Digital_Communications_Part_A

Fourier Analysis
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Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. There are so many flavors under this Fourier analysis that it becomes confusing to understand which one to use.

Fourier Types
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A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories illustrated below.

fourier_analysis_categories.png

Fourier Series
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A good portion of engineering topics (e.g., analog circuits, power, electromagnetics, signals and systems, engineering mathematics, digital communications), rely on Fourier Series as a tool to study the problems that are present in those domains.

Briefly, Fourier Series deals with periodic signals that can be written as an infinite sum of harmonically related sinusoids.

The confusion with Fourier Series, relies on the fact that there are 3 common forms to represent this infinite sum of sinusoids: trigonometric, phasor, and exponential form; depending on which problem in engineering you are trying to solve.

  • Analog Circuits and Power Electronics typically use the phasor (Amplitude/Phase) representation.
  • Signals and Systems, Electromagnetics, and Engineering Mathematics typically use the complex exponential.

fougier_series_forms.png

Fourier Series Spectrum
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Using the Amplitude/Phase or Complex Exponential Fourier Series representation, we can create the Fourier spectrum of a signal. The Fourier spectrum indicates the relative amplitudes (or magnitudes) and phases of the sinusoids that are required to synthesize that signal. It is also known as discrete frequency spectrum.

Info

The magnitude spectrum will always have positive magnitudes.

What if the signal is aperiodic? 🤔

Fourier Transform
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The Fourier series is a perfectly suitable construct for representing periodic functions, but what about nonperiodic (or aperiodic) functions? This is where Fourier Transform comes in place to answer this question.

Fourier Transform is the mathematical operation for converting a signal (periodic or aperiodic) from time domain into it's frequency domain. It is derived from the complex exponential Fourier Series representation and it has the following expression:
The inverse Fourier Transform converts a frequency domain signal to its time domain representation.
We may also use the symbolic form:

For Digital Communications topics, the Fourier Transform is normally used. However, it is rarely applied by definition. Instead, tables with the most common Fourier Transform pairs and properties are used.

Fourier Transform Pairs
fourier_transform_pairs.png

Fourier Transform Properties
fourier_transform_properties.png

Fourier Transform Spectrum
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The Fourier spectrum is identical to the spectrum presented during the Fourier Series. The main difference relies on a continuous spectrum instead of a discrete spectrum.

fourier_series_vs_transform_spectrum.png

Info

The Fourier Transform calculates the frequency components present in a signal, while the spectrum visually represents these components, often showing their amplitude, phase or power distribution across different frequencies.

The amplitude (or magnitude) spectrum will always have positive values, while the Fourier Transform will have signed real and imaginary numbers.

Fourier Transform Pairs and Properties
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The most common used Fourier Transform pairs in digital communications are:

fourier_transform_pairs_most_common.png

There is a small detail that is important to emphasize and redraw on the Fourier Transform pairs 8 and 9 figures, the y-axis changes according if it is a real or imaginary number.

The pair:

fourier_transform_pair_cos.png
The pair:

fourier_transform_pair_sin.png
This is going to be important later to understand more advance concepts (e.g.: Hilbert Transform).

The most common used Fourier Transform properties in digital communications are:

fourier_transform_properties_most_common.png

Sources of Confusion
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Carrier and Baseband Signals
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In digital communication literature, you will commonly find the carrier and the baseband signal . However, sometimes some resources use and or a combination of these signals.

Note

For simplicity I am going to set the amplitudes of and equal to 1

Which sinusoidal waveform should I use for the carrier and baseband, sin or cos? 🤔

To answer this question, it is important to be aware of a couple of concepts:

  • the difference between Fourier Transform and Spectrum
  • and are we working with real or complex numbers.

I will try to answer this question with some examples and then summarize the results at the end.

Example 1
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Let's start by analyzing the most common representation and multiply those two signals together:

There are two ways to approach this multiplication:

  1. Apply a trigonometric identity and then apply a Fourier Transform pair
    • the trigonometric identity can be used to represent all cases
      trig_product_identity.png
  2. Apply the modulation Fourier Transform property and then apply a Fourier Transform pair.

I am going to use option 1 but the math will yield the same results if we use option 2.

Apply the trigonometric identity to :
Then apply the Fourier Transform pair to :

fourier_transform_icon.pngFourier Transform Plot
cos_cos_fourier_transform.png

fourier_spectrum_icon.pngFourier Transform Magnitude Spectrum
cos_cos_fourier_spectrum.png

Note
  • Notice on how the Fourier Transform Plot has a real number on the y-axis
  • And the magnitude spectrum has positive magnitudes on the y-axis
  • For now these two plots are identical but we will see shortly the difference between them.

Example 2
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Now I will change the baseband signal to a sin waveform:

Apply the trigonometric identity to :
Then apply the Fourier Transform pair to :

fourier_transform_icon.pngFourier Transform Plot
sin_cos_fourier_spectrum.png

fourier_spectrum_icon.pngFourier Transform Magnitude Spectrum
cos_cos_fourier_spectrum.png

Note
  • Notice on how the Fourier Transform Plot has an imaginary number on the y-axis
  • and the magnitude spectrum has positive magnitudes on the y-axis
  • The spectrum of Example 2 is the same has Example 1 spectrum.

Example 3
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Now I will change the carrier and baseband signal to sin waveforms:

Apply the trigonometric identity to :
Then apply the Fourier Transform pair to :

fourier_transform_icon.pngFourier Transform Plot
sin_sin_fourier_spectrum.png

fourier_spectrum_icon.pngFourier Transform Magnitude Spectrum
cos_cos_fourier_spectrum.png

Note
  • Notice on how the Fourier Transform Plot has a real number on the y-axis
  • And the magnitude spectrum has positive magnitudes on the y-axis
  • The spectrum of Example 3 is the same has Example 1 and 2 spectrums.

Summary
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  • The spectrums are the same independently what type of sinusoidal waveform we used for the carrier or baseband signals.

  • This is why for analog modulations you can find different sinusoidal waveforms for the carrier and baseband and the spectrum results are the same.

  • However, for digital modulations or for some transforms used in analog modulations (e.g.: Hilbert Transform), quadrature information (or complex signals) is necessary and at this point the Fourier Transform plots can be a useful visualization tool to use.

References
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Table Of Contents

Fourier Analysis

Fourier Types

Fourier Series

Fourier Series Spectrum

Fourier Transform

Fourier Transform Spectrum

Fourier Transform Pairs and Properties

Sources of Confusion

Carrier and Baseband Signals

Example 1

Example 2

Example 3

Summary

References