add module 05

css/style.css

@@ -127,7 +127,7 @@ small {
 /* Add top and bottom paddings to the footer on larger devices */
 @media (min-width: 768px) {
   footer {
-      padding-top: 2rem;
+      padding-top: 1rem;
       padding-bottom: 2rem;
       margin-top: 2em;
   }

morea/03-wireless-channel-physical-model/reading-03-reflecting-wall-fixed-antenna.md

@@ -34,7 +34,7 @@ From the expression, we know that the received signal is a superposition of two
 
 Let us formulate this idea mathematically. The phase difference between the two sinusoids is
 \\[
-  \Delta \theta = \left( \frac{2 \pi f (2d-r)}{c} + \pi \right) - \left( \frac{2 \pi f r}{c} + \pi \right) = \frac{4 \pi f}{c} (d-r) + \pi.
+  \Delta \theta = \left( \frac{2 \pi f (2d-r)}{c} + \pi \right) - \left( \frac{2 \pi f r}{c} \right) = \frac{4 \pi f}{c} (d-r) + \pi.
 \\]
 
 The two sinusoids go from adding up constructively to canceling each other when the phase difference changes by \\(\pi\\), namely when
@@ -55,10 +55,8 @@ In terms of the frequency, the signal strength changes from a peak to a valley w
   \frac{1}{2} \left( \frac{2d-r}{c} - \frac{r}{c} \right)^{-1}.
 \\]
 Notice that the following quantity
-\\[
-  T_d = \frac{2d-r}{c} - \frac{r}{c},
-\\]
-is actually the propagation delay of the two paths. This is called **delay spread**.
+the difference between the propagation delay of the two paths
+is actually the difference between the propagation delay of the two paths. This is called **delay spread**.
 
 From the two equations above, we define \\(1/(2T_d)\\) as the **coherence bandwidth**. We can see that the coherence bandwidth is of the order of $1/T_d$.
 

morea/04-wireless-channel-input-output-model/assessment-04-input-output-model.md

@@ -0,0 +1,42 @@
+---
+title: "Input/output models"
+published: true
+morea_id: assessment-04-input-output-model
+morea_summary: "Physical models of wireless channels"
+# morea_outcomes_assessed:
+ # - outcome-CHANGE-ME
+morea_type: assessment
+morea_start_date: "2024-02-06T00:00"
+morea_end_date: "2024-02-12T23:55"
+morea_labels:
+---
+
+# Input/output models of wireless channels
+
+*Please submit your solutions in Laulima.*
+
+In this assessment, we look at a realistic channel model with its parameters determined by field measurements. Then we will implement a simplified version of this channel model.
+
+One feature of the 5th-generation (5G) wireless communication systems is the usage of the millimeter-wave (mmWave) band at 60 GHz. Therefore, it is crucial to understand the mmWave channels. The Millimetre-Wave Evolution for Backhaul and Access (MiWEBA) project is one of the efforts toward this goal. If you are interested, you might want to take a look at [the complete project report](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=e41ca51b3a590d267f5d09661b790c9002d3abbc).
+
+For this assessment, we will only need to refer to [this paper](https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-016-0568-6), which summarizes the main findings in channel modeling. You can read through the first two sections to get the background of the study.
+
+We will implement a quasi-deterministic (Q-D) channel model, as illustrated in Fig. 8 of [the paper](https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-016-0568-6). The Q-D channel consists of a few strong deterministic rays (D-rays) and some random rays (R-rays). Each ray or multipath is a cluster of rays with similar delays. You can look at Fig. 9 of [the paper](https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-016-0568-6) for an example with two D-rays (i.e., the line-of-sight (LoS) path and the ground reflection path) and two R-rays (i.e., one path due to reflection from a car and the other due to reflection from a building). 
+
+Below are the list of tasks to do.
+* Please look at Fig. 3 and Fig. 4, and identify four major multipaths in terms of delay (in ns) and power (in dB).
+  - Note that a major multipath should have a relatively high power (i.e., less attenuation). In addition, peaks that are very close to each other can be clustered into one multipath.
+* Identify two D-rays and three R-rays. Specify the delay and the power.
+* Direct ray
+* Ground ray
+* Random ray that is reflected from a random reflector (e.g., a car)
+* a far wall ray that is reflected from a far reflector (e.g., a building far away from the receiver)
+
+<figure style="text-align: center;">
+  <img src="04-baseband-spectrum.png" alt="Moving antennas with a reflecting wall at the transmitter" width="600">
+</figure>
+
+* (1 point) Derive the analytical expression of the received signal using the ray tracing method.
+* (2 points) Following [the procedure here](reading-03-reflecting-wall-fixed-antenna.html), derive the coherence distance, the delay spread, and the coherence bandwidth. Compare them with the example when the wall is at the receiver.
+* (2 points) Following [the procedure here](reading-03-reflecting-wall-moving-antenna.html), derive the Doppler spread and the coherence time. Compare them with the example when the wall is at the receiver.
+

morea/04-wireless-channel-input-output-model/module-wireless-channel-input-output-model.md

