Fourier Transform of Exponentials: Simple Explanation

Fourier Transform of Exponential Functions: A Simplified Breakdown

The Fourier transform is a powerful tool for analyzing signals and functions, decomposing them into their constituent frequencies. Understanding the Fourier transform of exponential functions is fundamental because many complex signals can be represented as a sum of exponentials (via the inverse Fourier transform). This document provides a clear and detailed explanation of how the Fourier transform operates on various exponential functions. The key focus is on explaining the concept surrounding the fourier transform exponential function.

1. The Fourier Transform: A Quick Review

Before diving into exponentials, let’s briefly revisit the Fourier transform itself. The Fourier transform converts a function of time, f(t), into a function of frequency, F(ω), where ω represents angular frequency.

1.1. The Formula

The Fourier transform is defined as:

F(ω) = ∫^∞_-∞ f(t)e^-jωt dt

where:

• F(ω) is the Fourier transform of f(t).
• f(t) is the original function in the time domain.
• ω is the angular frequency (ω = 2πf, where f is frequency in Hz).
• j is the imaginary unit (√-1).

1.2. Interpretation

This formula essentially calculates the correlation between the input function, f(t), and complex sinusoids, e^-jωt, for all frequencies ω. The resulting F(ω) indicates the amplitude and phase of each frequency component present in f(t).

2. Fourier Transform of the Complex Exponential e^jω₀t

The complex exponential function is fundamental. It serves as the building block for expressing other trigonometric functions and provides insights into the properties of the Fourier transform.

2.1. Calculation

Let’s find the Fourier transform of f(t) = e^jω₀t. Using the Fourier transform formula:

F(ω) = ∫^∞_-∞ e^jω₀t e^-jωt dt = ∫^∞_-∞ e^j(ω₀-ω)t dt

This integral is not straightforward to evaluate directly. Instead, we can recognize its relationship to the Dirac delta function. The Dirac delta function, δ(x), is zero everywhere except at x=0, where it is infinite, with the property that ∫^∞_-∞ δ(x) dx = 1.

The integral ∫^∞_-∞ e^j(ω₀-ω)t dt can be represented as 2πδ(ω – ω₀). Therefore:

F(ω) = 2πδ(ω – ω₀)

2.2. Interpretation

This result is crucial. The Fourier transform of a complex exponential e^jω₀t is a scaled Dirac delta function located at ω = ω₀. This means that the frequency content of e^jω₀t is concentrated at a single frequency, ω₀.

• The factor of 2π is a scaling factor arising from the definition of the Fourier transform.
• δ(ω – ω₀) signifies that the function has a non-zero value only when ω = ω₀, i.e., at the specific frequency ω₀.

3. Fourier Transform of Real Exponentials: e^-atu(t)

Now, let’s consider a real exponential function, where a is a positive real number and u(t) is the unit step function (u(t) = 0 for t < 0 and u(t) = 1 for t ≥ 0). The function is defined as f(t) = e^-atu(t).

3.1. Calculation

Using the Fourier transform formula:

F(ω) = ∫^∞_-∞ e^-atu(t) e^-jωt dt = ∫^∞₀ e^-(a+jω)t dt

The integral can now be evaluated:

F(ω) = [-e^-(a+jω)t / (a+jω)]^∞₀ = 0 – [-1/(a+jω)] = 1/(a+jω)

Therefore, the Fourier transform of e^-atu(t) is:

F(ω) = 1 / (a + jω)

3.2. Interpretation

The Fourier transform F(ω) = 1/(a + jω) is a complex-valued function. To fully understand it, we can examine its magnitude and phase.

• Magnitude: |F(ω)| = 1 / √(a² + ω²)

  • The magnitude decreases as frequency (ω) increases. This indicates that lower frequencies are more prominent in the signal.
  • The parameter ‘a’ controls the rate of decay. Larger ‘a’ means faster decay in the time domain and a wider frequency spectrum.

• Phase: ∠F(ω) = -arctan(ω/a)

  • The phase is a function of frequency and indicates the phase shift of each frequency component.
  • It is a negative arctangent function, indicating a phase lag that increases with frequency.

3.3. Symmetry considerations

Considering f(-t) = e^atu(-t), its Fourier transform becomes F(ω) = 1/(a – jω). The magnitude remains the same (|F(ω)| = 1 / √(a² + ω²)), but the phase changes sign (∠F(ω) = arctan(ω/a)).

4. Fourier Transform of Other Related Exponential Functions

Building upon the previous examples, we can find the Fourier transforms of several related exponential functions.

4.1. Two-Sided Exponential: e^-a|t|

Consider the two-sided exponential function, f(t) = e^-a|t|, where a > 0. This function decays exponentially as we move away from t = 0 in both directions.

The Fourier Transform can be derived as:
F(ω) = ∫^∞_-∞ e^-a|t|e^-jωt dt = ∫⁰_-∞ e^{a t}e^-jωt dt + ∫^∞₀ e^{-a t}e^-jωt dt
F(ω) = (1/(a – jω)) + (1/(a + jω)) = 2a / (a² + ω²)

This function is real and even in frequency domain.

4.2. Summary Table

Function f(t)	Fourier Transform F(ω)
ejω₀t	2πδ(ω – ω₀)
e-atu(t) (a > 0)	1 / (a + jω)
eatu(-t) (a > 0)	1 / (a – jω)
e-a|t| (a > 0)	2a / (a² + ω²)

So there you have it, a hopefully simple explanation of the fourier transform exponential function! Hopefully, you’re feeling a little more confident tackling these problems now. Keep practicing, and you’ll be a Fourier expert in no time!