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

Phase rotations impact a signal in a variety of ways. Therefore, understanding how a signal is rotated inphase is important for a DSP learner.

## Complex signals

In complex notation, rotating a complex number in the $IQ$ plane by a phase $\theta$ is very simple. 
- For a number $V$ and phase $\theta$, we write a phase rotation as:
$$
W= V\cdot e^{j\theta}
$$
Since $e^{j\theta}$ is a complex number with magnitude 1 and phase $\theta$, the magnitude of the result $W$ stays constant while $\theta$ is added to the angle of that complex number.

- For a signal $x(t)$ and phase $\theta$, we write a phase rotation as:
$$y(t)= x(t)\cdot e^{j\theta}$$

Again, the magnitude of the result $y(t)$ stays constant. This means that the individual $I$ and $Q$ values change, but their squared sum remains the same. The impact of $\theta$ appears for all time instants.

## Real signals

To understand the same process in terms of real signals, let's multiply a complex number $V$ by a complex number $U$ with magnitude $1$ and angle $\theta$. Using $|U| = 1$ and $\measuredangle U = \theta$:
$$
        U_I = \cos \theta \quad \textmd{and} \quad  U_Q = \sin \theta
$$

Plugging in the multiplication rule of complex numbers:
$$
          \begin{align*}
            W_I\: &= V_I \cdot \cos \theta - V_Q \cdot \sin \theta \\
            W_Q &= V_Q \cdot \cos \theta + V_I \cdot \sin \theta
          \end{align*}
$$
Using $V_I = |V| \cos \measuredangle V$ and $V_Q = |V| \sin \measuredangle V$:
$$          
\begin{align*}
            W_I\: &= |V| \left(\cos \measuredangle V \cdot \cos \theta - \sin \measuredangle V \cdot \sin \theta \right) \\
            W_Q &= |V| \left( \sin \measuredangle V \cdot \cos \theta + \cos \measuredangle V \cdot \sin \theta \right)
          \end{align*}
$$
Again, using the identities $\cos A \cos B - \sin A \sin B = \cos (A+B)$ and $\sin A$ $\cos B$ $+$ $\cos A \sin B = \sin (A+B)$,
$$          \begin{align*}
            W_I\: &= |V| \cos \left(\measuredangle V + \theta \right) \\
            W_Q &= |V|   \sin \left(\measuredangle V + \theta \right)
          \end{align*}
$$
which keeps the magnitude unchanged and adds the angle $\theta$ to the existing angle.

From the description above, we can derive a rule for phase rotation as follows. For a complex number $V$ and an angle $+\theta$, the phase rotation is given by:
$$    \begin{align*}
    I \cdot \cos \theta &- Q \cdot \sin \theta \\
    Q \cdot \cos \theta &+ I \cdot \sin \theta
    \end{align*}
$$

For a clockwise rotation, we get:
$$    \begin{align*}
    I \cdot \cos \theta &+ Q \cdot \sin \theta \\
    Q \cdot \cos \theta &- I \cdot \sin \theta
    \end{align*}
$$

These are the rules used in hardware implementation of complex signal processing algorithms, e.g., in field-programmable gate arrays (FPGAs) and DSPs.

Phase rotations impact a signal in a variety of ways. Therefore, understanding how a signal is rotated inphase is important for a DSP learner.

# Complex signals

In complex notation, rotating a complex number in the $IQ$ plane by a phase $\theta$ is very simple. 
- For a number $V$ and phase $\theta$, we write a phase rotation as:
$$
W= V\cdot e^{j\theta}
$$
Since $e^{j\theta}$ is a complex number with magnitude 1 and phase $\theta$, the magnitude of the result $W$ stays constant while $\theta$ is added to the angle of that complex number.

# Real signals

To understand the same process in terms of real signals, let's multiply a complex number $V$ by a complex number $U$ with magnitude $1$ and angle $\theta$. Using $|U| = 1$ and $\measuredangle U = \theta$:
$$
        U_I = \cos \theta \quad \textmd{and} \quad  U_Q = \sin \theta
$$

Plugging in the multiplication rule of complex numbers:
$$
          \begin{align*}
            W_I\: &= V_I \cdot \cos \theta - V_Q \cdot \sin \theta \\
            W_Q &= V_Q \cdot \cos \theta + V_I \cdot \sin \theta
          \end{align*}
$$
Using $V_I = |V| \cos \measuredangle V$ and $V_Q = |V| \sin \measuredangle V$:
$$          
\begin{align*}
            W_I\: &= |V| \left(\cos \measuredangle V \cdot \cos \theta - \sin \measuredangle V \cdot \sin \theta \right) \\
            W_Q &= |V| \left( \sin \measuredangle V \cdot \cos \theta + \cos \measuredangle V \cdot \sin \theta \right)
          \end{align*}
$$
Again, using the identities $\cos A \cos B - \sin A \sin B = \cos (A+B)$ and $\sin A$ $\cos B$ $+$ $\cos A \sin B = \sin (A+B)$,
$$          \begin{align*}
            W_I\: &= |V| \cos \left(\measuredangle V + \theta \right) \\
            W_Q &= |V|   \sin \left(\measuredangle V + \theta \right)
          \end{align*}
$$
which keeps the magnitude unchanged and adds the angle $\theta$ to the existing angle.

From the description above, we can derive a rule for phase rotation as follows. For a complex number $V$ and an angle $+\theta$, the phase rotation is given by:
$$    \begin{align*}
    I \cdot \cos \theta &- Q \cdot \sin \theta \\
    Q \cdot \cos \theta &+ I \cdot \sin \theta
    \end{align*}
$$

For a clockwise rotation, we get:
$$    \begin{align*}
    I \cdot \cos \theta &+ Q \cdot \sin \theta \\
    Q \cdot \cos \theta &- I \cdot \sin \theta
    \end{align*}
$$

These are the rules used in hardware implementation of complex signal processing algorithms, e.g., in field-programmable gate arrays (FPGAs) and DSPs.

Learn how to rotate a complex number by using phases.

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

Phase Rotations

Complex signals

Real signals