# Period of a complex exponential signal

In summary, the conversation discusses finding the period of a simple complex exponential signal of the form x(t)=ejωt by testing if x(t)=x(t+nT) for all n. Through this test, a general period expression of T=2πk/ωn is found. The period is defined as the least amount of time a signal repeats itself, with examples of T1=2π/ω and T2=3π/ω provided. However, since a signal should only have one period, the conversation raises the question of whether there could be multiple periods for the same signal.
I have a simple complex exponential signal of the form x(t)=ejωt. To find period of the signal I tested if x(t)=x(t+nT) for all n:

ejωt=ejω(t+nT) ⇒ ejωnT=1=ej2πk

where n and k are integers. Then I find a general period expression as

T=2πk/ωn​

Period T means it is the least time a signal repeat itself. As an example, pick k=1, n=1, then T1=2π/ω. Now pick k=3, n=2, then T2=3π/ω. These two periods seem valid for the same signal x(t) in contrary it shouldn't, since a signal should have only one period. Am I missing something here?

If x(t)=x(t+nT) for all n, then x(t)=x(t+nT)=x(t+n(2T)) for all n by using the same step twice. If T is a period, then 2T is a period as well, and 3T and so on. Usually, just the smallest value is called period of a signal.

## 1. What is a complex exponential signal?

A complex exponential signal is a type of mathematical function that has the form of Ae^(jωt), where A is the amplitude, e is the base of the natural logarithm, j is the imaginary unit, and ω is the angular frequency. It is commonly used in signal processing and telecommunications.

## 2. What is the period of a complex exponential signal?

The period of a complex exponential signal is the time it takes for the signal to complete one full cycle, or to repeat itself. It is given by the formula T = 2π/ω, where ω is the angular frequency.

## 3. How is the period of a complex exponential signal related to its frequency?

The period of a complex exponential signal is inversely proportional to its frequency. This means that as the frequency increases, the period decreases, and vice versa. This relationship is described by the formula T = 1/f, where f is the frequency.

## 4. Can the period of a complex exponential signal be negative?

No, the period of a complex exponential signal cannot be negative. It is always a positive value, as it represents the time it takes for the signal to complete one full cycle.

## 5. How is the period of a complex exponential signal affected by changes in its amplitude?

The period of a complex exponential signal is not affected by changes in its amplitude. The amplitude only affects the magnitude or strength of the signal, but not its period or frequency. The period remains constant as long as the angular frequency remains the same.

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