@@ -15,7 +15,7 @@ morea_experiences:
   - experience-04-baseband-passband-conversion
   - experience-04-sampling-theorem
 morea_assessments:
-  # - assessment-CHANGE-ME
+  # - assessment-04-input-output-model
 morea_type: module
 morea_icon_url: /morea/04-wireless-channel-input-output-model/04-module-icon-multipath.png
 morea_start_date: "2024-01-22"

morea/05-detection-in-fading-channel/experience-05-ber-vs-snr-curve.md

@@ -0,0 +1,11 @@
+---
+title: "BER vs SNR curves"
+published: true
+morea_id: experience-05-ber-vs-snr-curve
+morea_type: experience
+morea_summary: "BER versus SNR curves"
+morea_url: https://colab.research.google.com/drive/1eUt82xKu4llHabRWEYteLY6ZWdpKDXDO?usp=sharing
+morea_start_date: "2024-02-07"
+morea_labels:
+---
+

morea/05-detection-in-fading-channel/module-detection-in-fading-channel.md

@@ -0,0 +1,24 @@
+---
+title: "Detection in a Fading Channel"
+published: true
+morea_coming_soon: false
+morea_id: module-in-detection-fading-channel
+morea_prerequisites:
+morea_outcomes:
+  - outcome-05-detection-in-fading-channel
+morea_readings:
+  - reading-05-roadmap
+  # - reading-05-awgn-vs-fading
+morea_experiences:
+  - experience-05-ber-vs-snr-curve
+morea_assessments:
+  # - assessment-CHANGE-ME
+morea_type: module
+morea_icon_url: /morea/05-detection-in-fading-channel/05-module-icon-detection-2.webp
+morea_start_date: "2024-02-05"
+morea_end_date: "2024-02-07"
+morea_labels:
+morea_sort_order: 5
+---
+
+We analyze the performance degradation under a wireless fading channel.

morea/05-detection-in-fading-channel/outcome-05-detection-in-fading-channel.md

@@ -0,0 +1,11 @@
+---
+title: "Detection in a fading channel"
+published: true
+morea_id: outcome-05-detection-in-fading-channel
+morea_type: outcome
+morea_sort_order: 50
+---
+
+  * You understand intuitively why fading degrades the performance.
+  * You know the differences between the error rate versus SNR curve under fading channels and that under AWGN channels.
+  * You understand how to derive the error rate.

