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

With $N$ discrete time domain samples, there are only $N$ complex sinusoids that are orthogonal to each other for $k=0,1,2,\cdots, N-1$.

The other option is $k=N$, which we'll investigate next.

# Frequency index: $k=N$

Let's explore the option $k=N$ as a complex sinusoid.
$$
    \begin{equation*}
        e^{ j2\pi \frac{k}{N}n}\Big|_{k=N} =e^{j2\pi n} =\cos 2\pi n +j\sin 2\pi n = 1+j0
    \end{equation*}
$$
because $\sin 2\pi n = 0$ and $\cos 2\pi n = 1$. That's why the options $k=0$ and $k=N$ point toward the same complex sinusoid.

Next, we consider $k=N+1$.

# Frequency index: $k=N+1$

For $k=N+1$, we have
$$
    \begin{align*}
      e^{ j2\pi \frac{N+ 1}{N}n} &= e^{j2\pi n}\cdot e^{j2\pi \frac{1}{N}n} \\
            &= e^{j 2\pi \frac{1}{N}n}
    \end{align*}
$$
which is the same as $k=1$. We conclude that there are only $N$ distinct frequencies from $0$ to $N-1$. The rest is repetition of these frequencies
This leads to an interesting fact. 
For $N$ samples in the discrete time domain, there are $N$ samples in the discrete frequency domain! 

These $N$ samples represent discrete frequencies of complex sinusoids that are all integer multiples of the one fundamental frequency $1/N$. They are all orthogonal to each other.

# Construction of the frequency axis

The discrete frequencies $k/N$ give rise to a frequency axis of the following elements:
$$
    \begin{equation*}
      0, \frac{1}{N}, \cdots, \frac{N-1}{N}
    \end{equation*}
$$

Due to the sampling theorem, we'll work with an axis centered around $0$. 

After the signal sampling, the unique range of the continuous frequency $f$ is $-f_s/2$ to $+f_s/2$. or:
$$    \begin{equation*}
    -\frac{1}{2} \le \frac{f}{f_s} < +\frac{1}{2}
    \end{equation*}
$$

Next, we can simply chop this axis at $N$ intervals because there are $N$ frequency samples in total. This is effectively sampling the frequency axis.
$$
    \begin{align}
         -\frac{1}{2},-\frac{1}{2}+\frac{1}{N},\cdots,-\frac{1}{N},0,\frac{1}{N},\cdots,+\frac{1}{2}-\frac{1}{N}
    \end{align}
$$

The $k/N$ values are listed below. Consequently, the index $k$ of the discrete frequency axis is given by $[-N/2,N/2-1]$ or:
$$
    \begin{align*}
\frac{k}{N} &=  -\frac{1}{2},-\frac{1}{2}+\frac{1}{N},\cdots,-\frac{1}{N},0,\frac{1}{N},\cdots,+\frac{1}{2}-\frac{1}{N}\\ \\
        k &= -\frac{N}{2},-\frac{N}{2}+1,\cdots,-1,0,1,\cdots,\frac{N}{2}-1
    \end{align*}
$$

This range is shown in the figure below:

Investigate the details of the discrete frequency axis.

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

Discrete Frequency Axis

Frequency index: $k=N$

Demodulating an OFDM Signal Stream

Discrete Frequency Axis

Frequency index: k=Nk=Nk=N

Frequency index: $k=N$