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14.2.7.5: Fourier Series and Transform

  • Page ID
    67343
    • Franz S. Hover & Michael S. Triantafyllou
    • Massachusetts Institute of Technology via MIT OpenCourseWare
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    Fourier Series

    The Fourier transform is the underlying principle for frequency-domain description of signals. We begin with the Fourier series. Consider a signal \(f(t)\) continuous on the time interval \( [0, T] \), which then repeats with period \(T\) off to negative and positive infinity. It can be shown that

    \begin{align} f(t) &= A_o + \sum_{k=1}^\infty [A_n \, \cos (n \omega_o t) + B_n \, \sin (n \omega_o t)] & & \text{in which:} \\ \omega_o &= 2 \pi /T, \nonumber \\ A_0 &= \dfrac{1}{T} \int\limits_{0}^{T} f(t)\, dt, \nonumber \\ A_n &= \dfrac{2}{T} \int\limits_{0}^{T} f(t) \, \cos (n \omega_o t)\, dt, \nonumber \\ B_n &= \dfrac{2}{T} \int\limits_{0}^{T} f(t) \, \sin (n \omega_o t)\, dt. \nonumber \end{align}

    This says that the time-domain signal \(f(t)\) has an exact (if you carry all the infinity of terms) representation of a constant plus scaled cosines and sines. As we will see later, the impact of this second, frequency-domain representation is profound, as it allows an entirely new set of tools for manipulation and analysis of signals and systems. A compact form of these expressions for the Fourier series can be written using complex exponentials:

    \[ f(t) = \sum_{n = -\infty}^\infty C_n e^{i n \omega_o t}, \]

    where \(C_n = \dfrac{1}{T} \displaystyle \int\limits_{0}^{T} f(t) e^{-i n \omega_o t} \, dt \).

    Of course, \(C_n\) can be a complex number. In making these inner product calculations, orthogonality of the harmonic functions is useful:

    \( \displaystyle \int\limits_{0}^{2 \pi} sin (nt) \, sin (mt) \, dt = 0, \) for \(n \geq 1, \, m \geq 1, \, n \neq m \)

    \( \displaystyle \int\limits_{0}^{2 \pi} cos (nt) \, cos (mt) \, dt = 0, \) for \(n \geq 1, \, m \geq 1, \, n \neq m \)

    \( \displaystyle \int\limits_{0}^{2 \pi} sin (nt) \, cos (mt) \, dt = 0, \) for \(n \geq 1, \, m \geq 1. \)

    Now let’s go to a different class of signal, one that is not periodic, but has a finite integral of absolute value. Obviously, such a signal has to approach zero at distances far from the origin. We can write a more elegant transformation:

    \[ F(\omega) = \int\limits_{-\infty}^{\infty} f(t)e^{-i \omega t}\, dt, \]

    \[ f(t) = \dfrac{1}{2 \pi} \int\limits_{-\infty}^{\infty} F(\omega) e^{i \omega \tau}\, d\tau. \]

    Fourier Transform

    This is the real Fourier transform: a time-domain signal is transformed into a (complex) frequency-domain version, and it can be transformed back. On working it through, we see that derivatives and integrals look this way through the transform:

    \[ f(t) \longleftrightarrow F(\omega) \]

    \[\, \, \, \, \, \dfrac{d^n f(t)}{dt^n} \longleftrightarrow (i \omega)^n F(\omega) \]

    \[ \int\limits_{-\infty}^{t} f(\tau)\, d\tau \longleftrightarrow \dfrac{1}{i \omega} F(\omega). \quad \]

