Merge pull request #398 from jclsn/traktor-mk2-blogpost

Add a Traktor MK2 blog post

Author: Swiftb0y

content/images/avatars/jclsn.png

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+title: "How Does Timecode Vinyl Actually Work? (Pt. 3)"
+authors: Jan Claußen
+tags: traktor, timecode, dvs, vinyl control
+status: draft
+math: yes
+
+Since its release in 2011, the **Traktor Control Vinyl MK2** has sparked
+curiosity among digital DJs and audio developers alike. Its timecode format
+stands apart from Serato’s, which we explored in the previous posts. With the
+MK2 system, Native Instruments introduced a more advanced timecode that boosts
+resolution and accuracy by applying advanced cryptographic techniques.
+
+In this post, we’ll break down how it works at a basic level and how Mixxx is processing the signal.
+
+---
+
+## Recap: How Serato Timecode Works
+
+Serato’s timecode is built around a
+[Linear Feedback Shift Register](https://en.wikipedia.org/wiki/Linear-feedback_shift_register),
+modulated onto a 1 kHz carrier using
+[amplitude modulation (AM)](https://en.wikipedia.org/wiki/Amplitude_modulation) - a legacy
+technique from radio transmission.
+
+The demodulation process is relatively simple: when one stereo channel crosses
+the x-axis, the other hits a peak. If that peak exceeds a certain threshold,
+the system reads it as a **1**, if not, it’s a **0**.
+
+![Serato Timecode Signal]({static}/images/news/serato-control-cd.svg)
+
+We covered this in more detail in
+[DVS Internals Pt. 1]({filename}/news/2021-11-21-dvs-internals-pt1.md)
+and [Pt. 2]({filename}/news/2021-12-22-dvs-internals-pt2.md).
+
+---
+
+## The Traktor MK2 Signal
+
+Below is a signal that resembles what you’ll find on the Traktor MK2
+Control vinyl/CD, which has been specifically generated for this blog post by
+using a [Raised-Cosine Filter](https://de.wikipedia.org/wiki/Raised-Cosine-Filter) to
+modulate a random sequence onto the carrier.
+
+The carrier wave operates at **2500 Hz**, a significant increase from Serato’s
+**1000 Hz**.
+
+> **Advantage:** The higher carrier frequency allows for 2.5× greater resolution.
+
+![Offset-modulated Signal]({static}/images/news/traktor-mk2-signal.svg)
+
+Upon inspection, this waveform clearly doesn’t use amplitude modulation - the
+amplitude remains constant. Instead, it appears to be **offset-modulated**,
+where the signal is shifted vertically from the x-axis. This is a non-standard
+technique not commonly used in typical modulation schemes.
+
+On the original vinyl version (not shown here due to copyright), the offset
+can be so large that the signal floats entirely above the x-axis for multiple
+cycles - making zero-crossing detection impossible.
+
+Even when that doesn’t happen, the offset causes the time interval $\Delta t$
+between zero-crossings to become irregular, introducing audible pitch flutter.
+
+To decode the signal, we must solve:
+
+1. How to filter the signal to enable pitch detection
+2. How to demodulate this non-standard modulation
+3. How to decipher the code that is modulated onto the carrier
+
+---
+
+## Pitch Detection
+
+> **Note:** If you're unfamiliar with pitch detection in DVS systems, revisit
+> [DVS Internals Pt. 1]({filename}/news/2021-11-21-dvs-internals-pt1.md).
+
+Our goal is to produce a signal that oscillates evenly around the x-axis. This
+filtered waveform can then be processed by the existing pitch detection algorithm.
+
+A simple discrete derivative operation achieves this:
+
+$$
+y[n] = x[n] - x[n-1] \tag{1}
+$$
+
+$\text{where:}$<br>
+$\text{- x[n]: Input sample}$<br>
+$\text{- x[n-1]: Delayed input sample}$<br>
+$\text{- y[n]: Difference of both values}$<br>
+<br>
+
+When applied to the offset-modulated signal, we get:
+
+![Traktor MK2 signal with derivative]({static}/images/news/traktor-mk2-signal-with-derivative.svg)
+
+The resulting waveform oscillates cleanly around zero, which is ideal for
+analysis. It also makes it easier to pinpoint the half-cycle peaks needed for
+bit detection.
+
+---
+
+## Demodulation Techniques
+
+To extract bits from the signal, we detect the zero-crossings and sample the
+amplitude of the sinusoid at those moments.
+
+![Traktor MK2 signal with
+zero-crossings]({static}/images/news/traktor-mk2-signal-with-zero-crossings.svg)
+
+You may notice that the derivative’s zero-crossings don’t align perfectly with
+the original peaks. That’s due to a delay introduced by the filter. Smoothing
+the signal first, then compensating for the delay (e.g., by selecting
+$x[n-3]$), yields better results.
+
+For greater accuracy, one could analyze the phase response $\phi(\omega)$,
