Recall the receiver model that we created previously, where all non-idealities besides noise have been fixed up the chain.
where \(a[n]\) are symbols in a constellation and \(w[n]\) is complex Gaussian noise. We can write this complex noise as the following vector
\[ \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \sim N \left( \begin{bmatrix} 0 \\ 0 \end{bmatrix} , \begin{bmatrix} \frac{N_0}{2} & 0 \\ 0 & \frac{N_0}{2} \\ \end{bmatrix} \right) \]
We can look at a few examples to get familiar with performance analysis of these receivers.
We start with a 4-PAM constellation of real values.
import matplotlib.pyplot as plt
x = [-3, -1, 1, 3]
labels = [r"$\frac{-3d}{2}$", r"$\frac{-d}{2}$", r"$\frac{d}{2}$", r"$\frac{3d}{2}$"]
plt.figure(figsize=(6, 1))
plt.box(False)
plt.plot(x, [0, 0, 0, 0], 'ko-')
plt.xticks(x, labels)
plt.yticks([])
plt.show()
Notice that each point is equally spaced by a distance \(d\). We can write our constellation as follows
\[ A = \left\{ \frac{-3d}{2}, \frac{-d}{2}, \frac{d}{2}, \frac{3d}{2} \right\} \]
We could write the probability of \(\tilde{a}\) on our constellation diagram to see what we can expect to detect.
import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats
x = [-3, -1, 1, 3]
labels = [r"$\frac{-3d}{2}$", r"$\frac{-d}{2}$", r"$\frac{d}{2}$", r"$\frac{3d}{2}$"]
plt.figure(figsize=(6, 1))
plt.box(False)
plt.plot(x, [0, 0, 0, 0], 'ko-')
plt.xticks(x, labels)
for v in x:
mu = v
variance = 1
sigma = np.sqrt(variance)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
plt.plot(x, stats.norm.pdf(x, mu, sigma))
plt.yticks([])
plt.show()
Notice that at every point there is some \(a[n]\) which has a higher likelihood than any others. This value is the maximum likelihood estimate. We can thus split our constellation to determine how a received symbol will be detected. We call these separate regions decision regions \(D_a\). In this case, our decision regions are between the thresholds \(-d, 0, +d\).
These decision region tells us how a particular received symbol would be detected. For example, let the actual symbol \(\hat{A} = \frac{+d}{2}\). Then if the received symbol \(\tilde{A}\) falls within the correct range, i.e. \(0 \leq \tilde{A} \leq +d\), we would detect it correctly. We can thus describe our decision regions based on their range of acceptable values, i.e. \(D_{\frac{+d}{2}} = \{\tilde{a}: 0 \leq \tilde{a} \leq +d\}\).
We can similarly define an error in detection if the received symbol falls outside the region in which the correct symbol resides. Mathematically, we say an error occurs when \(\tilde{a} \notin D_{a}\), or \(\tilde{a} \in D^C_{a}\).
We can think of our received symbols as random variables. We can express the probability of an error as follows by using the total probability law
\[ \mathbb{P}[\hat{A}_{ML} \neq A] = \sum_{a \in A} \mathbb{P}[\hat{A} \neq A \mid A = a] \mathbb{P}[A=a] \]
Very often, we assume that symbols are uniformly distributed, so the term \(\mathbb{P}[A=a]\) is simply \(\frac{1}{|A|}\). We can then rewrite the above expression
\[ \mathbb{P}[\hat{A}_{ML} \neq A] = \frac{1}{|A|} \sum_{a \in A} \mathbb{P}[\tilde{A} \in D^C_a \mid A = a] \]
We can write this expression as an integral
\[ \mathbb{P}[\hat{A}_{ML} \neq A] = \frac{1}{|A|} \sum_{a \in A} \int_{D^C_a} f_{\tilde{a} \mid a} (\tilde{a} \mid a) d\tilde{a} \]
This equation is true in general for constellations with uniform distributions and Gaussian noise. If we look at the picture from before, though, we see that there is some symmetry to take advantage of.
Notice that the probability of error in the shaded region near \(-d\) is the same size as the shaded region near \(+d\), only mirrored. We can take advantage of this property to make our computations easier.
We can compute the probability of detecting \(\hat{A}\) when we know \(A=\frac{-3d}{2}\)
\[ \mathbb{P} [\hat{A} \neq A \mid A = \frac{-3d}{2}] = \int_{-d}^{\infty} N(\tilde{a}, \frac{-3d}{2}, \frac{N_0}{2}) d\tilde{a} \]
The first step whenever integrating a Guassian distribution with a non-zero mean and non-unit variance is to normalize it. We can say \(b=\frac{\tilde{a} - \frac{-3d}{2}}{\sqrt \frac{N_0}{2}}\) and then rewrite our equation using \(b\)
\[ \mathbb{P} [\hat{A} \neq A \mid A = \frac{-3d}{2}] = \int_{\frac{d / 2}{\sqrt{N_0 / 2}}}^{\infty} N(b, 0, 1) db \]
Recall the Q function:
\[ Q(x) = \int_x^\infty \frac{1}{\sqrt{2 \pi}} e^{\frac{-t^2}{2}} dt \]
then our equation from above becomes the Q function of the lower bound minus the Q function of the upper bound, which is the same as
\[ Q\left(\sqrt{\frac{d^2}{2 N_0}}\right) \]
We can use the same methodology for the next region.
\[ \mathbb{P} [\hat{A} \neq A \mid A = \frac{-d}{2}] = \int_{-\infty}^{-d} N(\tilde{a}, \frac{-3d}{2}, \frac{N_0}{2}) d\tilde{a} + \int_{0}^{\infty} N(\tilde{a}, \frac{-3d}{2}, \frac{N_0}{2}) d\tilde{a} \]
We can do the same change of variables to get \(b=\frac{\tilde{a} - \frac{-d}{2}}{\sqrt \frac{N_0}{2}}\), and then we can rewrite the above equation
\[ \mathbb{P} [\hat{A} \neq A \mid A = \frac{-d}{2}] = 2 \int_{\frac{d / 2}{\sqrt{N_0 / 2}}}^{\infty} N(b, 0, 1) db \]
which is the same as
\[ 2 Q \left(\sqrt{\frac{d^2}{2 N_0}}\right) \]
The final result is that the probability of a symbol error \(P_s\) is
\[ P_s = \mathbb{P}[\hat{A} \neq A] = \frac{1}{4} \left[ 2 Q(\sqrt{\frac{d^2}{2 N_0}}) + 4Q(\sqrt{\frac{d^2}{2 N_0}}) \right] \]
based on adding the error probability for each region, and because the Q function is the same, we can simplify
\[ \frac{3}{2} Q \left(\sqrt{\frac{d^2}{2 N_0}} \right) \]
There are a few conventions in performance analysis to be aware of.
\[ P_s = \frac{3}{2} Q \left(\sqrt{\frac{2}{5} \cdot \frac{E_s}{N_0}} \right) \]