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

**Time invariance** implies that the specific time at which the input is applied does not matter. A delay in the input simply reflects a delay in the output.

As an example of time variance, consider that the best time for our peak physical performance is late afternoon or early evening. A cyclist doesn’t cover the same distance at all times of the day. At midnight or noon, their performance suffers.


## Definition

A system is time invariant if a delayed input results in a delayed output and remains unchanged otherwise. Mathematically, for an input $x[n]$ and output $y[n]$:
$$
x[n] \qquad \rightarrow \qquad y[n]
$$
We should have the following for a delay of $n_0$ samples:
$$
x[n-n_0] \qquad \rightarrow \qquad y[n-n_0]
$$

## Algorithm

Follow these three steps to determine whether a system is time invariant or not.
1. Delay the input signal $x[n]$ by $n_0$ samples and find the output $y_0[n]$.
2. Now delay the original output signal $y[n]$ by $n_0$ samples to find $y[n-n_0]$.
3. The system is time invariant if:
$$
y_0[n] = y[n-n_0]
$$

In a time-invariant system, the order of delay does not matter, whether it appears at the input or the output.

## Linear time-invariant systems

A system that is both linear and time invariant is known as a **linear time invariant (LTI) system**. This is the class of systems on which most DSP applications are based.

# Definition

A system is time invariant if a delayed input results in a delayed output and remains unchanged otherwise. Mathematically, for an input $x[n]$ and output $y[n]$:
$$
x[n] \qquad \rightarrow \qquad y[n]
$$
We should have the following for a delay of $n_0$ samples:
$$
x[n-n_0] \qquad \rightarrow \qquad y[n-n_0]
$$

# Algorithm

Follow these three steps to determine whether a system is time invariant or not.
1. Delay the input signal $x[n]$ by $n_0$ samples and find the output $y_0[n]$.
2. Now delay the original output signal $y[n]$ by $n_0$ samples to find $y[n-n_0]$.
3. The system is time invariant if:
$$
y_0[n] = y[n-n_0]
$$

In a time-invariant system, the order of delay does not matter, whether it appears at the input or the output.

# Linear time-invariant systems

A system that is both linear and time invariant is known as a **linear time invariant (LTI) system**. This is the class of systems on which most DSP applications are based.

Discover what it means for a system to be time invariant.

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

Time Invariance

Definition