Exponential and Sinusoidal Signals

The exponential function is crucial for characterizing waveforms that rise and decay rapidly. This continuous-time exponential function is defined using exponential terms with constants α and A. When both constants are real, the function is represented as,

and can be graphically depicted to show exponential growth or decay. When the constant α is purely imaginary, the result is a complex exponential, expressed as,

where j is the imaginary unit and ω₀ is the angular frequency. This function is periodic if it maintains a magnitude of unity.

A continuous-time sinusoidal signal can be described in terms of frequency and time period. Euler's relation allows the sinusoidal signal to be expressed as periodic complex exponentials with the same fundamental period. Thus, a sinusoidal signal is represented as,

can be rewritten using complex exponentials as follows,

Similarly, the complex exponential function can be expressed in terms of sinusoidal signals, all sharing the same fundamental period. For instance, the sum of two complex exponentials can be written as the product of a single complex exponential and a single sinusoid, exemplified by,

​Both sinusoidal and complex exponential signals are extensively employed to describe energy conservation in mechanical systems, such as a mass connected to a stationary support via a spring, exhibiting simple harmonic motion. These signals provide a foundation for analyzing oscillatory behavior and resonance phenomena in such systems.