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

then

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(x/y)

log(1/x) = log(x

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 (P_{1} and P_{2}) a decibel (dB) is defined as

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

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((V

dB = 10 log((V

db = 20 log(V

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

then

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

where

The trigonometric identities are used in geometric calculations.

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.

Sin(θ_{1}) Cos(θ_{2})
= 1/2 Sin(θ_{1} + θ_{2})
+ 1/2 Sin(θ_{1} - θ_{2})

Sin(θ_{1}) Sin(θ_{2})
= 1/2 Cos(θ_{1} - θ_{2})
- 1/2 Cos(θ_{1} + θ_{2})

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

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.

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.

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

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

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

Copyright © 1997-2017 J.P. Hornak.

All Rights Reserved.