The Mathematics of Signal Processing | The z-transform, discrete signals, and more

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This video goes through an overview of what you would learn in a discrete time signals (or digital signal processing) course. Sampling, digital filters, the z-transform, and the applications of these are some of the things included. I had to over simplify a lot to barely keep this video under 30 minutes but hopefully it gives you an idea of how this type of math is used in the real world.
Pole/Zero plot and digital filter modeler used in video: www.micromodeler.com/dsp/
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[@vojtechstrnad1]

Don't be worried about making long videos. This video could be two hours long, and I would still watch it in one go.

[@alphasatari]

Vojtěch Strnad We*

[@sophiacristina]

Indeed, in fact, would be cool if he makes another episode of signal processing, since its the one i'm more interested...

[@alexanderquilty5705]

I watch 3 hour lectures that help me with the subject so I definitely don’t mind either.

[@Michael-vf2mw]

Haha, I take 2 hours to watch a 1/2hr video sometimes.

[@chuckkhubbard610]

Man, I was searching and searching, and so many videos just want to give "an overview", and then the people just wax poetic for 15 minutes about the subject, with no details. Thank you for jumping right in!! I hope to understand it better tomorrow after the alcohol wears off.

[@sagarjaju375]

Don't worry about the length. It's important that you keep the important points intact. And if the videos are longer than this, I would still watch.

[@Pupsi]

Audio rookie mistake 22:28. Using the filter wasn't the cause of distortion in your signal. What's actually causing your signal to distort is digital clipping that happens during the combination of your speech signal to that -0,0Db 1KHz signal 19:49. The majority of frequencies below 1KHz should be almost unaffected but above 1KHz they are more or less reduced to a square wave. The amplitude of a digital signal caps off at -0,0 Db beyond which no information can be stored and will result in distortion. Combining the two signals here would have required information to be stored above -0,0 Db amplitude. Try reducing both signals by 10-20Db and then combining them. You can also hear what the filter actually does by applying it to the pure signal of your speech.

[@roygalaasen]

Haha! I noticed the clipping too, but I assumed incorrectly that it wouldn’t matter, but of course it does! Edit: I was surprised about the fact you mentioned that it wouldn’t affect low frequency part of the signal, while the high frequency part would turn into a square wave like signal. It does make sense as the noise signal itself is the reason you do get clipping, and it only happens in the peaks of the noise signal, meaning frequencies that are higher than the signal that carries the speech signal into clipping.

[@davidjohnston4240]

Yep. "He's going to clip it" went through my head as he added a signal to a full amplitude sinusoid. And lo, clipping was heard.

[@zachstar]

Thanks for this comment! definitely should’ve mentioned the clipping.

[@LeoAr37]

Yeah I think the same thing happens with the two frequencies at 18:35. They sound like a square wave, when they should sound like 2 clean tones.

[@L0j1k]

Hay bro, your comment makes perfect sense. But your channel does not, and I would like to know what you are smoking so I can buy a pound.

[@clcsqueejy04]

As a radar engineer, this was a fun video to watch. Was really cool to see you tackle this topic, since it's pretty dang specialized, like you said.

[@DrDeuteron]

tornado radar?

[@majdavlog9966]

@@DrDeuteron ٠،

[@watcherofvideoswasteroftim5788]

I've watched this video multiple times at different points in my Signal Processing courses and every time I learn something new. Thank you Zach!

[@danielpetka446]

same, rewatching it the 3rd time, slowly it starts to click :D

[@AB-ol3fv]

This cleared the whole connection between z-transform and poles/zeros, and the link between circle and sinusoid. Much appreciated.

[@tammygoyal9334]

0:30 example 1:10 parabola, rational 4:10 moving average sample - cos 8:30 unit circle 10:00 vector z and y(z) 11:30 Manipulate algebra -average sample 12:50 algebra = pole and 0 to z 14:30 signals 0 average filtered out 15:30 fixing normalized frequencies 17:00 low pass filter - higher frequency filtered 17:45 frequency spectrum (what makes it up) spike = frequency, 19:00 audio 21:00 remove one tone 22:00 never ideal, removed original 23:00 using equation and notch filter 25:00 summary code poles change 26:30 filters change 27:00 applications

[@bolosantosi5398]

Thank you

[@HypnosisBear]

Great job!!!👍

[@johnyepthomi892]

This channel is a boon to humanity. Keep doing what you're doing.

