13.7 : 指数信号和正弦信号

The exponential function serves as an essential tool for characterizing waveforms that rise and decay rapidly. This function is defined using the constants m and A. When both the constants are real, the function can be graphically represented.

A complex exponential is achieved when the constant m is purely imaginary. It is periodic if it maintains a magnitude of unity.

A continuous-time sinusoidal signal can be expressed in terms of frequency and time period.

Euler's relation can be used to express the sinusoidal signal as periodic complex exponentials with the same fundamental frequency.

Similarly, the complex exponential function can be expressed in terms of sinusoidal signals, all sharing the same fundamental frequency.

For instance, the sum of two complex exponentials can be written as the product of a single complex exponential and a single sinusoid.

Both sinusoidal and complex exponential signals are extensively employed to describe energy conservation in a mechanical system where a mass, connected to a stationary support via a spring, exhibits simple harmonic motion.