Digital Communications: Signal Space - Part 1

Summary

In this video, Andrew introduces the concept of signal space, a key element in digital communication systems. He explains how signals representing binary values ('0' and '1') are projected onto an orthogonal basis to facilitate accurate decision-making. Using basic signals like pulses of varying heights, he illustrates how the received signal can be compared to the original signals through vector projection and dot products. By calculating the distance between vectors, the system can decide which signal was sent. This process is explained step-by-step, culminating in a decision-making strategy known as minimum distance decision or MAP (Maximum A Posteriori).

Takeaways

• 😀 Signal space is a fundamental concept used in digital communications for decision theory and signal representation.
• 😀 When sending binary signals, actual signals are used to represent zeros and ones, with examples such as pulse signals of different heights and lengths.
• 😀 The signals are projected onto a signal space to determine which signal is closest to the received signal.
• 😀 Vector space and distance calculations, like dot products, help determine which transmitted signal is closest to the received signal in signal space.
• 😀 In signal space, we use orthogonal basis vectors to represent signals, ensuring that signals are independent and distinct from one another.
• 😀 The dot product of two signals is defined as an integral over their period, used to calculate their relationship and orthogonality.
• 😀 Orthogonality of signals is crucial for making reliable decisions in communications, ensuring signals do not interfere with each other.
• 😀 The Gram-Schmidt orthogonalization process can be used to create orthogonal basis sets, although a simpler visual approach was employed in this example.
• 😀 Energy normalization of signals ensures that each basis signal has unit energy, represented by integral calculations over their periods.
• 😀 A minimum distance decision rule is applied to determine the most likely transmitted signal based on the received signal's distance in signal space.
• 😀 Signal space allows us to mathematically project received signals onto a known basis, simplifying the decision-making process in communications.

Q & A

What is the main focus of this video?

The main focus of the video is explaining the concept of signal space and how it is used to build a simple transmitter-receiver channel simulation in MATLAB.

What is the purpose of signal space in digital communication?

Signal space is used to represent the transmitted signals in a mathematical form, allowing engineers to decide which signal was sent based on the received signal by calculating the 'closeness' or 'distance' to known signal vectors.

What is the role of the dot product in signal space?

The dot product is used to measure the similarity or orthogonality between two signals. It helps in projecting received signals onto the basis vectors of the signal space to determine the closest transmitted signal.

What is an orthogonal basis in the context of signal space?

An orthogonal basis is a set of vectors in a signal space such that the dot product of any two distinct vectors equals zero. This property allows easy separation of signals in the signal space, which is crucial for reliable detection.

How are the signals for zero (S0) and one (S1) represented in this example?

In this example, S0 is a pulse of height 1 for 2 seconds, and S1 is a pulse of height 2 for 1 second. These pulse shapes are used to represent the binary signals 0 and 1 in the signal space.

What is the significance of creating a basis set in signal space?

Creating a basis set in signal space allows for the construction of all possible signal vectors within that space. This enables efficient decision-making about which signal was transmitted by projecting the received signal onto these basis vectors.

Why do we use the distance (or magnitude) squared in the decision process?

Using the square of the distance simplifies calculations by avoiding the need to take square roots, as the relative distance between vectors is preserved in the squared form.

What method does the video suggest for determining the orthogonal basis for signals?

The video suggests using an intuitive or 'eyeballing' method, where you analyze the signal shapes to determine orthogonal basis sets, rather than relying on more formal mathematical processes like Gram-Schmidt orthogonalization.

How do we determine the correct binary signal (0 or 1) based on the received signal?

The received signal is projected onto the signal space, and the closest basis vector is determined by calculating the distance between the received signal and the two possible transmitted signal vectors (S0 and S1). The signal corresponding to the closest vector is chosen as the transmitted signal.

What is the purpose of projecting the received signal onto the basis vectors?

Projecting the received signal onto the basis vectors helps to determine which transmitted signal (S0 or S1) is closest to the received signal, facilitating decision-making in the communication process.