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

Correlation is a measure of similarity between two signals and forms the foundations for most detection and estimation problems in signal processing. In our everyday life, we recognize something by correlating it with what we already know. Correlation plays a crucial role in many areas of our life such as economics, marketing, and psychology.

## The algorithm

As described in the beginning, the objective of correlation between two signals is to measure the degree to which those two signals are similar to each other. Mathematically, correlation between two signals $x[n]$ and $h[n]$ is defined as
$$y[n] = \sum \limits _{m = -\infty} ^{+\infty} x[m] h[m-n]$$
where the above sum is computed for each $n$ from $-\infty$ to $+\infty$. 


The process of correlation can be understood by reading the following steps:

**1. Select the time shift:** 

Observe that there are two indices, $n$ and $m$. The index $m$ is the summation index, and the index $n$ is the variable used for the signal time shift.
$$        \begin{equation*}
                h[m-n]
        \end{equation*}$$

**2. Sample-by-sample product:** 

Keep $x[m]$ where it is and compute the product, like so:
$$    \begin{equation*}
        x[m]\cdot h[m-n]
    \end{equation*}$$
The only nonzero values will be at those indices where the two signals overlap.

**3. Sum:** 

Sum these sample-by-sample products to generate the correlation output at the shift $n$.
$$    \begin{equation*}
      y[n] = \sum \limits _{m = -\infty} ^{+\infty} x[m] h[m-n]
    \end{equation*}$$

**4. Repeat:** 

Repeat the process above for **all** possible time shifts $n$.

Intuitively, this summation of sample-by-sample products computes the level of similarity between the two signals at each time shift. The more their samples match in magnitude and sign, the higher the output.

# The algorithm

As described in the beginning, the objective of correlation between two signals is to measure the degree to which those two signals are similar to each other. Mathematically, correlation between two signals $x[n]$ and $h[n]$ is defined as
$$y[n] = \sum \limits _{m = -\infty} ^{+\infty} x[m] h[m-n]$$
where the above sum is computed for each $n$ from $-\infty$ to $+\infty$. 


The process of correlation can be understood by reading the following steps:

**1. Select the time shift:** 

Observe that there are two indices, $n$ and $m$. The index $m$ is the summation index, and the index $n$ is the variable used for the signal time shift.
$$        \begin{equation*}
                h[m-n]
        \end{equation*}$$

**2. Sample-by-sample product:** 

Keep $x[m]$ where it is and compute the product, like so:
$$    \begin{equation*}
        x[m]\cdot h[m-n]
    \end{equation*}$$
The only nonzero values will be at those indices where the two signals overlap.

**3. Sum:** 

Sum these sample-by-sample products to generate the correlation output at the shift $n$.
$$    \begin{equation*}
      y[n] = \sum \limits _{m = -\infty} ^{+\infty} x[m] h[m-n]
    \end{equation*}$$

**4. Repeat:** 

Repeat the process above for **all** possible time shifts $n$.

Intuitively, this summation of sample-by-sample products computes the level of similarity between the two signals at each time shift. The more their samples match in magnitude and sign, the higher the output.

Understand correlation as a fundamental operation in DSP.

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

Correlation between Signals

The algorithm