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I am very new to Fourier analysis, but I understand that through the use of the Fourier transform a signal in the time domain is displayed in the frequency domain, where frequency values are normally displayed along the x-axis, and amplitude is displayed along the y-axis. However, at one point in the textbook I am using, the following is stated:

Let us assume that we have the function $f(t) = \cos(\omega_0 t)$. The spectrum then consists of two delta-functions

$$F(\omega) = \pi \delta(\omega - \omega_0) + \pi \delta(\omega + \omega_0)$$

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This confuses me. When we have $f(t) = \cos(\omega_0 t)$, then I would assume that the Fourier transform should yield an amplitude of $1$ at $\omega = \omega_0$ and $0$ elsewhere. But the delta function is defined as:

$$\delta(\omega - \omega_0) = \left\{ \begin{array}{1 1} \infty & \quad \omega = \omega_0 \\ 0 & \quad \omega \neq \omega_0 \end{array} \right.$$

So wouldn't this give an infinite value at $\omega = \omega_0$?

If anyone can explain the intuition behind the statement in my textbook, then I would be very grateful!

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  • $\begingroup$ Instead of assuming that the Fourier transform should yield an amplitude of $1$ at $\omega=\omega_0$, why don't you try to see if this actually happens? $\endgroup$
    – Dilip Sarwate
    Dec 19, 2013 at 18:59
  • $\begingroup$ I don't doubt that the textbook is right. But how do you get from the result above to an accurate representation in the frequency/amplitude domain? $\endgroup$
    – Kristian
    Dec 19, 2013 at 20:24
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    $\begingroup$ Learn about the difference between power and energy. As Hilmar has pointed out to you, a (mathematical) sinusoid has infinite energy although the sinusoids available to us mortals are always of finite duration and thus have finite energy. A mathematical sinusoid has finite power, though, and in this instance, the total power is $\frac{1}{2}$. $\endgroup$
    – Dilip Sarwate
    Dec 19, 2013 at 21:21
  • $\begingroup$ OK. Thanks. I will try to read more about this. $\endgroup$
    – Kristian
    Dec 19, 2013 at 21:36

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The textbook is right. A sine wave in the time domain has infinite energy since it continues over an infinite amount of time. When you transform into the Frequency domain all this energy is concentrated on a single (or two) frequency. Hence the value there is indeed infinite.

Of course these are all theoretical considerations. In the real world, ideal sine waves do not exist since they all have a beginning and an end.

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  • $\begingroup$ Thanks. I was under the impression that it is the coefficient before the cosine-function (in this case 1) which is the amplitude of the signal, and thus this is what should be displayed along the y-axis in the frequency domain. So I suppose this means that energy is not equal to amplitude? $\endgroup$
    – Kristian
    Dec 19, 2013 at 15:37

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