+which shows how filter delay varies with input frequency - but for this use case,
+a fixed delay works well enough.
+
+The filtered signal can cross the x-axis in two directions-positive to
+negative or vice versa. Based on the direction, we determine which half-cycle
+contains the encoded bit. In this example we only use the upper half-cycle for
+decoding, but you can also decode the signal from the lower half-cycle.
+
+![Traktor MK2 signal with readings]({static}/images/news/traktor-mk2-signal-with-readings.svg)
+
+Demodulation is then as simple as applying a threshold: amplitudes above it
+are **1**, and below it are **0**.
+
+![Traktor MK2 signal with timecodes]({static}/images/news/traktor-mk2-signal-with-timecodes.svg)
+
+On actual vinyl, the physical behavior of the needle causes the offset to
+decay over time, because the needle slowly drifts back to the middle. This decay complicates bit extraction.
+
+To compensate, we analyze the **slope** between subsequent readings by
+reusing the derivative equation in $\text{(1)}$.
+
+$$slope[n] = reading[n] - reading[n-1]$$
+
+$\text{where:}$<br>
+$\text{- x[n]: Current reading}$<br>
+$\text{- x[n-1]: Last reading}$<br>
+$\text{- y[n]: Difference of both values}$<br>
+<br>
+
+We then decode a positive slope to toggle the bit to **1** and a negative
+slope to toggle the bit to **0**.
+
+![Traktor MK2 signal with indicated slope]({static}/images/news/traktor-mk2-signal-with-slope.png)
+
+This method helps isolate the encoded signal from the floating zero line
+caused by mechanical drift.
+
+---
+
+## The Code
+
+> **Note:** A deeper explanation of LFSRs can be found in [DVS Internals Pt. 2]({filename}/news/2021-12-22-dvs-internals-pt2.md)
+
+Interestingly, the Traktor MK2 system also uses a [Linear Feedback Shift Register](https://en.wikipedia.org/wiki/Linear-feedback_shift_register) - but with different properties. While Serato’s LFSR has a
+**20-bit** length, Traktor’s uses a **110-bit** register with a minimum
+run length of two symbols. The generator polynomial of the the LFSR was found
+by using the [Berlekamp-Massey algorithm](https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm).
+
+The number of unique states an LFSR can generate is:
+
+$$n_{max} = 2^m -1$$
+
+Hence for the Serato timecode
+
+$$n_{serato} = 2^{20} -1 = 1\,048\,575$$
+
+and for the Traktor MK2 timecode
+
+$$n_{mk2} = 2^{110} -1 = 1.298 \cdot 10^{33} = 1\,298\,074\,214\,633\,706\,907\,132\,624\,082\,305\,023$$
+
+That’s an astronomically high number-far beyond what’s required for this
+application.
+
+But how many states are actually needed?  With a 2500 Hz carrier, you get 2500 bits per second.<br>
+
+For 12 minutes of timecode:
+
+$$12 \text{ min} \cdot 60  = 720 \text{ s}$$
+$$720 \text{ s} \cdot 2500 \text{ states/s} = 1\,800\,000 \text{ states}$$
+
+which exceeds the maximum state range of Serato’s 20-bit LFSR by far.
+
+However, a downside appears: each 110-bit state must be stored in 128 bits (4
+× 32-bit integers).
+
+So for the A-side with 12 minutes:
+
+$$1\,800\,000 \text{ states} \cdot 128 \text{ bit} = 230\,400\,000 \text{ bit} = 28\,800\,000 \text{ byte} = 27.46 \text{ MB}$$
+
+And for a 25-minute CD:
+
+$$4\,500\,000 \text{ states} \cdot 128 \text{ bit} = 576\,000\,000 \text{ bit} = 72\,000\,000 \text{ byte} = 68.66 \text{ MB}$$
+
+> **Disadvantage:** The memory footprint is large-even a single side of timecode can exceed 27 MB.
+
+This makes storing a full lookup table impractical in production software.
+
+It's important to point out that the current implementation is naive, because it treats the
+Traktor MK2 code as if it were Serato code. Since Mark Hills designed the xwax
+library, which is used by vinyl control in Mixxx, for exactly this style of
+timecode, changes would have to be made to make the decoder more modular.
+
+Nonetheless, the current technique works and it represents the current state
+of the decoder in Mixxx.
+
+---
+
+## Conclusion
+
+Fortunately, there are mathematical methods to reduce the memory requirements.
+This requires diving deeper into the crypthographic theory.
+
+First tests show that this can possibly be achieved by applying a fixed tap
+pattern (e.g. every 5th bit) to a 110-bit LFSR window - a form of structured
+decimation or undersampling. This collapses the sequence into a 22-bit
+[Gold code](https://en.wikipedia.org/wiki/Gold_code), whose two sequences alternate.
+The implementation of this technique is far more complex and not completed
+yet.
+
+We’ll explore those strategies in the next part of this series.

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