Complex Number

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z-Transform

where \(n\) is an integer index and \(z\) is a complex number.
Step Response of a Series RLC Circuit

The roots are complex.
Step Response of a Parallel RLC Circuit

The roots are complex.
Source-Free Series RLC Circuit

The roots are complex.
Source-Free Parallel RLC Circuit

The roots are complex.
Signal Measures

The absolute value \(\left|x(t)\right|\) allows this relation to be used for complex-valued signals.
Second-Order Circuits

Obtain the transient response \(x_{t}(t)\) depending on whether the response (damping) is overdamped (real and distinct roots), critically damped (real and repeated roots), or underdamped (complex roots).
Euler’s Formula

Euler’s formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Discrete Signal Measures

The absolute value \(\left|x[n]\right|\) allows this relation to be used for complex-valued signals.
Bode Plots

Since transfer function is a complex number, \(\displaystyle \mathbf{H}(\omega) = H(\omega)\angle\phi(\omega) = \textrm{Re}\left[\mathbf{H}(\omega)\right] + j\,\textrm{Im}\left[\mathbf{H}(\omega)\right]\).