Question

What is the Fourier transform of (a) the constant function \(f(x)=C\), and (b) the Dirac delta function \(\delta(x)\) ?

Short answer

The Fourier transform of the constant function \(f(x)=C\) is \(2\pi C \delta(\omega)\), and the Fourier transform of the Dirac delta function \(\delta(x)\) is 1.

Step-by-step solution

Fourier Transform of a Constant Function

The Fourier transform of a constant function \(f(x)=C\) can be obtained by applying the formula for the Fourier transform. Recall that the Fourier transform \(F(\omega)\) of a function \(f(x)\) is given by \(F(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx\). Therefore, we get: \[F(\omega) = \int_{-\infty}^{\infty} C e^{-i\omega x} dx\]This integral is non-zero only at \(\omega = 0\) and is otherwise \(0\) because we are integrating a complex oscillation over all space. So the Fourier transform of a constant function is a Dirac delta function at \(\omega = 0\), scaled by \(2\pi C\). Therefore, \[F(\omega) = 2\pi C \delta(\omega)\].

Fourier Transform of the Dirac Delta Function

The Dirac delta function \(\delta(x)\) is a special case where the Fourier transform can be immediately observed from the definition of the Fourier transform. Substituting \(f(x) = \delta(x)\) into the definition of the Fourier transform we get:\[F(\omega) = \int_{-\infty}^{\infty} \delta(x) e^{-i\omega x} dx\]But the property of the delta function notes that the integral of any function \(g(x)\) multiplied by \(\delta(x)\) over all space merely picks out the value of \(g(x)\) at \(x = 0\). Therefore, this integral results in: \[F(\omega) = e^{-i\omega*0} = 1\]. As such, the Fourier transform of the Dirac delta function is just 1.

Key concepts

Constant Function

A constant function is one where the output is the same for any input value, represented as \(f(x) = C\). This function assumes a uniform value at all points along the x-axis. This uniform behavior leads to some interesting properties when applying the Fourier transform.

In the case of a constant function, its Fourier transform reveals how this constant is distributed across different frequency components. You might imagine this as breaking down the function's "flatness" into simpler oscillations of different frequencies. However, unlike oscillating functions, a constant function has no variation along the x-axis, making its Fourier transform quite different.

When we apply the Fourier transform to a constant function, it boils down to the integral \( \int_{-\infty}^{\infty} C e^{-i\omega x} \, dx \).

• This value is non-zero only when \(\omega = 0\), meaning there are no oscillating components other than the "zero frequency" or constant component itself.
• The result is a Dirac delta function at \( \omega = 0 \), scaled by \(2\pi C\).

So, the Fourier transform of a constant is \(2\pi C \delta(\omega)\), emphasizing that it has zero frequency in the frequency spectrum.

Dirac Delta Function

The Dirac delta function, \(\delta(x)\), isn't a function in the traditional sense but a mathematical construct used to model a quantity occurring precisely at a single point. It's often visualized as a spike at \(x = 0\), with the property that integrating it over the entire space equals one. This makes it a powerful tool for analysis in mathematics and physics.

When computing the Fourier transform of the Dirac delta function \(\delta(x)\), we substitute it into the Fourier transform's definition:
\[ F(\omega) = \int_{-\infty}^{\infty} \delta(x) e^{-i\omega x} \, dx \]
According to the sifting property of the delta function, the above integral evaluates the function \(e^{-i\omega x}\) directly at \(x = 0\).

• This results in the simplified and insightful expression \( F(\omega) = 1 \).
• The result demonstrates that the Dirac delta function is equally composed of all frequencies, as it mirrors a perfect "flatness" across the entire frequency spectrum.

The Fourier transform of the Dirac delta function being equal to one highlights its unique properties in the context of frequency analysis.

Complex Oscillation

Complex oscillations are characterized by their oscillatory behavior, manifested in both real and imaginary components. They are typically expressed in terms of the exponential function \(e^{i\theta}\), where \(\theta\) is a function of time or space.

In the realm of Fourier analysis, complex oscillations are pivotal. They form the basis of expressing any function as a combination of simpler waves. The expression \(e^{-i\omega x}\) encountered in the Fourier transform introduces this concept as it signifies how different frequency components contribute to the overall behavior of a function.

• In the context of the Fourier transform of a constant function, the term \(e^{-i\omega x}\) effectively broadens to accommodate all frequencies yet collapses to zero except at \(\omega = 0\), maintaining the constant nature of the function in its frequency representation.
• For the Dirac delta function, \(e^{-i\omega x}\) works by "picking out" specific functional values over the entire spectrum, demonstrating the Dirac delta's all-encompassing frequency nature.

Thus, complex oscillations are integrally connected to understanding how Fourier transforms decompose functions into their essential frequency components.