Why digital audio isn't stair stepped

Because digital audio takes small snapshots of the incoming audio signal, why is the analog output smooth and not stair-stepped?

[@JoeStuffzAlt]

Fantastic explanation. Many audiophiles say THERE IS NO STAIR STEP but never can explain how the discreet audio samples get converted to a smooth waveform. You try to press them on how it happens, and they act like it's magic. You explained it beautifully.

[@CORVUSMAXYMUS]

PAUL CONGRATULATION. You learn me about LOUDNESS .That option is a MUST.Thank you very much.

[@blainemunro7520]

I’m not confused after 40 years of Hifi. Thank you Paul

[@michaelbuxton8947]

Lovely explanation to something that puzzled many of us. Thanks Paul.

[@BruceCross]

Most audiophiles don't understand digital audio (me included). You don't have to fully understand it to enjoy it.

[@nancy4don]

When I have explained this to friends (and customers when I was selling), I've put it this way. Yes, the samples look like stairsteps, but that's only part of the process. The rest of the process knows how to fill in the "blanks" of the space between the steps. The result is an output waveform that is EXACTLY like the input waveform - at least as far as audibility is concerned. If you drill down deep enough, you can see infinitesimal differences orders of magnitude below anything you can hear. But that's the key: you can't hear them. Other things like the analog output stage of the DAC will be the things that affect sound, not the digital process itself.

[@spentron1]

I really got it when I understood the almost equal importance of the input filter. You avoid reconstruction errors by not sampling anything it can't reconstruct.

[@jimw5165]

Ah, so beautifully explained. At last I understand why I need Tana leaves (filters) to properly unlock the sleeping digital information. It would be fun to see the frequency profiles of the various filters in PS Audio DAC’s.

[@danab7472]

I think you do a fine job explaining Paul. I learn a lot from your channel and it’s helped my system sound its best.

[@RandySmith-iz1ml]

Thanks for reducing the confusion Paul!

[@al5152001]

Thank you! Professor Paul😊👍

[@handsomehal142]

One thing that's interesting to note is those higher frequencies that get filtered off are just high frequency images or mirrors of the baseband signal So you could also recover the signal from the supersonic high pass as it would just alias back as the original signal

[@handsomehal142]

@nicksterj had never heard of the concept before but to a first approximation it sounds kinda smart lol "The images are all ultrasonic so just the ear be the filter" Until intermodulation gets a hold of it, I suspect

[@MarcoRistuccia]

To complete the explanation, there was a guy called Nyquist who proved that if you take snapshots very fast, when you apply the output smoothing (low pass) filter the result exactly matches the original analog signal. (still keeping this simple)

[@paulstubbs7678]

I would like to add that those hight frequency spikes Paul is refering to are way above the human hearing range, so inaudible to you, so removing them should have no impact in what you hear. If anything that high frequency noise has the potential to upset your amplifier etc., so we are mostly better off without them.

[@Fastvoice]

Although these high frequencies are indeed too high for the human ear, they do cause aliasing. And that can be audible.

[@glenncurry3041]

20CPS- 20KCPS (Hertz was assigned many years later) was established way before we had sound systems capable of being even close to flat at that bandwidth! It just stuck as a number range with no updates as technology progressed. But other areas of research overlap. The human auditory system is the fastest responding connection to our brain. Evolutionary to save our lives from noises in bushes around us. We can detect pulses arriving at the ears with difference over 10-20 microseconds. IOW our hearing responds to signals over 100Khz. (Microsecond temporal resolution in monaural hearing without spectral cues) (PubMed 2003) And as @Fastvoice states, even at higher sample rates PCM still produces many artifacts in thee audible range.

[@morbidmanmusic]

Everything changes everything. Even though we can't hear the spikes range, it would still effect some aspect of the signal we hear.

[@geocarey]

Hi Paul, do you think a white board would be handy when you explain things such as ADC and DAC conversion and 'filtering'? I taught physics for many years, and sometimes waving my hands in the air was quite sufficient, but some topics really need a drawing. Not a criticism, I love the videos. Just a thought.

