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

Until now, we haven’t included the impact of noise. The symbols sent as $-1$ and $+1$ have been received at the same amplitude. Let's include the additive white Gaussian noise and see how bits can be obtained from the equalized samples.

# Additive white Gaussian noise

**Additive white Gaussian noise (AGWN)** results from the random motion of electrons in the receiver frontend.
- This noise is added to the received signal.
- White light consists of all seven colors with equal intensity. Similarly, white noise implies a power spectral density that is equally spread across all frequencies.
- The noise amplitude is Gaussian distributed. This implies that the values close to the mean (the received signal itself) are more probable than values away from the mean.

# Minimum distance rule

When AWGN is added to the received samples, the equalized samples do not exactly map to the symbols $-1$ and $+1$ as before. Instead, they can assume any value around the mean, and the spread depends on the power of the noise. The symbol decisions are taken according to the minimum distance rule.

The minimum distance rule says that each sample should be mapped to the closest symbol. In terms of binary modulation, this implies that values greater than $0$ map to the symbol $+1$ and the values less than $0$ map to the symbol $-1$.
$$
\hat X[k] = \left\{ \begin{array}{l}
        -1, \quad r[k] < 0 \\
        +1, \quad r[k] > 0 \\
        \end{array} \right.
$$

Let's apply this rule and demodulate the transmitted bits in the presence of noise. As opposed to the previous cases, some symbol decisions can be wrong when, for instance, noise amplitude is strong enough to push a $-1$ symbol beyond the boundary of $0$ and vice versa.

Investigate the minimum distance rule and how received samples are mapped back to bits.

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

Digital Demodulation