Mathematics - Signals And Systems - Exercises on Modulation

Introduction

Hey it's a me again @drifter1!

Today we continue with my mathematics series about Signals and Systems in order to cover Exercises on Modulation.

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

Modulation Recap

Let's first refresh our knowledge on the concept of modulation. Modulation is one of the quite useful properties of the Fourier Transform, which tells us that the FT of two signals which get multiplied in the Time-Domain is turned into a convolution of their individual FTs whilst in the Frequency-Domain. If we consider x(t) (or x[n]) to be the signal to be "modulated", and c(t) (or c[n]) to be the signal that modulates it, or otherwise known as the carrier signal, we can specify the following:

In continuous-time systems and signals, the two basic modulation techniques are:

Amplitude Modulation (AM) = Transmission in the Frequency of the Carrier Signal
Frequency Modulation (FM) = Transmission in the Amplitude of the Carrier Signal

On the other hand, in the case of discrete-time, modulation is most commonly used in the case of modern, digital communication systems. One great example of this is Time-Division Multiplexing (TDM), where amplitude modulation with time-shifted pulse carriers can be used to transfer multiple signals (and so multiple data) using the same pulse carrier.

Continuous-Time Modulation Example (Based on 13.1 from Ref1)

Let's consider an continuous-time amplitude modulation system with input x(t) as shown below:

Determine the Fourier Transform of the output (only the magnitude), Y(ω), for each of the following carriers:

Solution

The Fourier Transform of the sinc function is a rectangular pulse, and as such the graph of X(ω) is:

For the first carrier its possible to use the shifting property of the Fourier Transform, which gives us:

And so Y(ω) can be plotted as follows:

For the second carrier we take the result of the previous case and multiply by e ^jπ/2, which gives us:

Of course this is the phase-shifting property and so the magnitude remains unaffected.

Next up are the sinusoidal carriers. The cosinus signal can be split into two complex exponentials, which makes the modulation a sum of two values:

The resulting plot is the following:

Similar to the first and second carrier, the third and fourth carrier also give us the same magnitude, but with a different phase spectrum. So, let's only split the sinus signal into the two complex exponentials:

The fifth and final carrier is a combination of the previous two sinusoidal signals, and so the 1 / 2 scaling of the magnitude of both cases gets undone, giving us the following graph:

The phase spectrum analysis gets quite complicated for this case, and so let's again skip it.

Discrete-Time Modulation Example (Based on 15.2 from Ref1)

Let's consider an discrete-time modulation system with an sinusoidal carrier signal, c[n] = cos[Ω₀ n], and the following input:

Sketch the output, Y(Ω), for Ω₀ = π / 2.

Solution

The Fourier Transform of the sinusodial carrier leads to the following impulses:

As such, based on the modulation theorem (or property), X(Ω) will be centered around each impulse of C(Ω) and scaled by 1 / 2. This results in the following graph:

RESOURCES:

References

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

Images

Mathematical equations used in this article were made using quicklatex.

Block diagrams and other visualizations were made using draw.io and GeoGebra

Previous articles of the series

Basics

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
Sinusoidal and Complex Exponential Signals → Sinusoidal and Exponential Signals in Continuous and Discrete Time

LTI Systems and Convolution

LTI System Response and Convolution → Linear System Interconnection (Cascade, Parallel, Feedback), Delayed Impulses, Convolution Sum and Integral
LTI Convolution Properties → Commutative, Associative and Distributive Properties of LTI Convolution
System Representation in Discrete-Time using Difference Equations → Linear Constant-Coefficient Difference Equations, Block Diagram Representation (Direct Form I and II)
System Representation in Continuous-Time using Differential Equations → Linear Constant-Coefficient Differential Equations, Block Diagram Representation (Direct Form I and II)
Exercises on LTI System Properties → Superposition, Impulse Response and System Classification Examples
Exercise on Convolution → Discrete-Time Convolution Example with the help of visualizations
Exercises on System Representation using Difference Equations → Simple Block Diagram to LCCDE Example, Direct Form I, II and LCCDE Example
Exercises on System Representation using Differential Equations → Equation to Block Diagram Example, Direct Form I to Equation Example

Fourier Series and Transform

Continuous-Time Periodic Signals & Fourier Series → Input Decomposition, Fourier Series, Analysis and Synthesis
Continuous-Time Aperiodic Signals & Fourier Transform → Aperiodic Signals, Envelope Representation, Fourier and Inverse Fourier Transforms, Fourier Transform for Periodic Signals
Continuous-Time Fourier Transform Properties → Linearity, Time-Shifting (Translation), Conjugate Symmetry, Time and Frequency Scaling, Duality, Differentiation and Integration, Parseval's Relation, Convolution and Multiplication Properties
Discrete-Time Fourier Series & Transform → Getting into Discrete-Time, Fourier Series and Transform, Synthesis and Analysis Equations
Discrete-Time Fourier Transform Properties → Differences with Continuous-Time, Periodicity, Linearity, Time and Frequency Shifting, Conjugate Summetry, Differencing and Accumulation, Time Reversal and Expansion, Differentation in Frequency, Convolution and Multiplication, Dualities
Exercises on Continuous-Time Fourier Series → Fourier Series Coefficients Calculation from Signal Equation, Signal Graph
Exercises on Continuous-Time Fourier Transform → Fourier Transform from Signal Graph and Equation, Output of LTI System
Exercises on Discrete-Time Fourier Series and Transform → Fourier Series Coefficient, Fourier Transform Calculation and LTI System Output

Filtering, Sampling, Modulation, Interpolation

Filtering → Convolution Property, Ideal Filters, Series R-C Circuit and Moving Average Filter Approximations
Continuous-Time Modulation → Getting into Modulation, AM and FM, Demodulation
Discrete-Time Modulation → Applications, Carriers, Modulation/Demodulation, Time-Division Multiplexing
Sampling → Sampling Theorem, Sampling, Reconstruction and Aliasing
Interpolation → Reconstruction Procedure, Interpolation (Band-limited, Zero-order hold, First-order hold)
Processing Continuous-Time Signals as Discrete-Time Signals → C/D and D/C Conversion, Discrete-Time Processing
Discrete-Time Sampling → Discrete-Time (or Frequency Domain) Sampling, Downsampling / Decimation, Upsampling
Exercises on Filtering → Filter Properties, Type and Output