Mathematics - Signals And Systems - Sinusoidal and Complex Exponential Signals.

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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!

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Sinusoidal Signals

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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_o: angular frequency
• 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_ot_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_ot 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_ot 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_o: angular frequency
• 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_on_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_oN 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:



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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_o which are separated by 2π result into identical signals.

So, in continuous-time, if ω₂ ≠ ω₁ then x₂(t) ≠ x₁(t). But, in discrete-time, if Ω₂ = Ω₁ + 2πm then x₂[n] ≠ x₁[n].

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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.



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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:

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RESOURCES:

References

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

Images

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4. https://www.sciencedirect.com/topics/computer-science/complex-exponential

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

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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

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Final words | Next up

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

See Ya!

Keep on drifting!