Behaviour of DT complex exponential

A discrete-time complex exponential is

$$x[n] = C\alpha^{n}$$

where $C$ and $\alpha$ are in general complex numbers. (With $\alpha = e^{\beta}$ it is also written $Ce^{\beta n}$.) Its behaviour depends on $C$ and $\alpha$:

1. Real exponential ($C$, $\alpha$ real)

• $\alpha > 1$: grows exponentially.
• $0 < \alpha < 1$: decays exponentially.
• $-1 < \alpha < 0$: alternates in sign and decays.
• $\alpha < -1$: alternates in sign and grows.
• $\alpha = 1$: constant; $\alpha = -1$: alternates between $+C$ and $-C$.

2. Purely imaginary exponent ($|\alpha| = 1$), sinusoidal

With $\alpha = e^{j\omega_0}$: $x[n] = Ce^{j\omega_0 n} = C(\cos\omega_0 n + j\sin\omega_0 n)$. The magnitude is constant; real and imaginary parts are sampled sinusoids. It is periodic only if $\omega_0/2\pi$ is rational, and frequencies $\omega_0$ and $\omega_0 + 2\pi$ give identical signals.

3. General complex exponential

With $C = |C|e^{j\theta}$ and $\alpha = |\alpha|e^{j\omega_0}$:

$$x[n] = |C||\alpha|^n\cos(\omega_0 n+\theta) + j|C||\alpha|^n\sin(\omega_0 n+\theta)$$

• $|\alpha| = 1$: constant-amplitude sinusoid.
• $|\alpha| < 1$: sinusoid with decaying envelope.
• $|\alpha| > 1$: sinusoid with growing envelope.

 0<a<1 decay        -1<a<0 alternating decay
 o                    o
 | o                  |   o
 | | o o .            | . | . .
-+-+-+-+-+-> n     ---+-+-+-+-+-> n
                        o