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7.1 Introduction toFourier Transforms

• frequency content and SNR enhancement• Fourier transform integrals• the relationship between time and frequency• the highest frequency of a signal can be estimated from the slope and curvature of a sine - a Gaussian peak example will be used• the frequency content of repetitive signals the use of basis set transforms and theorems

Using Frequency to Improve the SNR

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Most instrumental methods of signal-to-noise enhancement are based on electronic components or measurement strategies that are frequency dependent.

As an example consider a signal composed of a narrow, Gaussian-shaped band of frequencies and noise which has constant amplitude at all frequencies. The signal-to-noise ratio can be improved by restricting measurement to the region marked by the vertical lines.

In order to use frequency-dependent methods of signal-to-noise enhancement we need to be able to estimate both signal and noisefrequencies.

0

2

4

6

8

10

12

0 2000 4000 6000 8000 10000

frequency (Hz)

ampl

itude

measure only the frequencies between the lines

signalnoise

The Fourier Transform

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Ordinarily we make a measurement in the time domain (signal amplitude as a function of time). To obtain the frequency content we need to obtain the signal spectrum (signal amplitude as a function of frequency). Conversion between the two domains is obtained by a mathematical procedure called a Fourier transform.

The forward Fourier transform converts a temporal signal into spectrum.

The reverse (or inverse) Fourier transform converts a spectrum into a temporal signal.

Note that the sign of the exponent changes.

( ) ( ) ( ) ( ) ( ) ( )2 cos 2 sin 2i ftf F t e dt F t ft dt i F t ft dtπφ π π∞ ∞ ∞

−

−∞ −∞ −∞

= = −∫ ∫ ∫

( ) ( ) ( ) ( ) ( ) ( )2 cos 2 sin 2i ftF t f e df f ft df i f ft dfπφ φ π φ π∞ ∞ ∞

+

−∞ −∞ −∞

= = +∫ ∫ ∫

Sequential Transforms

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Let the symbol F represent the forward transform operation and F -1 represent the reverse transform operation.

( ) ( )( ) ( )( ) ( )

1

1

f F t

F t f

F t F t

φ

φ−

−

=

=

=

F

F

F F

As a result, sequential application of the forward and reverse transforms returns the initial function.

The original function will not be obtained unless the numeric sign of the exponent in the forward Fourier integral is different than the sign in the reverse integral.

Time and Frequency

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A cosine can be written using either a period (time between adjacent maxima) or a frequency (number of maxima occurring within one second).

Note that the period and frequency of a cosine are reciprocally related, f 0 = 1/t 0. This means that small time intervals are described by large frequencies. An important rule of thumb: a small parameter in one domain corresponds to a large parameter in the other domain.

A signal described by f 0 = 0 has a constant amplitude as a function of time. This is because the cosine goes to one for all values of t. Zero frequency signals are often called dc.

( ) ( )00cos 2 cos 2tF t C C f t

tπ π⎛ ⎞= =⎜ ⎟

⎝ ⎠

A time-varying signal has a dc component if the integral at the right is non-zero.

( )F t dt∞

−∞∫

Non-Zero Frequencies

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Any signal which changes with time has at least one non-zero frequency. This is because Fourier transform analysis requires that any temporal variation be represented by sines and cosines.

-1.5

-1

-0.5

0

0.5

1

1.5

115 120 125 130 135time

amplitude

slope changewith frequency

curvature changewith frequency

Fast-changing signals are characterized by large slopes and/or curvatures. The largest temporal slope of a unit amplitude sine is proportional to f 0. Likewise the greatest temporal curvature of a cosine is proportional to the square of the frequency.

Signals with steep slopes or tight curvature require high frequencies to describe them.

Gaussian Peak Example

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Shown below are two chromatographic peaks, both with a retentiontime of 125 s. The left peak has a FWHM of 11.8 s. The rapidlychanging portion of the signal near the top of the peak is reasonably approximated by an offset cosine with a frequency of 0.0426 Hz. The right peak has a FWMH of 4.7 s. It can be approximated by a cosine with a frequency of 0.106 Hz. The spectrum for the peaks would extent from approximately these frequencies to dc.

105 115 125 135 145105 115 125 135 145

Repetitive Signals

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All non-zero frequencies of a repetitive signal will be multiples of therepetition rate, f 0. A zero frequency may also be present. For a repetitive signal that lasts from -∞ to +∞, no other frequencies occur in the signal spectrum.

The graph shows a signal consisting of a train of Gaussian pulses with f 0 = 100 Hz. The spectrum has a zero frequency component. All other signal frequencies are multiples of 100 Hz. The graph shows a 400 Hz cosine, which matches the curvature of the pulses.

The spectral frequencies with substantial amplitude are 0, 100, 200, 300 and 400 Hz. Higher frequencies will exist, but with a much lower amplitude. -0.02 -0.01 0 0.01 0.02

time

Plan of Attack

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• The ability to estimate quickly the frequency content of a signal is invaluable in designing signal-to-noise enhancement strategies.

• Fourier transforms can also be used to understand the operationand/or limitations of many instruments.

• Many real signals can be approximated by a combination of nine basis set functions. Rules for combining the basis set functions are described by nine theorems (we’ll cover eight of them).

• By memorizing the basis set and theorems, the Fourier transformof rather complicated functions can be quickly estimated. This allows the conversion of a temporal signal into a spectrum. Since very little math is used, the process can be done in your head without recourse to paper and pencil.

• There will always be signals too complicated for this approach.For these cases, the Fourier transform is best accomplished by acomputer program.

Unrealistic Features of the Basis Set

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Several of the functions used in the basis set can not exist in the physical world. Fortunately, this does not limit their utility.

Some have discontinuities where the amplitude makes an instantaneous change between two values. This would require infinite frequencies to match the slope and/or curvature. Infinite frequencies are impossible since Planck's law, E = hf, states they require infinite energy to generate.

Others have existed since -∞ time and will exist until +∞ time. Frequencies that have existed for infinite time cannot convey information - it is changing signals that convey information.

These physically impossible functions are still useful. Many real signals have temporal changes so sharp that measuring equipment can not distinguish them from a discontinuity. Many signals last longer than the time scale of an experiment, making it impossible to distinguish them from signals of infinite extent.