[@zachstar]

Jeez I need to stop making these videos so long. Planned to make it shorter but there was so much I wanted to include and I still have some things I need to say lol. 1) I was really lazy with 'phase' in this video. I conveniently put the output samples in between the input points so that there was never any phase offset in the input or output functions. This just made the animations easier but no it doesn't work exactly like that. Also in discrete signals classes you usually see consider the x-axis to be just integer values representing the sample number (n=1,2,3, etc) but I kept time as the axis so I didn't have to change the variables from x to n. 2) Reconstruction is more complex than I was making it seem. 3) Discrete signals (that you saw in the this video) are not the same as digital signals. Digital signals are represented with just 1's and 0's (look up 'quantization' for more info on how continuous signals are made digital). 4) Towards the end with the complex numbers the animations weren't exactly done 'to scale' so just ignore the discrepancy between where some of the poles/zero's LOOK like they are vs what I label them as numerically (with the notch filter). 5) If you're wondering why I never crossed pi radians when sweeping the unit circle it's because after that something called 'aliasing' occurs. This is where you don't sample the signal fast enough and lose the information and cannot reconstruct the signal back (you must sample at least twice as fast as the highest frequency in the signal to avoid aliasing, aka a normalized frequency of .5). Edit: 6) As someone pointed out (and I can't believe I didn't mention this) the clipping that occurred when adding the tone to my audio signal is what really caused significant distortion when filtering and reconstructing the original audio signal.

[@501promo]

This video got me thinking outside the tesseract, and for that I thank you.

[@lucianociubotariu54]

Where can I learn more about this subject?

[@chinkeehaw9527]

I think you should make an animation for 5 because my friend didn't quite understand what 5 mean

[@regina1896]

I think this was perfect just the way it was. Don't stress .

[@morganandreason]

The lenght was perfect to me.

[@nonojak52]

How does this guy explains a whole semester-long course in a single video better than my professors did in 15 hours!?

[@chrism7574]

Because he doesn't prove anything he says.

[@nihargupte21]

That's the best ever explanation about Z transform I can ever get. Thanks a lot man, really made my day.

[@jaikumar848]

Thanks a lot Majorprep !! I requested this in previous video and you promised this... this video intuitively explain digital signal domain ..... specially Z- transform

[@SKARTHIKSELVAN]

I took several days to understand Z transform. You finished it in 30 min. Thanks.

[@arvindp551]

Sharing information is the best way to preserve it. Thank you so much!

[@gsp_admirador]

I wish I had professors like you at my college 😔

[@fast_and_curious9144]

Stop Blamin.. Start Learnin Gotcha!!?

[@govamurali2309]

Finally, I understand why Z transform is used. Thanks!!!

[@karthikmg3325]

Thank you sir im an engineering student n was struggling to understand these concepts of lapace n z transform u gave a clear intution thank u so much

[@tedsheridan8725]

Another cool video - really liking this channel. I did Mech. E years ago and even took some controls classes (continuous signals and Laplace transforms), but we never looked at digital filtering or z-transforms. I never even came across digital filtering until getting into music and exploring DSP.

[@mukuljobra3972]

It was really wonderful looking throughout the whole video, in every minute, you just simplified very in-depth topics, especially about what power the DSP is actually holding in the digital era. Make some more videos on similar stuffs like Laplace and Fourier transforms, the world must need to know what actually these two geniuses have given to us. Great 👏🏻👏🏻👍🏻👍🏻

[@JustinCao52]

Im taking DSP class rn, this video is really helpful with visualization. Very well constructed and easy to follow. Please keep up your work! Thank you

[@buh357]

I had never heard of signal processing, and after the first watch, I understand some parts; you did an excellent job here. Thanks, Zach.

[@victorgabrielmoreleduarte5999]

Please make more videos like these. It helps so much to get a grasp at real fields of engineering. Thank you very much! I just started a course in discrete time signal processing. With this whole Corona Virus thing though I don't know when I'll finish it but still I was finding it so cool.

[@rationalmind3567]

this topic is of my use i was waiting for it from you, Further videos on martingales, stochastic will be appreciated. Thanks for your efforts.

