The Fourier Series of a Signal: Analysis of x(t) 2cos(9t) 12sin(15t)

Often, when analyzing a signal, we turn to the exponential Fourier series to decompose it into its constituent frequencies. However, for the signal x(t)2cos(9t) 12sin(15t), the question arises whether it truly constitutes a series or if a simpler method like Euler's theorem might suffice.

Understanding the Signal Composition

The signal x(t) 2cos(9t) 12sin(15t) consists of only two cosine and sine components. These components are harmonically related as the second signal (12sin(15t)) is the second harmonic of the first (2cos(9t)). However, the fundamental frequency (3t) is not present in this signal, making the term 'series' somewhat misleading.

Why Fourier Analysis?

Fourier analysis is employed when a signal is composed of many frequency components over a wide range, whereas in this case, the signal is simpler. The fundamental reason to use Fourier analysis is to break down a complex signal into a sum of simpler sine and cosine components. While in this case, the signal can be more readily simplified using Euler's theorem, there are underlying reasons why Fourier analysis is a standard and powerful tool in signal processing.

Euler's Theorem for Signal Simplification

Euler's theorem, which states that eiycosy isiny, can indeed be utilized to convert the signal x(t) into imaginary exponentials. This transformation is particularly useful in digital signal processing and control systems, where the signal is often represented in the frequency domain.

Step-by-Step Conversion of the Signal

Let's consider the components of x(t) individually and then combine them using Euler's theorem.

Step 1: Decompose the Cosine Term

The first term, 2cos(9t), can be written as 2ei9t e?i9t2.

Step 2: Decompose the Sine Term

The second term, 12sin(15t), can be decomposed into imaginary exponentials as 12ei15t?e?i15t2i.

Step 3: Combine the Terms

Thus, the signal x(t) can be expressed as a sum of exponentials:

x(t)22ei9t 22e?i9t 122iei15t?122ie?i15t

This simplified form provides a more straightforward representation of the signal for various digital processing tasks, especially in the realm of communication systems and filter design.

Conclusion: When to Use Fourier Analysis

In summary, while Euler's theorem can simplify the representation of the signal x(t) 2cos(9t) 12sin(15t), the essence of Fourier analysis lies in its broader applicability. It is a cornerstone in signal and system analysis, allowing for a more comprehensive understanding of complex signals and systems. In scenarios involving a plethora of frequency components or requiring a deeper analysis of signal properties, Fourier series remain indispensable.