gHashTag · Oct 18, 2026 · Oct 18, 2026 · Oct 18, 2026 · Oct 18, 2026 · Oct 18, 2026
@@ -63,7 +63,7 @@ <h1>Arithmetic and DSP on the FPGA: from the carry chain to a filter</h1>
 <p class="goal">It picks up where the buses course ends. Every widget in it runs the fn bodies of a t27 spec compiled to WebAssembly: move a slider and the spec answers.</p>
 <p>Course 9, 27 lessons, a live widget and a t27 spec each: adders and carry chains, multipliers and the DSP48E1 slice, fixed point, CORDIC, sampling, FIR and CIC filters, the FFT, and a filter on the bench. Every number is computed by a compiled spec.</p>
 <p><a class="cta" href="/#/arithmetic-and-dsp">Open the interactive course →</a></p>
-<nav class="pager"><a class="chain" href="/learn/ternary-computing/">← Previous course: The ternary machine: from a trit to a running program</a><span></span></nav>
+<nav class="pager"><a class="chain" href="/learn/ternary-computing/">← Previous course: The ternary machine: from a trit to a running program</a><a class="chain" href="/learn/neural-networks/">Next course: Neural networks in t27: from one neuron to a transformer block →</a></nav>
 <section><h2>All lessons</h2><ul class="outline"><li><h3>Module 1 · Adding numbers</h3><p>Why addition is slow: the carry that walks up the word, the dedicated chain in every slice, prefix networks and carry-save trees.</p><ol><li><a href="/learn/the-carry-chain/">01 The carry chain</a></li><li><a href="/learn/prefix-adders/">02 Looking ahead: prefix adders</a></li><li><a href="/learn/carry-save/">03 Three numbers at once</a></li></ol></li><li><h3>Module 2 · Multiplying</h3><p>A product is a sum of shifted copies; Booth recoding halves them, and the DSP48E1 slice does the rest in one block.</p><ol><li><a href="/learn/shift-and-add/">04 Shift and add</a></li><li><a href="/learn/booths-trick/">05 Booth's trick</a></li><li><a href="/learn/the-dsp48e1-slice/">06 The DSP48E1 slice</a></li></ol></li><li><h3>Module 3 · Fixed point</h3><p>Where the binary point sits, what rounding does to a value and to its average, and what each bit of a quantizer buys.</p><ol><li><a href="/learn/q-formats/">07 Q formats</a></li><li><a href="/learn/rounding-and-saturation/">08 Rounding and saturation</a></li><li><a href="/learn/quantization-noise/">09 Quantization noise</a></li></ol></li><li><h3>Module 4 · Functions in hardware</h3><p>Sine, cosine, angle and length from shifts and adds, and when a table is the better answer.</p><ol><li><a href="/learn/rotating-by-shifts/">10 Rotating by shifts</a></li><li><a href="/learn/angle-and-magnitude/">11 Angle and magnitude</a></li><li><a href="/learn/tables-and-interpolation/">12 Tables and interpolation</a></li></ol></li><li><h3>Module 5 · Signals and sampling</h3><p>What sampling does to a frequency, how an oscillator is built from an adder, and what a DFT bin measures.</p><ol><li><a href="/learn/sampling-and-aliasing/">13 Sampling and aliasing</a></li><li><a href="/learn/the-nco/">14 A numerically controlled oscillator</a></li><li><a href="/learn/bins-of-a-spectrum/">15 Bins of a spectrum</a></li></ol></li><li><h3>Module 6 · FIR filters</h3><p>The moving average, a windowed-sinc design with integer taps, and folding the taps onto DSP slices.</p><ol><li><a href="/learn/the-moving-average/">16 The moving average</a></li><li><a href="/learn/designing-taps/">17 Designing taps</a></li><li><a href="/learn/folding-taps-onto-slices/">18 Folding taps onto slices</a></li></ol></li><li><h3>Module 7 · Multirate</h3><p>Lowering the sample rate without folding noise in: decimation, the CIC filter and the polyphase form.</p><ol><li><a href="/learn/lowering-the-rate/">19 Lowering the rate</a></li><li><a href="/learn/the-cic-filter/">20 The CIC filter</a></li><li><a href="/learn/polyphase/">21 Polyphase</a></li></ol></li><li><h3>Module 8 · The FFT</h3><p>N log N instead of N^2: butterflies, the bit-reversed input order and the bits each stage adds.</p><ol><li><a href="/learn/the-butterfly/">22 The butterfly</a></li><li><a href="/learn/twiddles-and-bit-reversal/">23 Twiddles and bit reversal</a></li><li><a href="/learn/growth-through-the-stages/">24 Growth through the stages</a></li></ol></li><li><h3>Module 9 · On the bench</h3><p>From the arithmetic to the board: a correlator from the modem spec, a budget of slices and timing, and the capstone filter.</p><ol><li><a href="/learn/a-correlator-finds-the-frame/">25 A correlator finds the frame</a></li><li><a href="/learn/does-it-fit-and-close-timing/">26 Does it fit and close timing?</a></li><li><a href="/learn/capstone-a-filter-on-the-board/">27 Capstone: a filter on the board</a></li></ol></li></ul></section>
 <footer><p><a href="/learn/arithmetic-and-dsp.t27">This course is itself a t27 spec: read it</a></p><p>Widgets keep their own English words: each one lives in its own t27 spec.</p></footer>
 </main>