Discover digital signal processing fundamentals, delve into signal design and analysis, learn about transforms, and develop practical OFDM systems focusing on modulation and demodulation. Gain insights into time and frequency domains.

Octave.tar.gz

octave-codewidget

octave-jupyter

We live in a world filled with signals which can be represented by sequences of numbers in a digital machine. These sequences of numbers can be manipulated by algorithms and have numerous applications in audio, images, video, wireless transmission and countless other domains. 

In this course, you’ll understand the field of digital signal processing with a focus on the design and analysis of signals, and algorithms. You’ll cover the DSP fundamentals of signals, systems and transforms. You’lll learn about the basic signal types and perform different operations. You’ll develop an understanding about the frequency domain and how to represent signals in time, and the transformation from one domain to the other using the Fourier transform. You’ll also learn about digital systems. Finally, you’ll develop a practical OFDM system focusing on modulation and demodulation.

By the end of this course, you’ll be able to model, analyze and design signals and systems in greater depth, in the time and frequency domains.

Fundamentals of Digital Signal Processing

## Frequency domain

From the DFT definition seen below, the DFT periodicity arises naturally:
$$
X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi\frac{k}{N}n},
$$
To see why, consider the DFT analysis of sinusoids at a frequency of $k+N$.
$$
e^{-j2\pi\frac{k+N}{N}n}=e^{-j2\pi\frac{k}{N}n}\cdot e^{-j2\pi n}=e^{-j2\pi\frac{k}{N}n}
$$
because $e^{-j2\pi n}=\cos 2\pi n-j\sin 2\pi n=1$.

Therefore, the DFT $X[k]$ is periodic with period $N$. This can be traced back to the fact that the discrete frequency axis is periodic, i.e., the frequency index $k$ and $k+N$ are essentially the same. This is a result of sampling the aliases outside the primary zone between $-\pi$ and $+\pi$.

## Time domain

The inverse DFT is defined as:
$$
x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}
$$

A similar derivation proves that the input signal $x[n]$ is also periodic.
$$
        x[n + N] = x[n]
$$

This can be understood as follows. While taking the DFT of an input signal $x[n]$, a finite number of samples are required, and there is no inherent periodicity visible.

However, because of the way these complex sinusoids are defined, the DFT output $X[k]$ would be the same if the input signal $x[n]$ was periodic with period $N$. This nature of periodicity is a matter of debate in the DSP community.

# Frequency domain

From the DFT definition seen below, the DFT periodicity arises naturally:
$$
X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi\frac{k}{N}n},
$$
To see why, consider the DFT analysis of sinusoids at a frequency of $k+N$.
$$
e^{-j2\pi\frac{k+N}{N}n}=e^{-j2\pi\frac{k}{N}n}\cdot e^{-j2\pi n}=e^{-j2\pi\frac{k}{N}n}
$$
because $e^{-j2\pi n}=\cos 2\pi n-j\sin 2\pi n=1$.

Therefore, the DFT $X[k]$ is periodic with period $N$. This can be traced back to the fact that the discrete frequency axis is periodic, i.e., the frequency index $k$ and $k+N$ are essentially the same. This is a result of sampling the aliases outside the primary zone between $-\pi$ and $+\pi$.

# Time domain

The inverse DFT is defined as:
$$
x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}
$$

A similar derivation proves that the input signal $x[n]$ is also periodic.
$$
        x[n + N] = x[n]
$$

This can be understood as follows. While taking the DFT of an input signal $x[n]$, a finite number of samples are required, and there is no inherent periodicity visible.

However, because of the way these complex sinusoids are defined, the DFT output $X[k]$ would be the same if the input signal $x[n]$ was periodic with period $N$. This nature of periodicity is a matter of debate in the DSP community.

Discover how a DFT gives rise to a periodic time-domain signal.

Introduction to Signals

Basic Signals and Operations

Complex Signals and the Frequency Domain

Sampling a Signal

The Discrete Fourier Transform

Fourier Transform Properties

Digital Systems

Putting It All Together: An OFDM Modem

Demodulating an OFDM Signal Stream

The Last Words

Bibliography

Periodicity in Frequency and Time Domains

Frequency domain