[@davidrippy1605]

Great explanation Paul.🎶

[@onepieceatatime]

It's maybe important to note that the sharp edged "stair step" isn't in the actual data. No sample was taken to get those sharp edges, and they're not a proper representation of the signal in any way. The samples are points, not stairs.

[@Jorge-Fernandez-Lopez]

Exactly. There's no information, line or step between points. There's great information out there. It's even possible to prove and test it at home. Unfortunately, even qobuz draw these misleading graphs (why?).

[@glenncurry3041]

So you have no clue as to how samples are created? YES the ADC accumulates voltage data during a sample duration that represents a flat step for that sample. The entire time period for that sample is one voltage for the entire duration. One STEP! Then the next accumulated STEP is sampled. Each sample starts with a sharp response when that gate is opened and a sharp edge when closed. A sample as a single point in time would require an infinite bandwidth. You've just been coned by the lollipop BS companies like TI through around!

[@davidstevens7809]

Yes but. It must be filterer.

[@fernandomoraledasamso750]

@@glenncurry3041 During the sampling of an analog signal, after passing through an appropriate antialiasing filter, an action called opening-retention-closing is performed. During the time that the signal is retained, its numerical evaluation is performed and it is quantified. The quantified value is at that moment what it was at the exact and specific moment of the opening, a point in time that has only lasted as long as necessary for its quantification. A point without a temporal dimension, but with an amplitude value, is what is stored, the temporal dimension is incorporated into the sampling rate and is somehow independent of the quantization.

[@onepieceatatime]

@@glenncurry3041 I'm trying to say that no instantaneous sample is taken at the sharp edge of the stair step, so that sharp edge is not really what the data says. The value corresponds to a *point at the end of the interval*. It's not a stair, it's a point. Meaning that sharp edge - if you draw a straight line down to its corresponding time on the X axis - wasn't taken at that level at that moment. So, presenting it as a continuous "step" across an entire interval is just wrong. It's like the census. It might take a year to conduct the census, but the number presented at the end isn't necessarily the number of people alive at that moment. It's the number of people you counted during the amount of time it took you to count them. It's the same thing for pixel data on digital cameras. The "pixels" (in the data and in the math used to interpret the data) are actually infinitesimal points, not pixels of a particular size or shape. (Though in this case they do have a corresponding physical component on the sensor - even though there are gaps between them - this is in contrast to film.) Once the samples are taken (which of course takes time), their value anywhere except at that point or anywhere *during* that interval doesn't exist, and the data itself definitely doesn't have "sharp edges." In the case of digital images, we then do then present the image using a stair step method, but it only works if the image has enough pixels and a high enough bit depth. But it's still wrong to look at the edges and conclude anything about them - the sharp edges are meaningless. With DACs we can't get away with presenting the edges because there are too few samples (although with DSD there are so many samples it will work).

[@yurodivy1]

The only thing that struck me as misleading is the description of Music as a collection of sine waves. There are many examples in acoustic and electronic music of non-sine waves, fuzz guitar for example. I think our ears are used to hearing these non-sine waves and high resolution audio does a better job of recreating these. I believe this why hi-res audio is more life-like, the accurate reproduction of the variety of wave shapes, not the inclusion of super high frequencies beyond human hearing.

[@immovableobjectify]

Mathematically, any arbitrary waveform can be created by (or decomposed into) a summation of sine waves of varying frequency, amplitude and phase. Google Fourier analysis. The "sharp" corners on a square or sawtooth wave, are due to the presence of high frequency components (harmonics.). The particular mix of harmonics is why different instruments have their own unique timbres when playing the same fundamental note.

[@Elkemper]

Shout out to Darko!

[@dilshodtojiddinzoda]

That was the best explanation!

[@ThinkingBetter]

When a DAC low-pass filter is filtering out frequencies above 20kHz (can be higher), it will make a 20kHz square wave PCM become like a 20kHz sine wave. A 10kHz square wave will also look like a 10kHz sine wave on the DAC output. So how about a 5kHz square wave? It will look like a 5kHz sine wave overlapped with a 15kHz sine wave. Reason is that a 5kHz square wave is having the base frequency of 5kHz and the 3rd harmonics of 15kHz still falling within the low pass filter bandwidth while the higher harmonics are above 20kHz. Any square wave signal combines the base frequency with odd harmonics (x3, x5, x7 etc.) as a sum of different sine waves. The 10kHz square wave has its 3rd harmonics at 30kHz above the filter frequency so you can never hear the difference between a 10kHz sine wave and a 10kHz square wave. If you go further down in frequency, you can start to see a square wave in PCM being appearing like a square wave on the analog output. A 100Hz square wave will retain many of the higher frequency harmonics and the analog representation of it on the DAC output will look much like an actual square wave. Our ears have a similar low pass filtering and that's why you can hear a big difference between a 100Hz sine wave and a 100Hz square wave.

