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 energy of a length-$N$ discrete-time signal is defined as:
$$
E_x = \sum_{n=0}^{N-1} |x[n]|^2
$$
Parseval's relation links the energy of a signal in the time domain to its energy in the frequency domain. In the equation form, it is expressed as:
$$
\sum _{n=0} ^{N-1} |x[n]|^2 = \frac{1}{N} \sum _{k=0} ^{N-1} |X[k]|^2
$$
In other words, the energy of a signal in the time domain is equal to its energy in the transform domain after scaling by $1/N$. 

## Derivation

From the definition of the inverse discrete Fourier transform, we have:
$$
x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}
$$

Using this in the expression for the energy of a signal:
$$
\begin{align*}
\sum _{n=0} ^{N-1} |x[n]|^2 &= \sum _{n=0} ^{N-1} x[n]\cdot x^*[n]\\
&= \sum _{n=0} ^{N-1} x[n]\cdot \left(\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}\right)^*
\end{align*}
$$

From here, we can rearrange the terms after conjugation.
$$
\begin{align*}
\sum _{n=0} ^{N-1} |x[n]|^2 &= \sum _{n=0} ^{N-1} x[n]\cdot \frac{1}{N}\sum_{k=0}^{N-1}X^*[k]e^{-j2\pi\frac{k}{N}n}\\
&= \frac{1}{N}\sum_{k=0}^{N-1}X^*[k]\underbrace{\sum _{n=0} ^{N-1} x[n] e^{-j2\pi\frac{k}{N}n}}_{X[k]}\\
&= \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2
\end{align*}
$$


## Intuition

Intuitively, going into the frequency domain doesn't alter the energy contents of the observed signal. In mathematics, this is known as a **unitary transformation**.

As far as the scaling factor of $1/N$ is concerned, recall that the DFT outputs a sum of complex sinusoids. For a constant input, this kind of sum magnifies the input by $N$. This is why a reduction by $1/N$ is necessary to balance the energy in the two domains. Due to this reason, many software routines define the DFT and inverse DFT with the same scaling factor of $1/\sqrt{N}$.

The energy of a length-$N$ discrete-time signal is defined as:
$$
E_x = \sum_{n=0}^{N-1} |x[n]|^2
$$
Parseval's relation links the energy of a signal in the time domain to its energy in the frequency domain. In the equation form, it is expressed as:
$$
\sum _{n=0} ^{N-1} |x[n]|^2 = \frac{1}{N} \sum _{k=0} ^{N-1} |X[k]|^2
$$
In other words, the energy of a signal in the time domain is equal to its energy in the transform domain after scaling by $1/N$. 

# Derivation

From the definition of the inverse discrete Fourier transform, we have:
$$
x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}
$$

Using this in the expression for the energy of a signal:
$$
\begin{align*}
\sum _{n=0} ^{N-1} |x[n]|^2 &= \sum _{n=0} ^{N-1} x[n]\cdot x^*[n]\\
&= \sum _{n=0} ^{N-1} x[n]\cdot \left(\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{+j2\pi\frac{k}{N}n}\right)^*
\end{align*}
$$

From here, we can rearrange the terms after conjugation.
$$
\begin{align*}
\sum _{n=0} ^{N-1} |x[n]|^2 &= \sum _{n=0} ^{N-1} x[n]\cdot \frac{1}{N}\sum_{k=0}^{N-1}X^*[k]e^{-j2\pi\frac{k}{N}n}\\
&= \frac{1}{N}\sum_{k=0}^{N-1}X^*[k]\underbrace{\sum _{n=0} ^{N-1} x[n] e^{-j2\pi\frac{k}{N}n}}_{X[k]}\\
&= \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2
\end{align*}
$$


# Intuition

Intuitively, going into the frequency domain doesn't alter the energy contents of the observed signal. In mathematics, this is known as a **unitary transformation**.

As far as the scaling factor of $1/N$ is concerned, recall that the DFT outputs a sum of complex sinusoids. For a constant input, this kind of sum magnifies the input by $N$. This is why a reduction by $1/N$ is necessary to balance the energy in the two domains. Due to this reason, many software routines define the DFT and inverse DFT with the same scaling factor of $1/\sqrt{N}$.

Discover the relation between signal energy in the time and frequency domains.

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

Parseval's Theorem

Derivation