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# Signals and Systems: Real and Complex Exponential Signals

## Real Exponential Signals

An **exponential signal** or **exponential function** is a function that literally represents an exponentially increasing or decreasing series.

## Continuous-Time Real Exponential Signal

A real exponential signal which is defined for every instant of time is called **continuous time real exponential signal**. A continuous time real exponential signal is defined as follows −

π₯(π‘) = π΄π^{πΌπ‘}

Where, A and πΌ both are real. Here the parameter A is the amplitude of the exponential signal measured at t = 0 and the parameter πΌ can be either positive or negative.

Depending upon the value of πΌ, we obtain different exponential signals as −

When πΌ = 0, the exponential signal x(t) is a signal of constant magnitude for all times.

When πΌ > 0, i.e., πΌ is positive, then the exponential signal x(t) is a growing exponential signal.

When πΌ < 0, i.e., πΌ is negative, then the signal x(t) is a decaying exponential signal.

The waveforms of these three signals are shown in Figure-1.

## Discrete-Time Real Exponential Signal

A real exponential signal which is define at discrete instants of time is called a discrete-time real exponential signal or sequence. A discrete-time real exponential sequence is defined as −

π₯(π) = π^{π } for all π

Depending upon the value of a the discrete time real exponential signal may be of following type −

When a < 1, the exponential sequence x(n) grows exponentially.

When 0 < a < 1, the exponential signal x(n) decays exponentially.

When a < 0, the exponential sequence x(n) takes alternating signs.

These three signals are graphically represented in Figure-2.

## Complex Exponential Signals

An exponential signal whose samples are complex numbers (i.e., with real and imaginary parts) is known as a **complex exponential signal**.

## Continuous-Time Complex Exponential Signal

A continuous time complex exponential signal is the one that is defined for every instant of time. The continuous time complex signal is defined as −

π₯(π‘) = π΄π^{π π‘}

Where,

A is the amplitude of the signal.

s is a complex variable.

The complex variable s is defined as,

π = π + ππ

Therefore, the continuous time complex function can also be written as

π₯(π‘) = π΄π^{(π+ππ)π‘} = π΄π^{ππ‘}π^{πππ‘}

βΉ π₯(π‘) = π΄π^{ππ‘}(cos ππ‘ + π sin ππ‘)

Depending upon the values of π and π, we obtain different waveforms as shown in Figure-3.

## Discrete-Time Complex Exponential Sequence

A complex exponential signal which is defined at discrete instants of time is known as discrete-time complex exponential sequence. Mathematically, the discrete-time complex exponential sequence is defined as,

$$\mathrm{x(n)=a^{n}e^{j(\omega _{0}n+\varphi )}=a^{n}\cos (\omega _{0}n+\varphi)+ja^{n}\sin (\omega _{0}n+\varphi)}$$

Depending on the magnitude of a, we obtained different types of discrete-time complex exponential signals as,

For |π| = 1, both the real and imaginary parts of complex exponential sequence are sinusoidal.

For |π| > 1, the amplitude of the sinusoidal sequence increases exponentially.

For |π| < 1, the amplitude of the sinusoidal sequence decays exponentially.

The graphical representation of these signals is shown in Figure-4.

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