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

We know that a phase shift is directly proportional to a time shift in a sinusoid.
$$
\begin{align*}
\cos \left(\omega t +\phi\right)&=\cos \left[\omega \left(t +\frac{\phi}{\omega}\right)\right]\\
&=\cos \left(\omega \left(t -\tau\right)\right)
\end{align*}
$$
Here, $\tau$ is a time shift. From this, we observe the exact relationship below:
$$
\phi = -\omega\tau
$$

Clearly, the time shift of each sinusoid depends on its phase shift normalized by *its own frequency*. This is a linear expression of the form $y=mx$ in which the slope $m$ is given by $-\tau$.

## Signal distortion

Almost all practical signals consist of multiple sinusoids. If there is a delay in the signal processing chain through a system, e.g., an amplifier or a filter, then this delay affects all the constituent sinusoids according to their own frequencies.

Let's see an example in the code below. A square wave is again generated by a sum of sinusoids, but their phases can be altered to observe their impact.

We know that a phase shift is directly proportional to a time shift in a sinusoid.
$$
\begin{align*}
\cos \left(\omega t +\phi\right)&=\cos \left[\omega \left(t +\frac{\phi}{\omega}\right)\right]\\
&=\cos \left(\omega \left(t -\tau\right)\right)
\end{align*}
$$
Here, $\tau$ is a time shift. From this, we observe the exact relationship below:
$$
\phi = -\omega\tau
$$

Clearly, the time shift of each sinusoid depends on its phase shift normalized by *its own frequency*. This is a linear expression of the form $y=mx$ in which the slope $m$ is given by $-\tau$.

# Signal distortion

Almost all practical signals consist of multiple sinusoids. If there is a delay in the signal processing chain through a system, e.g., an amplifier or a filter, then this delay affects all the constituent sinusoids according to their own frequencies.

Let's see an example in the code below. A square wave is again generated by a sum of sinusoids, but their phases can be altered to observe their impact.

Discover why the linear phase is preferred in most DSP algorithms.

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

The Significance of a Linear Phase