# Fourier transform

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

 {\displaystyle {\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}}} (12)

which is

 {\displaystyle {\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}}} (13)

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.

Table 1. Filters and transfer functions.
Causal FIR filter Corresponding transfer function
${\displaystyle B\left(Z\right)=b_{0}}$ ${\displaystyle B(f)=b_{0}}$
${\displaystyle B\left(Z\right)=Z}$ ${\displaystyle B\left(f\right)=e^{-i2\pi f\Delta t}}$
${\displaystyle B\left(Z\right)=b_{\rm {1}}Z}$ ${\displaystyle B\left(f\right)=b_{1}e^{-i2\pi f{\Delta }_{t}}}$
${\displaystyle B\left(Z\right)=b_{0}+b_{\rm {1}}Z}$ ${\displaystyle B\left(f\cdot \right)=b_{0}+b_{1}e^{-i2\pi f\Delta t}}$
${\displaystyle B\left(Z\right)=b_{0}+b_{\rm {1}}Z++b_{N}Z^{N}}$ ${\displaystyle B\left(j\right)=b_{0}+b_{1}e^{-i2\pi f{\Delta }t}+\dots +b_{\rm {N}}e^{-i2\pi f{\Delta t}N}}$

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

 {\displaystyle {\begin{aligned}B\left(Z\right)=b_{0}+b_{\rm {l}}Z+\dots +b_{N}Z^{N},\end{aligned}}} (14)

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

 {\displaystyle {\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}}} (15)

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

 {\displaystyle {\begin{aligned}B_{EE}\left(z\right)=b_{0}+b_{1}z^{-{l}}+\dots +b_{N}z^{-N},\end{aligned}}} (16)

and the Fourier transform is defined as

 {\displaystyle {\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}}} (17)

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

 {\displaystyle {\begin{aligned}B\left(Z\right)=b_{0}+b_{l}Z+\dots +b_{N}Z^{N},\end{aligned}}} (18)

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

 {\displaystyle {\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}}} (19)

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 ${\displaystyle Z=e^{-i2\pi f\Delta t}}$ 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 ${\displaystyle b_{0}}$, the transfer function always depends on the frequency f.

Much can be learned from considering the unit-delay filter, which can be represented as ${\displaystyle Z=e^{i\psi (f)}}$, where ${\displaystyle {\psi (f)}=-2\pi f\Delta t}$ is the phase lead. The phase lag ${\displaystyle \varphi \left(f\right)}$ is defined as the negative of the phase lead - that is, ${\displaystyle \phi \left(f\right)=-\psi (f)}$. Thus, the unit-delay filter can be written as ${\displaystyle Z=e^{-i2\pi f\Delta t}=e^{-i\phi (f)}}$. 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

 {\displaystyle {\begin{aligned}B\left(f\right)=|B\left(f\right){|}e^{-j\psi \left(f\right)},\end{aligned}}} (20)

where the magnitude ${\displaystyle {|}B\left(f\right){|}}$ and the angle ${\displaystyle \varphi (f)}$ 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 ${\displaystyle b_{0}+b_{1}Z}$ is

 {\displaystyle {\begin{aligned}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{aligned}}} (21)

Now B(f) (for a fixed value of f) is the vector that is the sum of the vectors ${\displaystyle b_{0}}$ and ${\displaystyle b_{1}e^{-i2\psi \Delta t}}$ (Figure 6).

Figure 6.  Depiction of the transfer function.

The length ${\displaystyle {|}B\left(f\right){|}}$ of the vector B(f) is

 {\displaystyle {\begin{aligned}{|}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_{0}^{2}+2b_{0}b_{1}{\rm {\ cos\ 2}}\pi f\Delta t+b_{1}^{2}}}.\end{aligned}}} (22)

This quantity is the magnitude spectrum of the filter ${\displaystyle b_{0}+b_{1}Z}$. The angle ${\displaystyle \varphi (f)}$, which is a function of f, is the phase lag (simply called the phase). Thus, the function

 {\displaystyle {\begin{aligned}\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{aligned}}} (23)

yields the phase-spectrum of the filter ${\displaystyle b_{0}+b_{1}Z}$. 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 ${\displaystyle 2\pi }$. When we use such a program, we do not necessarily obtain the phase ${\displaystyle \varphi (f)}$; instead, we might obtain ${\displaystyle \varphi (f)}$ reduced or augmented by ${\displaystyle 2\pi k}$, where k is an integer so determined that the computed value lies in the range from 0 to ${\displaystyle 2\pi }$. This result is called the wrapped phase spectrum; the true phase spectrum can be obtained by a computer process known as phase unwrapping.