Fourier transform
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| 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 Nth-order causal FIR filter now can be written down by induction as
$ {\begin{aligned}B\left(f\right)={\frac {\rm {Output}}{\rm {Input}}}={\frac {b_{0}e^{j2\pi jn\Delta t}+\dots \ +b_{\rm {l}}e^{i2\pi f\left(n-{\rm {l}}\right)\Delta t}}{e^{i2\pi fn}}},\end{aligned}} $ ()
which is
$ {\begin{aligned}B\left(f\right)=b_{0}+b_{\rm {l}}e^{-i2\pi f\Delta {\rm {t}}}+\dots +b_{N}e^{-i2\pi j\Delta tN}.\end{aligned}} $ ()
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 |
|---|---|
| $ B\left(Z\right)=b_{0} $ | $ B(f)=b_{0} $ |
| $ B\left(Z\right)=Z $ | $ B\left(f\right)=e^{-i2\pi f\Delta t} $ |
| $ B\left(Z\right)=b_{\rm {1}}Z $ | $ B\left(f\right)=b_{1}e^{-i2\pi f{\Delta }_{t}} $ |
| $ B\left(Z\right)=b_{0}+b_{\rm {1}}Z $ | $ B\left(f\cdot \right)=b_{0}+b_{1}e^{-i2\pi f\Delta t} $ |
| $ B\left(Z\right)=b_{0}+b_{\rm {1}}Z++b_{N}Z^{N} $ | $ B\left(j\right)=b_{0}+b_{1}e^{-i2\pi f{\Delta }t}+\dots +b_{\rm {N}}e^{-i2\pi f{\Delta t}N} $ |
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
$ {\begin{aligned}B\left(Z\right)=b_{0}+b_{\rm {l}}Z+\dots +b_{N}Z^{N},\end{aligned}} $ ()
and the Fourier transform is defined in the way originally given by Fourier, namely
$ {\begin{aligned}B_{M{\rm {A}}TH}\left(f\right)=b_{0}+b_{1}e^{i2\pi f\Delta t}+\dots +b_{N}e^{i2\pi f\Delta fN}.\end{aligned}} $ ()
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
$ {\begin{aligned}B_{EE}\left(z\right)=b_{0}+b_{1}z^{-{l}}+\dots +b_{N}z^{-N},\end{aligned}} $ ()
and the Fourier transform is defined as
$ {\begin{aligned}B\left(f\right)=b_{0}+b_{1}e^{-i2\pi f\Delta t}+\dots +b_{N}e^{-i2\pi f\Delta tN}.\end{aligned}} $ ()
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
$ {\begin{aligned}B\left(Z\right)=b_{0}+b_{l}Z+\dots +b_{N}Z^{N},\end{aligned}} $ ()
and the Fourier transform is defined as in electrical engineering, namely
$ {\begin{aligned}B\left(f\right)=b_{0}+b_{l}e^{-i2\pi f\Delta t}+\dots +b_{N}e^{-i2\pi j\Delta tN}.\end{aligned}} $ ()
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 $ Z=e^{-i2\pi f\Delta t} $ 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 $ b_{0} $, the transfer function always depends on the frequency f.
Much can be learned from considering the unit-delay filter, which can be represented as $ Z=e^{i\psi (f)} $, where $ {\psi (f)}=-2\pi f\Delta t $ is the phase lead. The phase lag Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi \left(f\right) is defined as the negative of the phase lead - that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi \left(f\right)=-\psi (f) . Thus, the unit-delay filter can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Z=e^{-i2\pi f\Delta t}=e^{-i\phi (f)} . 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} B\left(f\right) =|B\left(f\right){|}e^{-j\psi \left(f\right)}, \end{align} ()
where the magnitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}B\left(f\right){|} and the angle $ \varphi (f) $ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0+b_{{1}}Z is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} B\left(f\right)=b_0+b_{{1}} e^{-j2\pi f\Delta {\rm t}}=b_0+b_{{l}}{\rm \ cos\ 2}\pi f\Delta t-ib_{{l}}{\rm \ sin\ 2}\pi f\Delta t. \end{align} ()
Now B(f) (for a fixed value of f) is the vector that is the sum of the vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{{1}}e^{-i2\psi \Delta t} (Figure 6).
The length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}B\left(f\right){|} of the vector B(f) is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {|}B\left(f\right){|=}\sqrt{{\left(b_0+b_{{1}} {\rm \ cos\ 2}\pi f\Delta t\right)}^{2}+{\left(b_{{1}}{\rm \ sin\ 2}\pi f\Delta t\right)}^{2}}=\sqrt{b^{2}_0+2b_0b_{{1}}{\rm \ cos\ 2}\pi f\Delta t+b^{2}_{{1}}}. \end{align} ()
This quantity is the magnitude spectrum of the filter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0+b_{{1}}Z . The angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi (f) , which is a function of f, is the phase lag (simply called the phase). Thus, the function
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \varphi \left(f\right)=-{{\rm tan}} ^{-{1}}\left[\frac{-b_{{1}}{\rm \ sin\ 2}\pi f\Delta t}{b_0+b_{{1}}{\rm \ cos\ 2}\pi f\Delta t}\right]={{\rm tan}}^{-{1}}\left[\frac{b_{{1}}{\rm \ sin\ 2}\pi f\Delta t}{b_0+b_{{1}}{\rm \ cos\ 2}\pi f\Delta t}\right] \end{align} ()
yields the phase-spectrum of the filter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0+b_{{1}}Z . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi . When we use such a program, we do not necessarily obtain the phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi (f) ; instead, we might obtain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi (f) reduced or augmented by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi k , where k is an integer so determined that the computed value lies in the range from 0 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi . 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
External links
| find literature about Fourier transform/en |
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