB = CHAR(9) between each column of data. Using this method, you can write out the full N points of the Time/Freq, BROAD and FID arrays, and only N/2 points of the XREAL and XIMAG arrays. Remember however, that only the first N/2 values of XREAL(n) and XIMAG(n) represent the actual spectrum. Can you interpret what the values of XREAL(1 + N/2) through to XREAL(N) correspond to by inspection of the resultant graphs?

Transfer the file cricket.dat to a Macintosh Computer located in room 170 and plot the contents using the program Cricket Graph. See here on how to mount an Indigo file on a Macintosh

If you want to see how the output of the program should look, you can try running the "executable" only version of the program by typing

demoftnmr

from the console window. The project handed in should of course correspond to your own listing of this program.