The complex exponential signal is a generalization of the [[discrete-time real exponential signal]] where $\alpha$ becomes the complex number $z$: $ x(n) = A z^{n} $ if $z = e^{j \omega_{0}} = \cos(\omega_{0})+j\sin(\omega_{0})$: $ x(n) = A e^{j\omega_{0} n} $ and $A = |A| e^{j \phi}$: $ x(n) = |A| e^{j (\omega_{0} n + \phi)} $ i.e: $x(n) = |A| \cos(\omega_{0} n + \phi) + j |A| \sin(\omega_{0} n + \phi)$ in the simple case where $A=1$: $x(n) = e^{j\omega_{0}n}$ By analogy with the corresponding function in continuous time, $\omega_0$ is called the frequency of the complex sinusoid and $\phi$ its phase. The complex exponential signal can also be represented in polar coordinates as: $ x(n) = |A| e^{j\phi} e^{j\omega_{0} n} = |A| \angle \phi \cdot e^{j\omega_{0} n} $ where $|A|$ is the magnitude or amplitude of the signal, $\phi$ is its phase angle, and $\omega_{0}$ is the frequency in radians per sample. The term $e^{j\omega_{0} n}$ represents the phasor rotating with a constant angular frequency $\omega_{0}$. The frequency $\omega_{0}$ determines how many cycles of the sinusoid are completed in one sample period. For example, if $\omega_{0} = 2\pi f_0 /f_s$ where $f_0$ is the frequency in Hz and $f_s$ is the sampling rate in Hz, then one sample period corresponds to one cycle. The discrete-time complex is not always periodic in time (see the [[time periodicity of the discrete-time complex exponential signal]]) but is always periodic in frequency (see the [[frequency periodicity of the discrete-time complex exponential signal]]).