[@PaoloCaminiti]

this is actually interesting, this topic will never settle

[@glenncurry3041]

He is talking electronic circuits. Not simple mathematics. Regardless of frequency sampled, each output of a DAC is a stair step of that sample rate. One voltage out during the entirety of that clock sample. An RtoR is a specific voltage for the duration of the sample while a Sigma Delta is just ON yes or no for the duration of the sample. You are discussing the loss of information that makes Digital inferior to Analog.

[@ThinkingBetter]

@@glenncurry3041 No, I’m talking about that all sounds in nature are fundamentally constructed of sine waves and when you filter out above 20kHz your wave shapes will not be PCM steps but a sum of lower than 20kHz sine waves. More complex sounds simply contain more oscillations at different frequencies, stacked one upon another. Higher-frequency, oscillations which are tonally related to the fundamental frequency (the base note or tone) are known as harmonics.

[@glenncurry3041]

@@ThinkingBetter Actually no. Few sounds in nature are pure sine waves even with addition of other sine waves. They are complex nonrepeating waveforms. FTs might break out sine wave frequencies when converting from the time domain for simplicity of evaluation. But tossing any number of steady state sine wave generators together will not sound like a piano. We grasped at sine waves for audio analysis because back in the day even generating testing stead state sine waves was difficult. Forget time domain stuff! I've travelled giving clinics on waveform analyses of amps. Slew rates, rise time, FFTs, ...

[@ThinkingBetter]

@@glenncurry3041 Music or any sound is made of (or can be modeled as) a combination of sine waves that come and go with different timing and levels. There is no detail of a sound wave that escapes that logic and not being pure just means being complex composition of sine waves mathematically speaking. Obviously any wave is not constant for music. From Fourier analysis you can recall that all electrical or acoustical signals can be represented as the sum of one or more sine waves. Amps creating transient intermodulation distortion due to slow slew rate just means higher frequencies of a wave form end up being amplified more like open-loop amplification. It doesn’t defy the general theorem of audio signals being modeled as a composition of sine waves. And yes, I think you want to say that real musical instruments are making more complex waves than what comes out of a synthesizer trying to make piano sounds from tone generators. I can agree with that, but it doesn’t change what I said.

[@CORVUSMAXYMUS]

Im proud that Im not in that range of 80 per cent area of confuse audio client

[@paulgaerisch]

The more “steps” or samples the higher and accurate the sound will be. This is called resolution. Hence why in PCM many studios use higher sampling rates then the standard CD sampling rate of 44,100 samples per second!

[@Think_Up]

Bottom line is that well executed digital is at such a high resolution that the digital nature of the "stepping" is not audible. *But technically, if you're removing something that was there to smooth it out, you are removing data that was created as a representation of analog sound. So you have something missing, even if it's inaudible. What is audible are things like distortion, jitter, clipping and things that come along with poorly executed digital. But let's not forget - analog has it's issues as well. No format is perfect and neither are your ears. So pick your poison and enjoy the music because there is great music in both digital and analog. PSAudio is definitely well executed digital or analog and Paul is a legend.

[@mabehall7667]

Bottom line is: in this day and age, conversion of an audio analog signal to digital and back to analog, can, and is, done perfectly. Within the desired frequency range, regardless of what it is, no one, no equipment, can determine a difference in the original signal and the digital to analog converted signal. This does not mean someone cannot, in the analog to digital conversion, screw up a recording be it an original recording or remaster of an old analog tape. As Paul says, it’s complicated-but not really to the engineers working with it daily. It’s science and engineering, not brain surgery. Many years ago an appendectomy was considered state of the art human surgery-it was cutting edge-pardon the pun. That’s where we are today with digital audio-it’s just not that big of a deal despite the fact equipment manufacturers are still trying to sell $10,000 DACs.