[@yoshuanguerrero1381]

Man, you really make me understand what i was trying to learn all this last 6 months. So much thank you and god bless you!!! Greetings from Mexico

[@realcygnus]

Great stuff as always ! Perhaps even the best yet.

[@Frazzy87]

Dude. Honestly. Really good video. Please keep making these.

[@roygalaasen]

This is just great! A wonderful addition to the already good videos on Fourier transforms that are out there. Adds a lot of value to the topic of (digital) signal processing. Edit: (do I smell an introduction to filtering?)

[@MosesMatsepane]

Yo! Dude you took me back to my DSP class in varsity. I haven't seen this stuff in years. :) I enjoyed Z-Transforms especially in Engineering Mathematics, the lady who was giving it was the best.

[@thewhizkid3937]

Great Video. Learned a lot. I enjoy watching most of your videos as well.

[@victorvanderdrift5006]

I study mechanical engineering at Delft University of Technology, we also get a course about signal analysis and processing which covers most of what you talked about. Your videos are very entertaining to watch and are a great bonus to my studies as well, keep it up!

[@danielpaul8734]

Wat een terrorvak zeg, had je hem in een keer gehaald?

[@fjolublar]

amazing introduction to digital signal processing. keep up the good work 👌. Loved the animationz

[@jamesbra4410]

These are quality videos, thank you! You should be an engineering teacher, you're much better than the ones at university.

[@erikchen8897]

Awesome video Zach, I am currently in my second year of Electrical engineering at university taking a signal processing class and your video has given me a high level view of DSP. The content makes a lot more sense now. I like your skits on the other channel but please keep making Engineering videos. Thanks Zach

[@deekshas3936]

Wow thank you! I understood what actually is the z-transform and why it is used. In class, we only get "here is the definition of z transform and here's how you solve the sums". It was great to see the intuition behind it. I'll admit I didn't understand it entirely but now I have more of an idea than I did before.

[@drinductor8150]

Makes whole video about electrical engineering... uses i instead of j. smh

[@zoltankurti]

That's because electrical engineers are confusing as hell. For some funny reason, j is also the current density vector, so why electrical engineers like j for the imaginary unit is a mystery to me.

[@drinductor8150]

@@zoltankurti True, but that is a capital J with an arrow above so not too confusing. Plus EEs don't really care about current density unless we're doing emag stuff :)

[@drinductor8150]

@@zoltankurti Yeah actually emag was one of my favorites. Right behind DSP and control theory.

[@pasqualeredo]

??? What? I'm an Electrical Engineer, and whenever I deal with math in real life and I'm dealing with ELECTRICAL engineers - or, in higher level classes that were EE electives, again... dealing with EE's, the imaginary operator (-1) is ALWAYS written as " j " . This is because "I" or "i" is used in throughout the electrical industry and professions to denote current flow(amps or amperes), and I assure you that current flow and "i" is written and used FAR more in everyday life by engineers, laymen, and engineers than the imaginary operator, negative 1 is. Using "i" as the imaginary operator is very confusing to electrical engineers. The only time I've ever seen it used in such a manner was in core engineering mathematics classes, such as calculus 2(I think), or physics classes. Even in those cases, the professors were quite aware of the distinction and confusion it causes and most times would accept use of i or j in homework.

[@pasqualeredo]

@@zoltankurti it's confusing(not really unless your a complete moron), BUT, I, or I is used by electrical engineers, electronic techs, and electricians to denote current or amps. So, if you're looking for the angle that current is leading or lagging the village waveform,(phase angle) when trying to determine power factor, or dealing with power quality issues where knowing the angle of the current waveform is necessary, it can become quite confusing... That's why!

[@MrCordlu]

I loved your presentation, please, keep posting more videos on this subject.

[@TopAhmed1]

Thank you so much. As an Electrical Engineer it was so refreshing and revision

[@diyguild1327]

There's a few times when you log in just to like a video and subscribe because the content is so good. This was one of those times.

[@jenishpatel9433]

I normally don't comment on a video but this video is different, this video cleared out a lot of doubts and will help me with my DSP classes. Thank you.

[@yorch877]

Your videos are really great, even for us graduates. Thank you!

