Question

Sᵢ(t) = √2Eₛ / Tₛ cos(2π fₑt - π / 4 - i π / 2), 0 ≤ t ≤ Tₛ = n / fₛ, i = 0,1,2,3 n integer,

is the received signal is r(t)= sᵢ(t) + n(t). The noise n(t) has double sided power spectral density N₀ / 2. The four possible 4-PSK signals are equiprobable.

Draw the optimal basis function receiver for 4-PSK signal. Plot signal space diagram. What are decision regions?

Answer 1

The question involves constructing an optimal basis function receiver for 4-PSK signals, plotting a signal space diagram, and identifying decision regions for signal detection in digital communications. Basis functions are orthogonal functions used for projection, and decision regions are the quadrants in the signal space where the receiver classifies the received signal despite the presence of Gaussian noise.

Step-by-step explanation:

The question relates to the optimal basis function receiver for a 4-PSK (Phase Shift Keying) signal system and involves understanding signal space diagrams and decision regions in the context of digital communication. In 4-PSK, four possible signals (s0(t), s1(t), s2(t), s3(t)) corresponding to i=0,1,2,3 are equidistant from the origin in the signal space diagram and are separated by 90 degrees from each other. Their projections onto the basis functions are used to decode the received signal r(t), which includes noise n(t).

To draw the optimal basis function receiver for 4-PSK, one would create two orthogonal basis functions, which can be cos(2πfet) and sin(2πfet), and then the receiver will project the received signal, r(t), onto these basis functions to recover the original signal, sᵢ(t). The noise n(t) has a constant power spectral density of N0/2, which affects the received signal and is represented as a Gaussian noise contribution in the signal space diagram.

A signal space diagram for 4-PSK would show four points, each representing one of the possible signals, at equal radius from the origin, on the axes defined by the basis functions. The decision regions for 4-PSK signal detection are typically demarcated by the lines x=0 and y=0, which divide the signal space into four quadrants. Each region corresponds to one of the 4-PSK signals and is where the receiver decides which signal was most likely transmitted based on the projection of the received signal onto the basis functions. Any received signal falling into one of these regions will be interpreted as the signal corresponding to that region.