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

A **rectangular signal**, also known as a rectangular window, is the signal that is most frequently encountered in DSP. This is because the fundamentals of DSP are built on the basis of continuous-time and infinite-duration signals. However, all practical signals are limited in time (say, $T$ seconds) that can be taken as a product between the same infinite-duration signal and a rectangular window of length $T$. This process is known as **windowing**.

## Definition

In discrete time, an odd-length rectangular sequence $L$ is given by:
$$  \begin{equation*}
        x[n] = \left\{ \begin{array}{l}
        1, \quad -\frac{L-1}{2} \le n \le \frac{L-1}{2} \\
        0, \quad \textmd{otherwise} \\
        \end{array} \right.
    \end{equation*}
$$

This is drawn in the figure below for $L=7$ and $N=16$:

## Spectrum

The spectrum of such a signal is not really intuitive. Let’s derive and plot this spectrum with the help of the DFT.

Since the DFT is periodic over $N$ samples, the summation can either be from $0$ to $N-1$ or $-N/2$ to $N/2$, even for $N$. We have
$$
X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}
$$
because the signal is $1$ for the given values of $n$ only. Here, we can use the well-known geometric series formula below:
$$
\sum_{n=N_1}^{N_2}a^n=\frac{a^{N_1}-a^{N_2+1}}{1-a}
$$
This produces the following result:
$$
X[k]=\frac{e^{+j2\pi\frac{k}{N}\frac{L-1}{2}}-e^{-j2\pi\frac{k}{N}\frac{L-1}{2}}}{1-e^{-j2\pi\frac{k}{N}}}
$$
Factoring the common expression out,
$$
\begin{align*}
X[k]&=\frac{e^{+j\pi\frac{k}{N}L}-e^{-j\pi\frac{k}{N}L}}{e^{+j\pi\frac{k}{N}}-e^{-j\pi\frac{k}{N}}}\\
&= \frac{\sin\left( \pi\frac{k}{N}L\right)}{\sin \left(\pi\frac{k}{N}\right)}
\end{align*}
$$
where we have used Euler's identity above. This last expression is known as the **aliased sinc signal**. For a large sample rate, $L\rightarrow \infty$ within the same window, and the numerator becomes much larger than the denominator. Using the sine identity $\sin \theta\approx \theta$ for small $\theta$, we get
$$
X[k]=\frac{\sin\left( \pi\frac{k}{N}L\right)}{\pi\frac{k}{N}}
$$
which is the well-known **sinc signal** when scaled by $L$. 

Let's run some code now. We'll start with the width $L=N$.

A **rectangular signal**, also known as a rectangular window, is the signal that is most frequently encountered in DSP. This is because the fundamentals of DSP are built on the basis of continuous-time and infinite-duration signals. However, all practical signals are limited in time (say, $T$ seconds) that can be taken as a product between the same infinite-duration signal and a rectangular window of length $T$. This process is known as **windowing**.

# Definition

In discrete time, an odd-length rectangular sequence $L$ is given by:
$$  \begin{equation*}
        x[n] = \left\{ \begin{array}{l}
        1, \quad -\frac{L-1}{2} \le n \le \frac{L-1}{2} \\
        0, \quad \textmd{otherwise} \\
        \end{array} \right.
    \end{equation*}
$$

This is drawn in the figure below for $L=7$ and $N=16$:

# Spectrum

The spectrum of such a signal is not really intuitive. Let’s derive and plot this spectrum with the help of the DFT.

Since the DFT is periodic over $N$ samples, the summation can either be from $0$ to $N-1$ or $-N/2$ to $N/2$, even for $N$. We have
$$
X[k]=\sum_{n=-\frac{N}{2}}^{\frac{N}{2}}x[n]e^{-j2\pi\frac{k}{N}n}=\sum_{n=-\frac{L-1}{2}}^{\frac{L-1}{2}}e^{-j2\pi\frac{k}{N}n}
$$
because the signal is $1$ for the given values of $n$ only. Here, we can use the well-known geometric series formula below:
$$
\sum_{n=N_1}^{N_2}a^n=\frac{a^{N_1}-a^{N_2+1}}{1-a}
$$
This produces the following result:
$$
X[k]=\frac{e^{+j2\pi\frac{k}{N}\frac{L-1}{2}}-e^{-j2\pi\frac{k}{N}\frac{L-1}{2}}}{1-e^{-j2\pi\frac{k}{N}}}
$$
Factoring the common expression out,
$$
\begin{align*}
X[k]&=\frac{e^{+j\pi\frac{k}{N}L}-e^{-j\pi\frac{k}{N}L}}{e^{+j\pi\frac{k}{N}}-e^{-j\pi\frac{k}{N}}}\\
&= \frac{\sin\left( \pi\frac{k}{N}L\right)}{\sin \left(\pi\frac{k}{N}\right)}
\end{align*}
$$
where we have used Euler's identity above. This last expression is known as the **aliased sinc signal**. For a large sample rate, $L\rightarrow \infty$ within the same window, and the numerator becomes much larger than the denominator. Using the sine identity $\sin \theta\approx \theta$ for small $\theta$, we get
$$
X[k]=\frac{\sin\left( \pi\frac{k}{N}L\right)}{\pi\frac{k}{N}}
$$
which is the well-known **sinc signal** when scaled by $L$. 

Let's run some code now. We'll start with the width $L=N$.

Derive the DFT of a rectangular signal that appears often in DSP applications.

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

A Rectangular Signal

Definition

Spectrum