[@ayushkumar-zr1lp]

I am an electrical engineering graduate and I love this In college we are taught the theory that makes things kind a boring but only thing that isn't boring is understanding the meaning of these things in reality Thank you so much for such a video.

[@anishsharma6702]

god , this is a gift to humanity such amazing explainations , i can only stand in awe

[@AQWraghd99]

Thanks for the video ! Started the course yesterday :)

[@prashantnook]

Omg thank you soo much we just started this chapter yesterday and our college lecturer was confusing the hell out of me Love from India

[@its_me_kirankumar]

Bro, what are you studying. And I'm also from india

[@devagyayadav7061]

Best 30 minutes of whole DSP lectures.

[@zeroxdan]

As a music producer studying computer science, you connected the dots perfectly haha. Thank you for this video! Now off to passing Signals and Systems!

[@roberthoward5852]

Excellent explanation! Thanks.

[@happyhayot]

Your videos are really great, thanks!

[@Nandaaditya92]

Thank you for sharing this wonderful video with us..

[@starsmasher568]

I watched this video when it first came out. Didn't realize it would be this useful in my junior year of college where we're studying signal processing.

[@Zxymr]

Great video! I wish I had this to refer to 5 years ago when I was learning DSP! I feel that the z-transform is a really rough place to start from, though. While it's true that the pole-zero plot is a good method to determine the frequency response of the system, its use mainly comes in when designing feedback (IIR) filters. However, it is a good topic to cover since very little people actually talk about it. Effectively, z is an eigenvalue of the forward time step operation (hence why ∃z: |z| > 1 leads to instability), and the z-transform is the inverse of the Laurent series. When you're just designing feedforward (FIR) filters like the moving average, simpler math like the inverse DTFT and convolution would be enough.

[@marinam3579]

Your videos are amazing. Thanks a lot !

[@user-ct2fz4go4x]

Thank's a lot Mr Zach

[@attila3028]

This was what is going on behind the maths of ee. I am so grateful

[@pankajchauhan3797]

Its just so amazing.

[@mutalasuragemohammed6954]

Your animations are splendid...Thumps up! 👍👍👍👍👍

[@Appl_Jax]

I wish it was explained to me this way when I took my classes. Much more intuitive this way than, "do it this way because the math works"

[@abdulhameedyerimaadamu1020]

Thanks for this video brother its wonderful keep it up and a wonderful channel👍👍👍

[@cbranalli]

magnificent. i'm starting to understand what went by me in a blur 50 years ago !

[@imadeddinerabia9541]

Bro what is this Thank you so much seeing your videos makes me go and study all the signal processing module

[@xxbongobazookaxx7170]

Watching Zach star videos to procrastinate and then the exact thing I was tryna procrastinate comes as recommended... Guess I have no excuse now

[@goldfishdoc1999]

Great video

[@antoinetteamal5839]

Awesome should team up with AhmadBazzi 🙌🏻

[@captainkite]

You really make education come to life.

[@HassanSelim0]

At Uni I picked a very interesting major, they call it Digital Media Engineering and Technology (DMET), but it's more like the Computer Science and Engineering (CSEN) major with a bit of media tech sprinkled on top. During our 10 semesters we only diverted significantly in the last 3 semesters. Both CSEN and DMET took a Signals course that was like an overview of continuous and discrete signals that I think ended with basic fourier analysis. We also took 2 Electric Circuits courses, first one being just DC circuits, and the second had AC madness and we applied both Fourier and Laplace in some messy stuff I really can't remember. However one of the big differences in the DMET major is the Digital Signal Processing course. It was a very interesting course for me because first of all there was no continuous signals making the math simpler (integrals become summations :D), and we really started to understand what Fourier does to a signal, we also studied the z-transform in that course and I remember something that have to do with damping (and growth which would lead to an "unstable" signal processing system), and we looked at the impulse response of different systems. DMET students also later had an Audio and Acoustics course but it didn't have any math, and we also took courses that explained how audio and video are digitally encoded (MPEG4, MP3, and all that). All that was mandatory, the elective courses weren't that much (we often took computer science electives instead), but one interesting elective was Digital Video Broadcasting where we studied different standards like DVB-T and DVB-S with their contellations, it was very interesting :D

[@zf164]

Great video. I’d like to see more on KZbin. When I took signals and systems the material was so abstract and seemed unrelated to the field of electronics until I took further courses

[@markuscwatson]

Great video.