[@Think_Up]

@@mabehall7667 I generally agree with you, but have learned first hand that amps and dacs sound different, so it's not an automatic given that it's done well. I'm not a snake oil believer and quite skeptical with most audio claims, but some are valid. I tried a well respected class D amp and it was annoyingly harsh with my titanium dome tweeters I had at the time. A nice class A/AB amp was a night and day improvement. I tried some DACs and most are fine, but some were amazing - A ladder DAC I tried had a sound that was very pleasing to listen to vs another which many consider high end, which was shrill and artificially bright. It's nice to have choices.

[@PSA78]

"FL Studio" here on KZbin have video showing visually and explaining quite well. It's a bit much to do just verbally in 5 minutes like Paul do, though I think he's doing a great job. 😄 Edit: look for videos with Monty Montgomery, might be on other channels as well, like "Audio University".

[@philiptong4978]

kzbin.info/www/bejne/mXWainmLjrGjesU

[@glenncurry3041]

I believe it was Audio University that uses that bogus lollipop crap to hide the stair steps. And he refused to show any actual images of that lollipop existing at any point anywhere in any circuit! What a waste of time that vid was!

[@PSA78]

@@glenncurry3041 Watch videos with Monty.

[@philiptong4978]

voltages measured (sampled) at a certain time happen at a particular instant, imagine a column of voltage values stepped by time, the raw output is shown as dots in x (voltage) y (time) axis without lines connecting them Horizontal lines would imply the signal voltages stays the same before next measurement took place The lollipop vertical lines merely visually indicate the magnitude of the signal voltage at an instant when measurement happened, again not to confuse the vertical lines with the wave form being re-created The dots (voltages at a certain time) goes through another step to re-create the analog signal

[@glenncurry3041]

@@philiptong4978 Please show us all the physic behind voltage values than can change instantly to one point and then return instantly back to zero which is what you describe. Horizontal lines prove that voltage changes happen over a PERIOD of time. Not an INSTANT. It violates every known law of physics to claim anything changes in less than one instance of time. It is known as Planck time. Given the voltage range and bandwidth of any specific circuit we can calculate the maximum rate of change it can provide from one voltage point to another and it is NEVER in an instant! And in a properly performing RtoR DAC it will put out X voltage as fast as it can and stay at that level, a horizontal line, until the next different sample.

[@user-kf7oz6nu4g]

For some reason it seems to me that people are confused by the word "digital"; if the more correct name PULSE CODE MODULATION were used, then for many people it would not be so difficult?

[@Fastvoice]

Not all digital audio is PCM. There are also other formats.

[@user-kf7oz6nu4g]

@@Fastvoice The name of any format that is called "digital" always contains two defining words: pulse and modulation. The modulation can be different, but it is the PULSE MODULATION that is most important.

[@Fastvoice]

@@user-kf7oz6nu4g Why do you adress that to me? I answered the thread starter that not all digital audio is *PCM*. Of course there's always some kind of pulse modulation, but e. g. for DSD it's not PCM but PDM (pulse-density) or to be more precise DSM (delta-sigma). And that's the format Octave Records uses.

[@user-kf7oz6nu4g]

@@Fastvoice It is difficult for me to discuss something with you since I am not a native English speaker and I have to use Google translator.

[@tan143danh]

Digital is math and Analogue is physical and that all you need really

[@thinkIndependent2024]

Sample Rate= RPM & Feet Per Inch all motors have a rotation speed. Analog simply means 1:1

[@vanhetgoor]

I was never worried that the conus of the speaker would exactly make stairway steps, concusses don't believe in that nonsense. Whatever what you do the speaker is always producing analogue sounds because that is what ears prefer.

[@rvin2105]

Paul, I am amazed how wrong y are, those voltages are sample points that are converted by a mathematical transformation in the original smooth analog curve by the dac, no jagies anyware, the filters are to eliminate the high frequency samples that are less that the at least double the sample rate than the frequency, 40k hz for 20k hz for human hearing, please correct. Marvin

[@JJ-no2ob]

Eh ? Whatever Paul says sounds- good but does it have to be so darn expensive?

