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

Varying sample rates (e.g., $f_s=5, 10, 20$ samples/second) for a continuous-time signal determine the spacing between the samples. The question is how close or far we can sample in time. Here, we explore some interesting phenomena observed when the sample rate is changed beyond certain values. 

After sampling, a complex sinusoid has a frequency scaled by $f_s$ because
$$
    \begin{align}
      x[n] &= A e^{j \left(2\pi f t + \theta\right)}| _{t=nT_s}\nonumber \\
           &= A e^{ j\left(2 \pi \frac{f}{f_s} n + \theta\right)}    
\end{align}
$$
where $f_s=1/T_s$. The frequency $\frac{f}{f_s}$ in the equation above is the frequency of the discrete-time signal $x[n]$. 

# Aliasing

Now let's sample another complex sinusoid at the same rate $f_s$ and with the continuous frequency $f + k f_s$, where $k = \pm 1, \pm 2,etc$.
$$
    \begin{equation*}
      x(t) = A e^{ j\left\{2 \pi (f +kf_s) t + \theta\right\}}
\end{equation*}
$$
The sampled waveform becomes
$$
\begin{align*}
      x[n] &= A e^{ j\left\{2 \pi (f + kf_s) nT_s + \theta\right\}} \\
           &= A e^{j \left(2 \pi \frac{f + kf_s}{f_s} n + \theta \right)} \\&= A e^{j\left(2 \pi \frac{f}{f_s} n + 2\pi kn + \theta\right)} \\
           &= A e^{j \left(2 \pi \frac{f}{f_s} n + \theta\right)} 
    \end{align*}
$$
due to the $2\pi$ periodicity of the sinusoid. This expression is exactly the same as that of the discrete-time sinusoid in Eq (1). 

It can be concluded that at the output of the sampling process, it’s impossible to distinguish between two discrete-time signals whose frequencies are $f_s$ Hz apart. This phenomenon is known as **aliasing**.

# Time domain

A time domain demonstration of aliasing is shown in the figure below. Observe that both sinusoids pass through the same set of points in the discrete domain. Due to this reason, they cannot be distinguished from one another after the sampling is done.

Explore the sampling and aliasing effects 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

Why Do Spectral Aliases Arise?