Random variables section (#137)

* Random variables (#136)

* "Add real-time packet detection section to detection chapter"

* Add full E2E QPSK simulation to Sync chapter

* Add Random Variables chapter and improve Detection flowchart

---------

Co-authored-by: Marc Lichtman <marcll@vt.edu>

* cleanups

* cleanups

* cleanups

* asd

* asd

* asd

---------

Co-authored-by: Vishwaksen Reddy Dhareddy <73452224+vishwaksen-1@users.noreply.github.com>

Author: 777arc

_images/central_limit_theorem.svg

@@ -22,6 +22,7 @@
     'sphinx.ext.mathjax',
     'sphinx.ext.autosectionlabel',
     #'sphinxcontrib.tikz', #added for dutch
+    "sphinxcontrib.mermaid"
 ]
 
 mathjax_path = "mathjax/tex-mml-chtml.js" # so that the textbook can work offline

_images/detection_realtime.png

@@ -436,40 +436,16 @@ Implementation
 
 Our detector will follow this workflow:
 
-.. code-block:: text
-
-    ┌─────────────────────────────────────────────────────────────┐
-    │  Continuous IQ Stream from SDR (e.g., 1 MHz sample rate)    │
-    └────────────────────┬────────────────────────────────────────┘
-                         │
-                         ▼
-    ┌─────────────────────────────────────────────────────────────┐
-    │  Buffer Accumulation (e.g., 100k samples = 0.1 sec)         │
-    └────────────────────┬────────────────────────────────────────┘
-                         │
-                         ▼
-    ┌─────────────────────────────────────────────────────────────┐
-    │  Cross-Correlation with Known Preamble                      │
-    │  → Produces correlation vs. sample index                    │
-    └────────────────────┬────────────────────────────────────────┘
-                         │
-                         ▼
-    ┌─────────────────────────────────────────────────────────────┐
-    │  CFAR Threshold Computation                                 │
-    │  → Adaptive threshold that tracks noise floor               │
-    └────────────────────┬────────────────────────────────────────┘
-                         │
-                         ▼
-    ┌─────────────────────────────────────────────────────────────┐
-    │  Peak Detection (correlation > threshold)                   │
-    │  → List of candidate packet start indices                   │
-    └────────────────────┬────────────────────────────────────────┘
-                         │
-                         ▼
-    ┌─────────────────────────────────────────────────────────────┐
-    │  Packet Extraction & Validation                             │
-    │  → Extract samples, pass to demodulator                     │
-    └─────────────────────────────────────────────────────────────┘
+.. mermaid::
+
+ flowchart TD
+    A("Continuous IQ Stream from SDR<br/>(1 MHz sample rate)")
+    B("Buffer Accumulation<br/>(100k samples = 0.1 sec)")
+    C("Cross-Correlation with Known Preamble")
+    D("CFAR Threshold Computation")
+    E("Peak Detection<br/>(correlation > threshold)")
+    F("Packet Extraction & Validation")
+    A --> B --> C --> D --> E --> F
 
 To avoid missing packets that straddle buffer boundaries, we use an **overlap-save** approach, where each buffer includes the last ``N_preamble`` samples from the previous buffer.  This ensures any packet starting near the end of buffer ``i`` will be fully contained in buffer ``i+1``.  This requires a small additional computational overhead but we don't want to miss packets just because they straddle buffer boundaries.
 
@@ -825,7 +801,6 @@ Step 7: Visualize Results
     axes[2].legend()
     
     plt.tight_layout()
-    plt.savefig('../_images/detection_realtime.svg', bbox_inches='tight')
     plt.show()
 
 The visualization should show:
@@ -834,9 +809,9 @@ The visualization should show:
 2. **Middle plot**: Correlation output with adaptive CFAR threshold tracking the noise floor
 3. **Bottom plot**: Detected packets highlighted as peaks above threshold
 
-.. image:: ../_images/detection_realtime.svg
-   :align: center 
-   :target: ../_images/detection_realtime.svg
+.. image:: ../_images/detection_realtime.png
+   :align: center
+   :scale: 50 % 
    :alt: Real-time packet detection results
 
 Practical Considerations and Tuning