[@brianhginc.2140]

Your distorted playback after filtering the 1khz tone is due to signal clipping. (IE, the sum of the 2 waveforms together exceeded the maximum and minimum values of 16bit audio's +/-32767 values) Lower the volume of the sine wave tone and your speech by 50% (-6db) BEFORE mixing the 2 audio samples together, then filter out your tone again. Your recovered audio will almost sound perfect as a really narrow removed slice of speech at 1Khz wouldn't do much.

[@djmips]

Yes this was apparent when he zoomed into the summed version and the characteristic chopped sinusoids were present. This brings in a lot of unwanted frequencies as we know these squared off sine waves are now more like square waves.

[@balajisriram6363]

man you are a genius!!!

[@philippezevenberg1332]

Me: "thinks of Fourier for some reason" Zach: "speaks about Fourier" Me: "Am... am I learning? o_o"

[@coreymiller2201]

Yep, computer engineers have to take DSP at The University of Akron. Gotta say, I really wish I had this video when I was taking signals, DSP, and embedded scientific computing. I kind of went through the motions treating the z-transform like a magical discrete laplace transform but I did not fully understand it. I love your videos-- keep em coming!

[@Dynamic_games_]

Thanks for this videos bro

[@RohitSingh-go9kc]

thanks .. that was awesome and motivating

[@ivod.2602]

This video is the reason I am becoming better friend with signal processing course at my uni.

[@Amine-gz7gq]

visualization is everything !

[@matthewolanrewaju2440]

This guy is good!

[@jlleluc4s]

Nice video to watch.

[@phonezawthein1990]

thank u very much for this video.

[@joshuasizer1709]

This is awesome

[@LieutenantAwesom3]

great video! subbed

[@TheReligiousCrap]

While I was studying in engineering, I was ok with Fourier and Laplace transforms. Then we went into Z transform and convolution. This is where I decided to go in power engineering.

[@jacobpenn2266]

Wow this finally makes sense

[@tanishqsingh4347]

Very good videos of yours

[@obibellowme]

Please support this man on Patreon!

[@avikdas4055]

Ted-ed and majorprep uploaded at the same moment. Coincidence, I think not.

[@avikdas4055]

I'm third

[@randomdude9135]

I'm fourth

[@palrob1714]

Fun video. Thanks.

[@BusterofGlitch]

I study electrical engineering. I have an exam on this topic tomorrow lol. Thanks for the video, you are explaining it better than the lectures! :)

[@sonuyadav1182]

You are awesome dude .. don't forget that .

[@terapode]

I wish I could give more than one like to this video.

[@jiezhang5325]

Fantastic vedio!

[@TheGaetanomariadigio]

Soft music and analysis The top for a sucker!

[@juliusrobertoppenheimer9104]

I can’t really understand how you’re able to make long and high quality videos in such short period of time, share with us mortals your secret

[@zachstar]

haha well I appreciate the comment! Recently I hired an animator so that has helped a lot. Also I work on the channel a lot and I really enjoy writing scripts, explaining the concepts, and seeing the videos come together so that enjoyment definitely helps.

[@hudasedaki1409]

Thanks for the clear explanation. I got a note though; when you divided by z you said since it's magnitude is one it won't change your output. However, it will be worthy to point out that this step is necessary to get proper transfer function as we can't access future samples in the case of having higher order function at the numerator of the transfer function. Y(z)/X(z)

[@g7sky]

Thanks!

[@angurajarenganathan6860]

I love watching your video , Greetings from Tamilnadu, India

[@georgeh1352]

Yep fascinating subject & video, I kind of get it (I think). Need to watch a few times more to be confident in knowing the subject (not easy at my age of 75), keeps my mind alert if nothing else. Thanks for your effort in passing on your knowledge.

[@wickedstereo7193]

*watched one of your videos.. . . . . . . . . . . SUBSCRIBED!

[@agod5608]

I am nots mathematician, buy o understand graphing and sampling. Excellent video. I could understand your presentation without understanding the math.

[@jp8159]

it was pretty good

[@MrKiemsi]

more videos about this please !!