This is a detailed review, measurements and comparison two portable DAC and headphone amplifiers: the Cypher Labs Theorem 720 and Fiio Q1 DAC. The Theorem 720 is on kind loan from a member. It is a circa 2013 product and quick look on third-party sellers on Amazon shows a retail price of $1,000. I own the Fiio Q1 and it retails for just $69 (plus $8 for shipping on Amazon). Obviously this is not an even match from price point of view but the Q1 was the only portable DAC+headphone amp I had that includes a battery like the Theorem.Physically, the Fiio Q1 is quite a bit more slick and pocketable than the Theorem 720:The Theorem 720 on the other hand looks like a hobby box. Definitely not for the pocket but portable nonetheless.The Fiio Q1 does have a few faults of its own. Its volume control has no markings. The switch operation and LED lights are not easily decipherable from the poorly written manual.Both products are plug-and-play from Windows driver point of view. I did have to install the ASIO drivers for the Q1 for some of my testing (see later) which was a disaster from the point of view of finding them. We are talking about dropbox links that don't let you download things to other drivers with Mandarin in file names and no instructions for help!No doubt you are interested in how they measure. So let's get into that. As usual, if you are not familiar with what these graphs mean, refer to my tutorial on understanding audio measurements: https://www.audiosciencereview.com/forum/index.php?threads/understanding-audio-measurements.2351/ Note: some of these measurements are made from the Audio Precision APx525 which is on kind loan from the company (you can see it in the above picture). You can tell them from the white background. I am still in the process of "porting over" my measurements to this new model. So the measurements may not be identical to same ones performed by my unit.For all the measurements I powered the units using USB from my computer. I did not attempt to measure them while operating on battery. I did do a quick test on Q1 and it seemed to have same performance either way.Let's start with test of jitter and noise:Both of these outputs are from line out, i.e. treating them as DACs. The Theorem has some random low frequency jitter that is causing the broadening of the shoulders on our 12 kHz tone. The Q1 is much cleaner in this regard but has higher absolute noise floor.Measuring THD+N using my analyzer, I saw an odd behavior in Theorem 720. Namely, even though the volume control does NOT change the level of line out, it serves to significantly degrade its performance if you turn up the level! You can see that in this output:Using minimum volume control, the Theorem 720 produces distortion and noise that is 83 dB down from reference signal. The Q1 outperforms it by 3 dB at -86 dB. But as seen there is a sharp drop to just -67 dB when I turned up the level to max on Theorem 720. This is very odd and not proper.Drilling into it using spectrum analysis the problem is very clear:We see massive increase in harmonic distortion at max volume. If you are going to use the Theorem 720 as a DAC, be sure to keep the volume control at minimum.Looking at the same test but using headphone output with the output of Theorem 720 matched to the lower output of Q1 we see:The Theorem 720's distortion is dominated in the low order second and third harmonics which is perceptually more masked than later distortions.Test of -90 dB sine wave shows this for each DAC:The Fiio Q1 shows a pretty reasonable facsimile of a sine wave. The Theorem 720 though is a disaster with noisy output and sudden spikes.Measuring the headphone output voltage and hence power (P = Voltage * Voltage/impedance) we get these sequence of graphs for Theorem 720:Sadly this is nothing to write home about. Best case distortion is 0.02%. There is decent output power though with onset of clipping around 2.6 volt for 300 ohm load.Here are the same results for Fiio Q1:Distortion is an order of magnitude lower at 0.0026% beating the manufacturer's spec (0.005%). Just as well varying the output impedance made no difference in distortion profile! The only limitation is an output voltage that is limited to just 2 volts.Let's look at intermodulation distortion which shows high frequency non-linearity better than THD:I have shown the headphone output of the RME ADI-2 Pro as reference. We see that the ADI-2 Pro easily beats both of these products and by good margin. The Fiio Q1 though, performs predictably and again with nice control of distortion at maximum level. As seen in other measurements the Theorem 720 starts to distort early and becomes highly distorted at max volume.Here are the output impedance measurements and how they compare to all others I have measured in the past:The Fiio Q1 falls in the excellent category (although higher than the 0.3 ohm spec). The Cypher Labs is in the good category. So no real worry for either one in this front as far as changing the frequency response of the headphone in use.I matched the levels on both DACs at a loud but still comfortable level using the Sony MDRV6 headphone. There was a tiny amount of high frequency exaggeration on Theorem 720 but this was pretty slight. In non-critical listening this would not observable.Unfortunately my AB switch causes ground loop noise and made the above comparison a bit more challenging. So I took the output of both DACs and fed them to my Stax SRM-007t headphone amp powering the Omega pro headphones. There, the Fiio Q1 was subtly cleaner. Again, this was a very small difference.For the third test, I connected the Sennheiser HD650 using the AB box and cranked the volume up as much as needed. On some tracks with low average levels, the Theorem 720 managed to produce noticeably more output resulting in better fidelity compared to max volume on Fiio Q1. On other tracks such as Pop/Rock, max level was not needed and the two again were neck and neck. In my scan of my library, 90% of the tracks could be played with the HD650 and Q1 below its max volume.The Cypher Labs Theorem 720 DAC measures like yet another "boutique" product. Anomalies such as headphone volume control changing its line out performance significantly, and poor low level linearity show that care was not used in the design of the unit. Its saving grace though is the higher headphone output which I needed 10% of the time with my library of music with the high-impedance and hence inefficient sennheiser HD-650. If I had both of these units for free, and the headphone I used was the HD-650, my vote would be for the Theorem 720. Otherwise there is little to hang your hat on at $1,000 retail price.The Fiio Q1 on the other hand measures like a well-engineered and well-behaved device at a bargain price of just $69+ shipping. It is slick with nice industrial design and doesn't suffer from any obvious warts. I am listening to it using the HifiMan HE-400i and it sounds really nice. Wish it had a tad more output and then it would be perfect.As always, questions, comments, corrections, etc. are all welcome.----If you like this review,for these types of hardware purchases using Patreon ( https://www.patreon.com/audiosciencereview ), or upgrading your membership here though Paypal ( https://audiosciencereview.com/foru...eview-and-measurements.2164/page-3#post-59054 ).