    Another very important property of the Fourier transform is Parseval’s Relation:

    \[ \int\limits_{-\infty}^{\infty} f^2(t) \, dt = \dfrac{1}{2 \pi} \int\limits_{-\infty}^{\infty} F(\omega) F^*(\omega) \, d\omega = \int\limits_{-\infty}^{\infty} |F(\omega)|^2 \, d\omega, \]

    where the \(\ast\)-superscript indicates the complex conjugate. We will give more properties for the related Laplace transform in a later section. But as is, the Fourier transform is immediately useful for solving linear differential equations with constant coefficients (LTI systems):

    \[ mx'' + bx' + cx = u(t) \longleftrightarrow [-m\omega^2 + i \omega b + k] X(\omega) = U(\omega), \]

    so that \(X(\omega) = \dfrac{1}{-m \omega^2 + i \omega b + k} U(\omega).\)

    Hence, the action of the differential equation to relate \(f(t)\) with \(x(t)\) is, in the frequency domain, captured by the function

    \[ H(\omega) = \dfrac{1}{-m \omega^2 + i \omega b + k} \]

    Putting two and two together, we then assert that \( X(\omega) = H(\omega)U(\omega) \); in the Fourier space, the system response is the product of impulse response function, and the input! To back this up, we show now that convolution in the time-domain is equivalent to multiplication in the frequency domain:

    \begin{align*} X(\omega) &= \mathcal{F} \left[ \int\limits_{-\infty}^{\infty} u(\tau) h(t - \tau)\, d\tau \right] \\ &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} u(\tau) h(t - \tau) e^{-i \tau t} \, dt d\tau \\ &= \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} u(\tau) h(\xi) e^{-i \omega (\xi + \tau)} \, d\xi d\tau \, && \text {because \(e^{-i \omega t} = e^{-i \omega(t - \tau + \tau)}\)} \\ &= \displaystyle \int\limits_{-\infty}^{\infty} e^{-i \omega \xi} h(\xi) \, d\xi \int\limits_{-\infty}^{\infty} e^{-i \omega \tau} u(\tau)\, d\tau \\ &= H(\omega) U(\omega). \end{align*}

    The central role of the impulse response should be reiterated here. It is a complete definition of the system, and for systems of differential equations, it is a specific function of the parameters and of the frequency \(\omega\). The Fourier Transform of the impulse response is called the system transfer function, and we often refer to the transfer function as “the system,” even though it is actually a (transformed) signal.

    By way of summary, we can write

    \[y(t) = h(t) * u(t) ,\] and

    \[Y(\omega) = H(\omega) U(\omega). \]

    Fast Fourier Method (FFT)

    This method of Fourier transforms is very good when not using a computer, but converting this as is for computers is very cumbersome. The solution to this is the Fast Fourier Method (FFT) which is really a Discrete Fourier Transform (DFT). The main idea of the FFT is to do a couple of "tricks" to handle sums faster. Gauss had developed just such a method where you divide your signal into many smaller discrete Fourier transforms. This technique is referred to as the Cooley-Tukey algorithm based off the paper that James Cooley and John Tukey developed using Gauss' method. Note the Cooley-Tukey algorithm is designed so that other similar algorithms can be included to avoid problems with the original algorithm.

    The FFT is used extensively in engineering and science for analysis and other purposes. Almost all higher level languages have access to a FFT algorithm as will as some other Discrete Fourier Transform (DFT) algorithms. This topic will be discussed in your numerical methods class (signals and systems).

    Note that the generalized form of DFT leads to the chirp Z-transform (CZT). This transform is in the z-space as opposed to the frequency space of the Fourier transform and the s-space of the Laplace transform. The algorithms associated with this transform is Bluestein's algorithm which uses FFTs.

    Other FFT algorithms include the Rader-Brenner's algorithm, Bruun's algorithm, Winogard's FFT algorithm, etc. While the study of these algorithms are very interesting, from a practical point-of-view it is only the final product that is of interest to engineers and scientists as the FFT is a method that allows us to use Fourier transforms on a computer to maximum effect. FFTs are demonstrated in the data analysis section of this book/course.

    Contribution

    • Franz S. Hover & Michael S. Triantafyllou
    • FFT addition by Scott D. Johnson, Joshua Halpern, and Scott Sinex


    14.2.7.5: Fourier Series and Transform is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Franz S. Hover & Michael S. Triantafyllou (MIT OpenCourseWare) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.