# Fourier transform

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 6 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

The transfer function of the *N*th-order causal FIR filter now can be written down by induction as

**(**)

which is

**(**)

Our results can be tabulated in the form of Table 1.
Definitions are important. Unfortunately, the *Z*-transform and the Fourier transform are defined in different ways depending on the convention used. Now we must take a moment to reflect on conventions. We will discuss three conventions: (1) the *mathematics convention*, (2) the *electrical engineering convention*, and (3) the *hybrid convention*.

Causal FIR filter | Corresponding transfer function |
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Their counterparts are the gasoline automobile, the electric automobile, and the hybrid auto-mobile.

1) Under the mathematics convention, the generating function (i.e., the *Z*-transform with a capital Z) is defined in the way originally given by Euler, namely

**(**)

and the Fourier transform is defined in the way originally given by Fourier, namely

**(**)

In the mathematics convention, the exponents in both transforms (equations **14** and **15**) are positive.

2) Under the electrical engineering convention, the *z*-transform with a lowercase *z *is defined as

**(**)

and the Fourier transform is defined as

**(**)

In the electrical engineering convention, the exponents in both transforms (equations **16** and **17**) are negative.

3) Under the *hybrid convention*, the *Z*-transform with a capital *Z *is defined as in mathematics, namely

**(**)

and the *Fourier transform* is defined as in electrical engineering, namely

**(**)

In the hybrid convention, the exponents in the *Z*-transform (equation **18**) are positive, and the exponents in the Fourier transform (equation **19**) are negative.

In this book, we use the hybrid convention. Working with negative powers of *Z* is cumbersome. The hybrid convention avoids this contingency but keeps the form of Fourier transform that electrical engineers use. Under the hybrid convention, the *transfer function* of a filter is obtained formally by the substitution of in the filter’s *Z*-transform. In other words, *the transfer function is the (electrical engineering) Fourier transform of the impulse-response function*. We notice that except for the case of the constant filter , the transfer function always depends on the frequency *f*.

Much can be learned from considering the unit-delay filter, which can be represented as , where is the phase lead. The *phase lag* is defined as the negative of the phase lead - that is, . Thus, the unit-delay filter can be written as . Physical systems involve delay, so phase lag rather than phase lead becomes the natural choice. Electrical engineers sometimes but not always refer to phase lag simply as *phase*. However, wherever we use the word *phase*, we explicitly mean phase lag.

As we have seen, the transfer function is the (electrical engineering) Fourier transform of the impulse-response function. In polar form, the transfer function can be written as

**(**)

where the magnitude and the angle represent, respectively, the magnitude spectrum and phase-lag spectrum (or simply the phase spectrum) of the filter. For example, the transfer function of the filter is

**(**)

Now *B*(*f*) (for a fixed value of *f*) is the vector that is the sum of the vectors and (Figure 6).

The length of the vector *B*(*f*) is

**(**)

This quantity is the magnitude spectrum of the filter . The angle , which is a function of *f*, is the phase lag (simply called the phase). Thus, the function

**(**)

yields the phase-spectrum of the filter . We see that both the magnitude and the phase spectra are functions of the frequency *f*.

At this point, we must introduce a word of warning. The arctangent function is not a one-to-one function but instead is a many-valued function. Thus, a computer usually picks the value of the arctangent function that lies in the range from 0 to . When we use such a program, we do not necessarily obtain the phase ; instead, we might obtain reduced or augmented by , where *k* is an integer so determined that the computed value lies in the range from 0 to . This result is called the *wrapped phase spectrum*; the true phase spectrum can be obtained by a computer process known as phase unwrapping.

## Continue reading

Previous section | Next section |
---|---|

Magnitude spectrum and phase spectrum | Minimum-phase spectrum |

Previous chapter | Next chapter |

Filtering | Wavelets |

## Also in this chapter

- Frequency spectrum
- Magnitude spectrum and phase spectrum
- Minimum-phase spectrum
- Inverse Fourier transform
- Appendix F: Exercises