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.

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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

The idea of a complex sinusoid can easily be extended to discover negative frequencies. This will help us establish the concept of the frequency domain, which we will discuss in detail later.

# Direction and negative numbers

Think about negative numbers. They were shrouded in mystery in the early years of mathematical exploration. before it was discovered that they can be associated with the idea of direction.
 
- For example, you owe me $-100$ dollars if I am the one in debt. 
- Similarly, heading $+4$ km or $-4$ km determines whether we travel towards the east or the west.

# Direction and negative frequencies

The question is how can a frequency be negative if it is defined as an inverse period $F=1/T$ of a sinusoid. The idea of associating the negative value with direction is also helpful here. 

The frequency of a complex sinusoid is related to the time period through its magnitude. The direction, on the other hand, is defined through the direction of rotation.
- A positive frequency implies an anticlockwise rotation. The complex sinusoid
$$
e^{j2\pi Ft}=\cos 2\pi Ft +j\sin 2\pi Ft
$$
has a positive frequency $+F$. 
- A negative frequency implies a clockwise rotation. The complex sinusoid
$$
\begin{align*}
e^{j2\pi (-F)t}&=\cos 2\pi (-F)t +j\sin 2\pi (-F)t 
&= \cos 2\pi Ft -j\sin 2\pi Ft \\
\end{align*}
$$
has a negative frequency $-F$. In terms of real signals, the $I$ and $Q$ parts are:
$$          
\begin{align*}
            \cos 2\pi (-F) t &= \cos 2\pi F t \\
            \sin 2\pi (-F) t   &= -\sin 2\pi F t
          \end{align*}
$$
Here, we have used the identities $\cos (-\theta) = \cos \theta$ and $\sin (-\theta) = -\sin \theta$.
This is in line with the sign associated with an angle in the geometrical plane.

Run the code below in which we have chosen a negative frequency. Verify the direction of rotation as clockwise—the cosine wave without any change and sine wave with a negative amplitude.

Discover negative frequencies and where they are located.

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

Negative Frequencies

Direction and negative numbers

Direction and negative frequencies