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

If an input signal $x[n]$ is applied to a discrete-time system with the impulse response $h[n]$, the output $y[n]$ is given by convolution that is implemented through the following expression:
$$
y[n] = x[n]*h[n] = \sum_m x[m]h[n-m]
$$



## Time indexes

There are two time indexes here: $m$ and $n$.
- $m$ represents the summation index running over the samples to be multiplied.
- $n$ represents the current index 
of the *output* signal in which sample-wise products and summations are being performed.

## Algorithm

From the convolution expression above, the following algorithm can be devised.

Rearrange the equation as:
$$
          y[n] = \sum _m x[m] h[-m+n]
$$

- **Flip:** Here, the impulse response is in terms of the time index $m$, which is being flipped indicated by the minus sign to obtain $h[-m]$.
- **Shift:** The target is to obtain $h[-m+n]$, i.e., shift $h[-m]$ by $n$ indexes to the right for positive $n$ and to the left for negative $n$.
- **Multiply:** Perform the sample-by-sample multiplication of the two signals $x[m]$ and $h[-m+n]$.
- **Sum:** Add the values of the product sequence above, which generates the convolution output at time $n$.

This process has to be repeated for all possible values of $n$.

## Example

Let's convolve the following two discrete-time signals:
$$
\begin{align*}
x[n] &= \{2, -1, 1\}\\
h[n] &= \{1, 1, -1\}
\end{align*}
$$

The figure below illustrates the procedure explained above:

# Time indexes

There are two time indexes here: $m$ and $n$.
- $m$ represents the summation index running over the samples to be multiplied.
- $n$ represents the current index 
of the *output* signal in which sample-wise products and summations are being performed.

# Algorithm

From the convolution expression above, the following algorithm can be devised.

Rearrange the equation as:
$$
          y[n] = \sum _m x[m] h[-m+n]
$$

- **Flip:** Here, the impulse response is in terms of the time index $m$, which is being flipped indicated by the minus sign to obtain $h[-m]$.
- **Shift:** The target is to obtain $h[-m+n]$, i.e., shift $h[-m]$ by $n$ indexes to the right for positive $n$ and to the left for negative $n$.
- **Multiply:** Perform the sample-by-sample multiplication of the two signals $x[m]$ and $h[-m+n]$.
- **Sum:** Add the values of the product sequence above, which generates the convolution output at time $n$.

This process has to be repeated for all possible values of $n$.

# Example

Let's convolve the following two discrete-time signals:
$$
\begin{align*}
x[n] &= \{2, -1, 1\}\\
h[n] &= \{1, 1, -1\}
\end{align*}
$$

The figure below illustrates the procedure explained above:

Learn how to implement convolution in digital systems.

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

How to Implement Convolution

Time indexes