[@davidstevens7809]

Problem is..multiple frequencies all at once and their interaction in the audio rhealm..yes.. harmonics..interact. next the filter changes the steps to look like analog.ITS NOT ANALOG..its a zerox copy..in time that represents ANALOG ANALOG

[@davidstevens7809]

@nicksterj uh..no they are coupled from stage to stage..some direct some with coupling cap..but actually a living analog sonic gizmo.. being so it has harmonic signatures from the interaction of the frequencies..thats whats missing in modern amps.

[@davidstevens7809]

@nicksterj well its more than sound pressure..its a union of energy. I understand that theres benefits to using a computer in the digital rhealm.

[@CORVUSMAXYMUS]

YES JUST LISTEN MUSIC , DAILY.

[@CORVUSMAXYMUS]

Please dont waste time by searching too much , close your eyes and LISTEN MUSIC

[@bernardodon7501]

MoFi, the "Most Fidelity" company in audio sold over ten years vinyl from digital master tapes and no audiophile heard it. Just enjoy music, or search for your own mystic Hotel California in the rabbit hole!

[@glenncurry3041]

So you actually ignored the reviewers of the time! NO! The reviewers did not miss it! While they expected pure analog and thus based their reporting on it, may actually used the terms of it sounding more digital and ir less analog. Many Michael Fremer vids says so.

[@intothevoid9831]

They absolutely did. Many people have complained about mofi releases from the last 10 years.

[@sidesup8286]

BUT Michael Fremer called the digital copied Mofi Abraxas lp, "Simply the best sound I ever heard." If they really did at times pick up on the sound resembling digital, they also were fooled by it at other times.

[@tubefreeeasy]

Can a tubed DAC create a smoother and naturally cleaner sine wave than a digitally processed filtered ‘analog’ signal?

[@KCarlWhite]

No, because the stairstep is only a visual reference for the audio engineers creating the audio. The stairstep really does not exist. All DACS uses complicated to create an exact copy of the original, and since it's math, it can be proven. Finally, with math, there is only one answer. Adding a tube will not make the answer better.

[@KCarlWhite]

Sorry, I meant to say complicated math...

[@PaoloCaminiti]

@@KCarlWhite what a random answer this is

[@lasskinn474]

no. the tube on tube dacs acts just a filter/preamp, basically as a gimmick. you could incorporate tube(s) into the dac circuitry and some crt but nobody really dos it that way, maybe it would make sense if you didn't have transistors at all.

[@mitchtaylor6512]

​@@lasskinn474there is actually a small manufacturer that has a tube dac that uses tubes in the up sampling circuit and the output stage.

[@Piglet6256]

i am confused since the day i was born 🤣

[@glenncurry3041]

THANK YOU Paul! The number of times I get into an argume... discussion with some digihead claiming it's some lollipop instead of stair steps because TI shows their Sigma Delta DAC that way for some reason. Show them actual scope images and they reject the very existence of it! So anxious to deny any possible artifacts from the reality of such shaping.

[@Heke97]

it actually is stair step :D --> "drawing a digital waveform as a stair step was wrong to beging with. a stair step is a continuous time function it's jagged and it's piecewise but it has the defined value at every point in time. sampled signal is entirely different. it's discrete time, it's only got a value right at each instantaneous sample point and it's undefined there, there is NO value at all everywhere in between... a discrete time signal is properly drawn as lollipop graph." more info in --> kzbin.info/www/bejne/mXq0anyOiLqtq68

[@fullranger3435]

A 20KHz signal, sampled at 40KHz rate will be represented simply by two values: Highest positive and lowest negative. So, digitally, it is a double triangle or a square meander of 50 microseconds periodic duration. How the heck is this turned into a SINEWAVE of 50 microseconds periodic duration? This is not explained!