morea/05-detection-in-fading-channel/reading-05-awgn-vs-fading.md

@@ -0,0 +1,111 @@
+---
+title: "AWGN vs fading channels"
+published: true
+morea_id: reading-05-awgn-vs-fading
+morea_summary: "A failed attempt in the fading channel"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 51
+---
+
+# AWGN channel versus fading channel
+
+The input/output model of a wired channel is usualy modeled as an additive white Gaussian noise (AWGN) channel, namely
+
+\begin{align}
+  y[m] = x[m] + w[m],
+\end{align}
+where $w[m] \sim \mathcal{CN}(0, N_0)$ is a circularly symmetric Gaussian random variable. It is reasonable to assume that the noises over different time slots are independent (i.e., "white").
+
+In contrast, the input/output model of a wireless Rayleigh fading channel with only one path is
+
+\begin{align}
+  y[m] = h[m] x[m] + w[m],
+\end{align}
+where $h[m]$ is a circularly symmetric Gaussian random variable. Therefore, the channel has a magnitude following the Rayleigh distribution, and a phase following a uniform distribution in $[0, 2\pi]$.
+
+Next, we use a simple signaling scheme, Binary Phase-shift keying (BPSK), to illustrate the key differences of the AWGN channel and the Rayleigh fading channel.
+
+
+## Performance in the AWGN channel
+
+In BPSK, we assign the binary bits of 0 and 1 to $x[m]=-a$ and $x[m]=+a$, respectively. Our goal is to determine which signal was sent base on the received signal $y[m]$. Therefore, we need to derive the optimal detection rule. 
+
+### Maximum likelihood detector
+
+A commonly-used principle of a detector is to maximize the likelihood, namely the conditional probability $\mathbf{P}(y[m] \vert x[m])$ of the received signal $y[m]$. Specifically, denoting the estimate of the transmit signal by $\hat{x}[m]$, the maximum likelihood detector is
+
+$$
+\hat{x}[m] = \left\{ 
+  \begin{aligned} 
+    &+a, && \text{if } \mathbf{P}(y[m] \vert x[m]=+a) \ge \mathbf{P}(y[m] \vert x[m]=-a), \\ 
+    &-a, && \text{if } \mathbf{P}(y[m] \vert x[m]=+a) < \mathbf{P}(y[m] \vert x[m]=-a).
+  \end{aligned} 
+\right.
+$$
+
+Since
+
+For any signal $s(t)$ with a bandwidth limited in $[-W/2, W/2]$, the sampling theorem asserts that we can sample it at a sampling frequency equal to its bandwidth $W$, namely at a sampling interval of $1/W$, and perfectly reconstruct it by
+
+\begin{align}
+  s(t) = \sum_{n} s[n] \text{sinc}(Wt-n),
+\end{align}
+where $s[n] = s(n/W)$ is the $n$-th sample, and $\text{sinc}(t)$ is defined as
+
+\begin{align}
+  \text{sinc}(t) \triangleq \frac{\sin(\pi t)}{\pi t}.
+\end{align}
+
+We can see that to perfectly reconstruct the continuous-time signal from its samples, we must sample it at a frequency at least as large as its bandwidth. This is intuitive. A signal with a larger bandwidth contains components at higher frequencies, which translate to faster variations in the time domain. Therefore, we need to sample it at smaller intervals to avoid missing any information.
+
+This is also part of the reason why we want to do signal processing in the baseband. The passband signal has the highest frequency up to $f_c+W/2$, requiring us to sample it at a sampling frequency of $2 f_c + W$, which is much higher than the sampling frequency of $W$ required for a baseband signal.
+
+### Modulation
+
+The sampling theorem also suggests how we should convert the discrete-time data stream $x[n]$ to the continuous-time baseband signal $x_b(t)$:
+
+\begin{align}
+  x_b(t) = \sum_{n} x[n] \text{sinc}(Wt-n)
+\end{align}
+
+In other words, we pass the data stream $x[n]$ through a filter whose impulse response is the sinc function. We can also say that we "modulate" the data stream by the sinc function. 
+
+In practice, we usually use other functions (e.g., raised cosine) to modulate the discrete-time signals.
+
+## Discrete-time baseband channel model
+Now that we know how to convert between the discrete-time signal and the continuous-time signal, we can derive an equivalent input/output model of the channel in the baseband and also in discrete time. As you may have guessed, the equivalent channel model is also a FIR filter, but in discrete time. 
+
+Using the [equivalent channel model in the baseband](reading-04-baseband-equivalent-model.html), we can write the baseband receive signal as
+
+$$
+\begin{align}
+  y_b(t)  & = \sum_i a_i^b(t) x_b(t - \tau_i(t)) \notag\\
+          & = \sum_i a_i^b(t) \sum_{n} x[n] \text{sinc}\left[W(t-\tau_i(t))-n\right] \notag\\
+          & = \sum_{n} x[n] \sum_i a_i^b(t) \text{sinc}\left(Wt-W\tau_i(t)-n\right). \notag
+\end{align}
+$$
+
+The $m$-th sample of the baseband receive signal $y_b(t)$ is then
+
+\begin{align}
+  y[m] = y_b(m/W) = \sum_{n} x[n] \sum_i a_i^b(m/W) \text{sinc}\left[m-W\tau_i(m/W)-n\right] \notag
+\end{align}
+
+Defining the delay $\ell \triangleq m-n$, we have
+
+\begin{align}
+  y[m] = \sum_{\ell} x[m-\ell] \sum_i a_i^b(m/W) \text{sinc}\left[\ell-\tau_i(m/W) W\right] \notag
+\end{align}
+
+In conclusion, we can define the **discrete-time baseband equivalent channel model** as
+\\[
+  h_\ell[m] = \sum_i a_i^b(m/W) \text{sinc}\left[\ell-W\tau_i(m/W)\right],
+\\]
+so that the discrete-time baseband receive signal can be written as
+$$
+  y[m] = \sum_{\ell} h_\ell[m] x[m-\ell].
+$$
+
+Therefore, the discrete-time baseband equivalent chanenl model is *also a complex-valued FIR filter!*

