Assuming my DAC is 1 million samples per second 12-bit, with the goal of increasing the effective number of bits beyond 12 bit at the expense of reduced sample rate and bandwidth, why would I use a TPDF or noise shaping dither over a simple triangle or saw oscillator an order of magnitude above the corner frequency of the analog RC low pass reconstruction filter? I imagine this works like a PWM DAC with the amplitude of the LSB voltage. If it works for a 1-bit PWM DAC with RC low pass, why wouldn't this be the preferred method for increasing a few extra bits from a DAC that already exists in the micro-controller? This device is for audio playback DC to 16KHz. Ideally I could get it to sound as good as a high quality 14-bit DAC with 44.1KHz sample rate. According to the datasheet, settling time is 2uS min to max and VDD is 1.8v.

If yuor saw frequency is exact n times the sound frequency there is a "look up" so there will be no more bits.

Forgive me for not knowing the acronym TPDF.  Just adding dither does not "improve" the signal-to-noise ratio (SNR).  It just randomizes the quantization noise.  So yes, it may "sound better" but the objective SNR is not improved.

TPDF is Triangular Probability Density Function. The idea is to mix a random signal with peak to peak amplitude two times that of DAC LSB and probability density function that is triangular. Without noise shaping, the frequency content of the TPDF is assumed to be mostly white noise or some approximation of white noise.

My question is regarding a possible scenario where there is no randomness in the dither. Take for example a 1 bit digital output without dither and without PWM. The effective number of bits is 1. If you mix the digital audio signal with a digital SAW oscillator equal in amplitude that is above the corner frequency of the analog RC low pass filter, you create a DAC with the bit depth of MIN(sizeof(datatype),sizeof(datatype)) for the digital audio and the digital SAW oscillator.

Is it not reasonable to suggest that in this contrived example we have increased the effective number of bits of the DAC from 1 to 8 without using a random dither? A similar argument can be made for 1 bit delta sigma DAC with an analog reconstruction filter.

My understanding is that noise shaping pushes the noise up into frequency bands that will be attenuated by an RC low pass filter and thus improving signal to noise ratio by attenuating only noise.

So it that is true, why not use a periodic oscillator that will have greater than -48dB attenuation assuming 24dB octave analog RC low pass filter. Most audio equipment has filtering above 20KHz anyway. Every gain stage after this device provides additional attenuation of a 160KHz Saw oscillator. You can't even record that signal at 44.1KHz.

You can also do a thought experiment about TPDF processed with noise shaping that has band pass filter with extremely high Q factor and center frequency at 160KHz. How are they the same and how are they different?