[@maidsandmuses]

As an engineer myself, I agree that simply saying that they use a filter does not explain it well. Unfortunately, explaining _how_ an electronic low-pass filter works is difficult without going into the detail of the maths and physics involved. The required fundamental understanding here is that the 20kHz _triangle_ wave and 20kHz _square_ wave are actually made up of the _same_ 20kHz sine wave that they share, but with different higher frequency waves superimposed on top of those, so that the total adds up to either a 2kHz triangle or 20kHz square wave. What the filter does is let the fundamental 20kHz sine wave pass through, but filter out (think of it as cancelling out) all the higher frequency superimposed waves. That's what a low-pass filter does in essence; it allows the slow moving component to pass, but it blocks the faster moving components (idealised, a practical low pass filter attenuates the faster moving components rather than blocking them completely, but I wanted to keep it simple)

[@Jorge-Fernandez-Lopez]

It's not a triangle, nor a square wave. It's just two points and nothing in between. It's like the line defined by two points, no stairs in between. You will find a good explanation if you look for it. You can even test this at home with your own software and ears.

[@fullranger3435]

@@Jorge-Fernandez-Lopez yes and if you connect the points with lines, it's a saw's teeth pattern. Not a sinewave. So, who turns it into a sinewave? Thanks for the effort.

[@Jorge-Fernandez-Lopez]

@@fullranger3435 You can draw lines, but there's no correspondance with physics or mathematics. There's only two points, nothing in between. These lines you draw are representing wrong points that same voltage that reality. Instead, try to find the sine wave (you have to modify length of wave and amplitud) that cross these points: only one solution. An analogy would be a circle defined by center and radius, or defined by three points on the circumference. You can't draw a line and create a squared circle, because definition is precise, while interpretation is wrong. The solution would be to draw arcs, or square lines to find the center, and then the circles. The maths that define that wave and the real solution won't understand your lines, only the amplitude and distance of the sine wave that include these points.

[@Jorge-Fernandez-Lopez]

@@fullranger3435 You can do yourself the test at home, see that there's no lines anywhere, and listen the sound. Only close to Nyquist you will find some strange sounds. If you really want to test further, you will how and why that problem is solved and the sound will be perfect even at the limit. Stairs, or lines, are wrong and have nothing to do with reality.

[@rodm1949]

I am humbled by your knowledge, but in retrospect can you please listen to a 845 tube amplifier and depart your insight. Purely to open your neural pathways and provide insight.

[@gilgalaad80]

this time I have to disagree with Paul. the "stair step" representation of the sound is totally misleading. there is actually no stair step. this is just a way (one of the ways) to graphycally present the data is a comprehensible way, too bad that this was is misleading. and yes, even if you need a filter, that filter doesn't actually "smooths the steps". the real answer is: given a correct sampling rate that follows the Nyquist-Shannon theorem, there is only a single solution to the equation that matches 100% of the samples, and that solution is the original waveform. you don't "smooth the steps", you take out other solutions that would match the samples, creating aliasing. but I understand that sometimes, when you speak about science to people, and that is quite and advanced topic in signal theory, you have to (over) simplify things, even if you say something that is not technically 100% true. and I wanna thank Paul for what he does, this channel is amazing, even if we disagree sometimes! :)

[@Pete.across.the.street]

You don't have to, you choose too

[@cheekibreeki9515]

Yeah, except when you put in scope and zoom in for some million times you'll see it produce stair step.

[@gilgalaad80]

@@cheekibreeki9515 considering the sample rate of about 44 THOUSAND times a seconds, if the stair step occurs when you zoom in SOME MILLION times, it's obviously a limitation of the instrument. no instrument has infinite resolution. but hey! "science doesn't have all the answers", isn't it?

[@Projacked1]

80 procent still doesn't get it? Do people actually watch the stuff before they ask questions like that? Or is the attention span WAY too short these days? They invented 'shorts' for the people with the shortest attention spans, plus lazy content creators, so. I guess its logical.

[@jonfoss3437]

Bought an acom gafa 555. Wow huge step up. Screw digital

[@glenncurry3041]

555 or 555-II? The original 555 is a Nelson Pass design. It is what I am currently (pun intended?) using! Mine has an added 2nd power supply included 2nd toroidal transformer. Basically upgraded to full dual mono. Not even shared transformer. I'm thinking of upgrading it to two power cords in fact. Incredible amp! But I added active cooling fans to drive my Maggies in my way too large of a room!

[@762gunr]

That being said it will always lack the definition of a good analog signal.

[@Pete.across.the.street]

sounds made up