add module 07

morea/07-antenna-diversity/assessment-CHANGE-ME.md

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+---
+title: "CHANGE ME"
+published: false
+morea_id: assessment-CHANGE-ME
+morea_summary: "CHANGE ME"
+morea_outcomes_assessed:
+ # - outcome-CHANGE-ME
+morea_type: assessment
+morea_start_date: "2021-07-16T09:00"
+morea_labels:
+---
+# CHANGE ME
+
+TBD

morea/07-antenna-diversity/module-07-antenna-diversity.md

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+---
+title: "Antenna Diversity"
+published: true
+morea_coming_soon: false
+morea_id: module-antenna-diversity
+morea_prerequisites:
+morea_outcomes:
+  - outcome-07-antenna-diversity
+morea_readings:
+  - reading-07-roadmap
+  - reading-07-receive-diversity
+  - reading-07-transmit-diversity
+  - reading-07-space-time-codes
+  # - reading-05-coherent-detection
+morea_experiences:
+  # - experience-07-time-diversity
+morea_assessments:
+  # - assessment-CHANGE-ME
+morea_type: module
+morea_icon_url: /morea/07-antenna-diversity/module-07-icon-space-time-codes.webp
+morea_start_date: "2024-02-21"
+morea_end_date: "2024-02-28"
+morea_labels:
+morea_sort_order: 7
+---
+
+We study how to achieve antenna diversity with multiple antennas.

morea/07-antenna-diversity/outcome-07-antenna-diversity.md

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+---
+title: "Time diversity"
+published: true
+morea_id: outcome-07-antenna-diversity
+morea_type: outcome
+morea_sort_order: 60
+---
+
+  * You understand what is antenna/spatial diversity.
+  * You understand space-time codes, and in particular, the Alamouti scheme.
+  * You understand the concept of degrees of freedom.
+  * You understand the tradeoff between diversity gain and spatial multiplexing in MIMO systems.

morea/07-antenna-diversity/reading-07-receive-diversity.md

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+---
+title: "Receive diversity"
+published: true
+morea_id: reading-07-receive-diversity
+morea_summary: "Receive diversity is similar to time diversity"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 71
+---
+
+# Receive diversity
+
+We start from the simplest case with one transmit antenna and multiple receive antenna. This is also called single-input-multiple-output (SIMO).
+
+The signal model can be written as
+
+$$
+y_\ell[m] = h_\ell[m] x[m] + w_\ell[m] \qquad \ell=1,\ldots,L,
+$$
+
+where $L$ is the number of receive antennas, $m$ is the index of the time slot, $h_\ell[m]$ is the channel gain from the transmit antenna to the $\ell$-th receive antenna, $x[m]$ is the transmit signal, $w_\ell[m] \sim \mathcal{CN}(0,N_0)$ is the additive Gaussian noise, and $y_\ell[m]$ is the receive signal at the $\ell$-th receive antenna.
+
+In this case, we essentially send multiple copies of the transmit signal to the receive antennas. The signal model is fundamentally the same as that in [repetition coding](../06-time-diversity/reading-06-performance-gain-time-diversity.html). 
+
+Therefore, assuming that the channel gains to different receive antennas are independent, we can achieve the same diversity gain of $L$ as in time diversity. Such a diversity gain can be realized by either coherent or noncoherent detection.

morea/07-antenna-diversity/reading-07-roadmap.md

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+---
+title: "Roadmap of this module"
+published: true
+morea_id: reading-07-roadmap
+morea_summary: "Overview"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 60
+---
+
+<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 3.3 of [the textbook](https://web.stanford.edu/~dntse/papers/book121004.pdf).
+</div>
+
+
+In this module, we explore antenna/spatial diversity in communication systems with multiple antennas. 
+
+We introduce space-time codes that achieve antenna diversity, and discuss the tradeoff between diversity gain and spatial multiplexing in multiple-input-multiple-output (MIMO) systems.

morea/07-antenna-diversity/reading-07-space-time-codes.md

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+---
+title: "Space-time codes"
+published: true
+morea_id: reading-07-space-time-codes
+morea_summary: "Utilize spatial and temporal dimensions"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 73
+---
+
+# Space-time codes
+
+In [the previous discussion](reading-07-transmit-diversity.html), we achieved the diversity gain with a data rate of one symbol per time slot by using rotation codes across multiple transmit antennas and multiple time slots. But with multiple antennas, our question is how can we further improve this scheme?
+
+## Alamouti scheme
+
+To illustrate the Alamouti scheme, we consider the case of two transmit antennas and single receive antenna.
+
+## General space-time codes
+

morea/07-antenna-diversity/reading-07-transmit-diversity.md

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+---
+title: "Transmit Diversity"
+published: true
+morea_id: reading-07-transmit-diversity
+morea_summary: "Need to be clever to achieve transmit diversity"
+# morea_url: https://github.com/airbnb/javascript
+morea_type: reading
+morea_labels:
+morea_sort_order: 72
+---
+
+# Transmit Diversity
+
+We have learned that [receive diversity is essentially time diversity](reading-07-receive-diversity.html). Specifically, by sending multiple copies of the transmit signal to $L$ receive antennas, we can achieve a diversity gain of $L$ when the channels to different receive antennas are independent. 
+
+Now consider the case of multiple transmit antennas and single receive antenna, namely multiple-input-single-output (MISO). Can we achieve the same diversity gain by copying the transmit signal to different transmit antennas?
+
+## Copy and paste does not work
+
+Suppose that in time slot $m$, we send $L$ copies of the transmit signal $x[m]$ to the $L$ transmit antennas. Then the receive signal $y[m]$ will be
+
+$$
+y[m] = h_1[m] x[m] + \cdots + h_{L}[m] x[m] + w[m] = \left( \sum_{\ell=1}^L h_\ell[m] \right) x[m] + w[m].
+$$
+
+Essentially, the transmit signal goes through a channel with channel gain $\sum_{\ell=1}^L h_\ell[m] \sim \mathcal{CN}(0, LN_0)$. It is still a Rayleigh fading channel, but with $L$ times the gain.
+
+Therefore, we do not get any diversity gain. The bit error rate (BER) will improve by the increased signal-to-noise ratio (SNR). But the BER versus SNR curve is simply shifted to the left, but has the same slope.
+
+This example shows that we need to carefully design the transmission scheme to obtain the transmit diversity.
+
+## Repetition code and rotation code
+
+The simple "copy and paste" scheme does not work, because the copies of the transmit signals are combined at the single receiver. This is different from the case of multiple receive antennas, where we get independent copies at the receiver. So the natural idea is to create indepedent copies at the receiver in the MISO case.
+
+Since there is only one receive antenna, we have to utilize the temporal dimension again. For example, we can transmit the signal $x$ at one transmit antenna in each time slot while keeping all the other transmit antennas idle. More specifically, we have
+
+$$
+y[m] = h_m[m] x + w[m] \qquad m=1,\ldots,L.
+$$
+
+It is clear that this scheme is equivalent to time diversity and will result in a diversity gain of $L$. However, since we are repeating the same signal $x$ in $L$ time slots, the data rate is reduced $L$ times.
+
+To achieve the same diversity gain without sacrificing the data rate, we can use the [rotation code](../06-time-diversity/reading-06-beyond-repetition-coding.html).
+
+However, we could have achieved the diversity gain of $L$ with one symbol per time slot using time diversity only. With multiple transmit antennas, we would like to achieve more.