The Delta Function, also known as the Dirac Delta Function, is a fundamental concept in signal processing and Fourier analysis. The Delta Function \( \delta(x) \) is a "function" with the property of being zero everywhere except at \( x = 0 \), where it is infinite.
In essence, it acts like an "impulse" and is extremely useful in breaking down and representing signals in terms of their frequency components:
- The key property of the Delta Function is: \( \int_{-\infty}^{\infty} \delta(x) \, dx = 1 \)
- It selectively picks out values at specific points when integrated inside other functions.
In the Fourier Transform context, a sinusoidal function like \( \cos(\omega_0 t) \) can be represented using Delta Functions due to its precise frequencies. The Delta Function simplifies this representation by focusing only on the principal frequencies of interest.
For this problem, the Fourier Transform of \( f_1(t) \) leads to Delta Functions at \( \omega = \omega_0 \) and \( \omega = -\omega_0 \), pinpointing the exact frequencies involved without any spread, unlike the Sinc Function. This serves to highlight how specific frequencies contribute exactly to the overall signal spectrum in mixed time-frequency analyses.