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

In this chapter, we’ll look at the time shift from an $IQ$ viewpoint.

# A shift in impulse

Let's assume that a unit impulse signal $\delta[n]$ is time-shifted by $n_0$. From the DFT definition, we get
$$
X[k]=\sum_{n=0}^{N-1}\delta[n-n_0]e^{-j2\pi\frac{k}{N}n}=e^{-j2\pi\frac{k}{N}n_0}
$$
because $\delta[n-n_0]$ is $1$ at $n_0$ and zero everywhere else. What exactly is this output?

Since $k$ is the variable and $n_0$ is the constant, the DFT output is a *frequency-domain complex sinusoid* with an amplitude of $1$.

Consider the following signals that we’ll find the DFT for:
- A unit impulse signal. 
- A circularly time-shifted unit impulse.

These signals are shown in the figure below:

Explore the impact of a time shift on the signal spectrum.

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

Spectrum After the Time Shift II

A shift in impulse