The Basics of NMR

Chapter 2 THE MATHEMATICS OF NMR

Logarithms and Decibels
Exponential Functions
Trigonometric Functions
Differentials and Integrals
Vectors
Matrices
Coordinate Transformations
Convolutions
Imaginary Numbers
The Fourier Transform

Logarithms and Decibels

Scientists have many shorthand ways of representing numbers. These representations make it easier for the scientist to perform a calculation or represent a number. A logarithm (log) of a number x is defined by the following equations. If

x = 10^y

then

log(x) = y.

You will see in the next section, logarithms do not need to be based on powers of 10.

Logarithms are useful, in part, because of some of the relationships when using them. For example,

log(x) + log(y) = log(xy)
log(x) - log(y) = log(x/y)
log(1/x) = log(x^-1) = -log(x)
log(x^y) = y log(x)

A useful application of base ten logarithms is the concept of a decibel. A decibel is a logarithmic representation of a ratio of two quantities. For the ratio of two power levels (P₁ and P₂) a decibel (dB) is defined as

dB = 10 log(P₁/P₂).

Sometimes it is necessary to calculate decibels from voltage readings. The relationship between power (P) and voltage (V) is

P = V²/R

where R is the resistance of the circuit, which is usually constant. Substituting this equation into the definition of a dB we have

dB = 10 log((V²₁/R)/(V²₂/R))
dB = 10 log((V²₁)/(V²₂))
dB = 10 log((V₁)/(V₂))²
db = 20 log(V₁/V₂).

Exponential Functions

The number 2.71828183 occurs so often in calculations that it is given the symbol e. When e is raised to the power x, it is often written exp(x).

e^x = exp(x) = 2.71828183^x

Logarithms based on powers of e are called natural logarithms. If

x = e^y

then

ln(x) = y,

Many of the dynamic NMR processes are exponential in nature. For example, signals decay exponentially as a function of time. It is therefore essential to understand the nature of exponential curves. Three common exponential functions are

y = e^-x/t

y = (1 - e^-x/t)

y = (1 - 2e^-x/t)

where x is a constant.

Trigonometric Functions

The basic trigonometric functions sine

and cosine

describe sinusoidal functions which are 90^o out of phase.

The trigonometric identities are used in geometric calculations.

Sin(θ) = Opposite / Hypotenuse
Cos(θ) = Adjacent / Hypotenuse
Tan(θ) = Opposite / Adjacent

Csc(θ) = 1 / Sin(θ) = Hypotenuse / Opposite
Sec(θ) = 1 / Cos(θ) = Hypotenuse / Adjacent
Cot(θ) = 1 / Tan(θ) = Adjacent / Opposite

Three additional identities are useful in understanding how the detector on a nuclear magnetic resonance spectrometer operates.

Cos(θ₁) Cos(<θ₂) = 1/2 Cos(θ₁ - <θ₂) + 1/2 Cos(θ₁ + θ₂)

Sin(θ₁) Cos(θ₂) = 1/2 Sin(θ₁ + θ₂) + 1/2 Sin(θ₁ - θ₂)

Sin(θ₁) Sin(θ₂) = 1/2 Cos(θ₁ - θ₂) - 1/2 Cos(θ₁ + θ₂)

The function sin(x) / x occurs often and is called sinc(x).

Differentials and Integrals

A differential can be thought of as the slope of a function at any point. For the function

the differential of y with respect to x is

An integral is the area under a function between the limits of the integral.

An integral can also be considered a sumation; in fact most integration is performed by computers by adding up values of the function between the integral limits.

Vectors

A vector is a quantity having both a magnitude and a direction.

The magnetization from nuclear spins is represented as a vector emanating from the origin of the coordinate system. Here it is along the +Z axis.

In this picture

the vector is in the XY plane between the +X and +Y axes. The vector has X and Y components and a magnitude equal to

( X² + Y² )^1/2

Matrices

A matrix is a set of numbers arranged in a rectangular array.

This matrix has 3 rows and 4 columns and is said to be a 3 by 4 matrix.

To multiply matrices the number of columns in the first must equal the number of rows in the second.

Click sequentially on the next start buttons to see the individual steps associated with the multiplication.

Coordinate Transformations

A coordinate transformation is used to convert the coordinates of a vector in one coordinate system (XY) to that in another coordinate system (X"Y").

Convolution

The convolution of two functions is the overlap of the two functions as one function is passed over the second.

The convolution symbol is

. The convolution of h(t) and g(t) is defined mathematically as

The above equation is depicted for rectangular shaped h(t) and g(t) functions in this animation.

Imaginary Numbers

Imaginary numbers

are those which result from calculations involving the square root of -1. Imaginary numbers are symbolized by i.

A complex number is one which has a real (RE) and an imaginary (IM) part. The real and imaginary parts of a complex number are orthogonal.

Two useful relations between complex numbers and exponentials are

e^+ix = cos(x) +isin(x) and
e^-ix = cos(x) -isin(x).

Fourier Transforms

The Fourier transform (FT) is a mathematical technique for converting time domain data to frequency domain data, and vice versa.

The Fourier transform will be explained in detail in Chapter 5.