morea/05-detection-in-fading-channel/reading-05-noncoherent-detection.md

@@ -0,0 +1,83 @@
+---
+title: "Roadmap of this module"
+published: true
+morea_id: reading-05-noncoherent-detection
+morea_summary: "Overview"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 50
+---
+
+# The discrete-time baseband model
+
+The signal processing is not only done in the baseband, but also in discrete time. This is because the digital chips ultimately work in the discrete time. However, the signal transmitted in the air is continuous-time. 
+
+How do we convert the discrete-time signal in devices to a continuous-time signal ready for transmission, and how we convert the continuous-time signal received to a discrete-time signal for processing in the receiver?
+
+## Conversion between discrete time and continuous time
+
+### The sampling theorem
+
+For any signal $s(t)$ with a bandwidth limited in $[-W/2, W/2]$, the sampling theorem asserts that we can sample it at a sampling frequency equal to its bandwidth $W$, namely at a sampling interval of $1/W$, and perfectly reconstruct it by
+
+\begin{align}
+  s(t) = \sum_{n} s[n] \text{sinc}(Wt-n),
+\end{align}
+where $s[n] = s(n/W)$ is the $n$-th sample, and $\text{sinc}(t)$ is defined as
+
+\begin{align}
+  \text{sinc}(t) \triangleq \frac{\sin(\pi t)}{\pi t}.
+\end{align}
+
+We can see that to perfectly reconstruct the continuous-time signal from its samples, we must sample it at a frequency at least as large as its bandwidth. This is intuitive. A signal with a larger bandwidth contains components at higher frequencies, which translate to faster variations in the time domain. Therefore, we need to sample it at smaller intervals to avoid missing any information.
+
+This is also part of the reason why we want to do signal processing in the baseband. The passband signal has the highest frequency up to $f_c+W/2$, requiring us to sample it at a sampling frequency of $2 f_c + W$, which is much higher than the sampling frequency of $W$ required for a baseband signal.
+
+### Modulation
+
+The sampling theorem also suggests how we should convert the discrete-time data stream $x[n]$ to the continuous-time baseband signal $x_b(t)$:
+
+\begin{align}
+  x_b(t) = \sum_{n} x[n] \text{sinc}(Wt-n)
+\end{align}
+
+In other words, we pass the data stream $x[n]$ through a filter whose impulse response is the sinc function. We can also say that we "modulate" the data stream by the sinc function. 
+
+In practice, we usually use other functions (e.g., raised cosine) to modulate the discrete-time signals.
+
+## Discrete-time baseband channel model
+Now that we know how to convert between the discrete-time signal and the continuous-time signal, we can derive an equivalent input/output model of the channel in the baseband and also in discrete time. As you may have guessed, the equivalent channel model is also a FIR filter, but in discrete time. 
+
+Using the [equivalent channel model in the baseband](reading-04-baseband-equivalent-model.html), we can write the baseband receive signal as
+
+$$
+\begin{align}
+  y_b(t)  & = \sum_i a_i^b(t) x_b(t - \tau_i(t)) \notag\\
+          & = \sum_i a_i^b(t) \sum_{n} x[n] \text{sinc}\left[W(t-\tau_i(t))-n\right] \notag\\
+          & = \sum_{n} x[n] \sum_i a_i^b(t) \text{sinc}\left(Wt-W\tau_i(t)-n\right). \notag
+\end{align}
+$$
+
+The $m$-th sample of the baseband receive signal $y_b(t)$ is then
+
+\begin{align}
+  y[m] = y_b(m/W) = \sum_{n} x[n] \sum_i a_i^b(m/W) \text{sinc}\left[m-W\tau_i(m/W)-n\right] \notag
+\end{align}
+
+Defining the delay $\ell \triangleq m-n$, we have
+
+\begin{align}
+  y[m] = \sum_{\ell} x[m-\ell] \sum_i a_i^b(m/W) \text{sinc}\left[\ell-\tau_i(m/W) W\right] \notag
+\end{align}
+
+In conclusion, we can define the **discrete-time baseband equivalent channel model** as
+\\[
+  h_\ell[m] = \sum_i a_i^b(m/W) \text{sinc}\left[\ell-W\tau_i(m/W)\right],
+\\]
+so that the discrete-time baseband receive signal can be written as
+$$
+  y[m] = \sum_{\ell} h_\ell[m] x[m-\ell].
+$$
+
+Therefore, the discrete-time baseband equivalent chanenl model is *also a complex-valued FIR filter!*

morea/05-detection-in-fading-channel/reading-05-roadmap.md

@@ -0,0 +1,22 @@
+---
+title: "Roadmap of this module"
+published: true
+morea_id: reading-05-roadmap
+morea_summary: "Overview"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 50
+---
+
+<div class="alert alert-success" role="alert" markdown="1">
+<i class="fa-solid fa-book fa-xl"></i> **Reading in the textbook**
+<hr/>
+
+This module covers Section 3.1 of [the textbook](https://web.stanford.edu/~dntse/papers/book121004.pdf).
+</div>
+
+
+In this module, we analyze the performance of communications systems in wireless fading channels. 
+
+Through the performance analysis, we will see how fading significantly degrades the performance (in terms of the error rate). This challenges call for special techniques to combat fading in wireless channels.