[Image1]

## Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about **Signals and Systems** in order to cover **Sinusoidal and Complex Exponential Signals**.

So, without further ado, let's get straight into it!

## Sinusoidal Signals

[Image 2]

Sinusoidal signals are periodic signals, which are based on the trigonometric functions sine (sin) and cosine (cos).

### Continuous-Time Sinusoidals

In continuous-time, the generic form of a sinusoidal signal is:

where:

*A*: amplitude*ω*: angular frequency_{o}*φ*: phase shift

#### Period

The period of a sinusoidal signal is given by:

#### Present Ampltiude

The signal's amplitude at present (*time = t _{o} = 0*) can be easily calculated using:

because only the phase shift

*φ*affects the present value.

#### Time Shift - Phase Shift Relationship

Time shifting a sinusoidal signal is related to phase shifting as follows:

Thus, in this example, the time shift by *t _{o}* is equal to a phase shift by

*φ = ω*.

_{o}t_{o}
Or, the other way around, a phase shift by any *φ* implies a time shift by some unknown multiple of *ω _{o}*.

#### Even Sinusoidal

When the phase shift is *φ = 0*, the sinusoidal signal *A cos ω _{o}t* is falling into the even category.

For a signal to be even the following must be true: *x(t) = x(-t)*, which is of course true for the cosine function, as:

#### Odd Sinusoidal

In a similar way, its also possible to prove that the sine function is odd, because:

Thus, the signal *A sin ω _{o}t* is considered an odd signal.

Its worth noting that the sine and cosine functions/signals differ by *φ = -π/2*.
So, its easy to conclude that phase shifting by *φ = -π/2*, the cosine signal is also considered an odd signal:

### Discrete-Time Sinusoidals

In discrete-time, the sinusoidal signal is given by:

where:

*A*: amplitude*Ω*: angular frequency_{o}*φ*: phase shift

#### Time Shift - Phase Shift Relationship

In the case of discrete-time, time shifting again implies a phase shift:

So, a time shift by *n _{o}* samples is equal to a phase shift by

*Ω*.

_{o}n_{o}
Let's note that phase shifting now doesn't imply a time shift, as the sample rate affects the outcome of phase shifting.
As a result:

#### Even-Odd Sinusoidal

Similar to continuous-time, the cosine signal is again considered an even signal, whilst the sine signal an odd signal:

#### Requirements for Periodicity

Any sinusoidal signal is considered an periodic signal in continuous-time, but in discrete-time things change slightly.

In general, in discrete-time a signal is considered periodic only when their exists a small integer *N* for which:

In discrete-time, a sinusoidal signal:

is periodic when *Ω _{o}N* is a multiple of

*2π*, and so the following is true:

The period

*N*is the smallest natural number for which this equation is true. If none exists then the sinusoidal is aperiodic.

The following is a visualization of how the sample-rate affects the periodicity of a sinusoidal signal:

[Image 3]

### Sinusoidals in Continuous- and Discrete-time at Distinct Frequencies

In addition to the issue of periodicity, continuous- and discrete-time sinusoidal also differ in other aspects.

In continuous-time, distinct values of the frequency *ω _{o}* result into completely distinct signals.
However, in discrete-time, values of

*Ω*which are separated by

_{o}*2π*result into identical signals.

So, in continuous-time, if *ω _{2} ≠ ω_{1}* then

*x*. But, in discrete-time, if

_{2}(t) ≠ x_{1}(t)*Ω*then

_{2}= Ω_{1}+ 2πm*x*.

_{2}[n] ≠ x_{1}[n]## Exponential Signals

Exponential signals can be defined as:

where both *C* and *a* are real numbers.

### Time Shift - Scale Change Relationship

Time shifting an exponential signal implies scale change, as follows:

### Complex Exponentials

Replacing *C* and *a* with complex numbers results in complex explonentials, which can be easily related to sinusoidal signals using Euler's relation.

#### Continuous-Time

In continuous-time, an complex exponential is defined as:

where *C* and *a* tend to be defined as:

which results in the following representation for complex exponentials:

Euler's relation allows us to replace the second exponential with a sum of cosine and sine:

And so, the final representation of complex exponentials is:

where the real and imaginary parts are clearly separated.

Complex exponentials can be thought of as exponentially growing or decaying sinusoidal signals, as shown below.

[Image 4]

#### Discrete-Time

In discrete-time, an complex exponential is defined as:

where *C* and *a* are defined as:

therefore resulting into the following representation:

Using Euler's relation, the last exponential can be replaced by sinusoidal signals, giving us the following, now final, form:

## RESOURCES:

### References

- Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.
- https://www.tutorialspoint.com/signals_and_systems/

### Images

- https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif
- https://commons.wikimedia.org/wiki/File:Sine_wave_amplitude.svg
- https://commons.wikimedia.org/wiki/File:Aliasing_sinusoidal.gif
- https://www.sciencedirect.com/topics/computer-science/complex-exponential

Mathematical equations used in this article, where made using quicklatex.

## Previous articles of the series

- Introduction → Signals, Systems
- Signal Basics → Signal Categorization, Basic Signal Types
- Signal Operations with Examples → Amplitude and Time Operations, Examples
- System Classification with Examples → System Classifications and Properties, Examples

## Final words | Next up

And this is actually it for today's post! Till next time!

See Ya!

Keep on drifting!