Howtosignalprocessing
- How do you find the phase of a complex number?
- How do you find the period of a complex exponential function?
- How is complex exponential defined?
- Why is complex exponential a circle?
How do you find the phase of a complex number?
The angle or phase or argument of the complex number a + bj is the angle, measured in radians, from the point 1 + 0j to a + bj, with counterclockwise denoting positive angle. The angle of a complex number c = a + bj is denoted c: c = arctanb/a.
How do you find the period of a complex exponential function?
II. Periodicity of complex the exponential. Recall the definition: if z = x + iy where x, y ∈ R, then ez def = exeiy = ex(cosy + i siny). It's clear from this definition and the periodicity of the imaginary exponential (§I) that ez+2πi = ez, i.e.: “The complex exponential function is periodic with period 2πi.”
How is complex exponential defined?
A complex exponential is a signal of the form. (1.15) where A = ∣A∣ej θ and a = r + j Ω 0 are complex numbers. Using Euler's identity, and the definitions of A and a, we have that x(t) = A eat equals. We will see later that complex exponentials are fundamental in the Fourier representation of signals.
Why is complex exponential a circle?
Exponential Behavior
Therefore any point on the unit circle can be reached by raising i to the appropriate power. This power is not unique, any displacement of an integer multiple of four will also be a solution. This is why the complex unit circle can be seen as being exponential.
