You're right, they do. And I'm certainly no expert, but I have a more difficult time imagining a tire coming off the wheel intact without some catastrophic "blowout" impeding it's ability to smoothly roll than I do imagining a whole wheel coming off.
Leak, low pressure becomes no pressure, various lateral forces encouraging the tire to fold this way and that under mass, and soon enough the tire alone escapes its tormentor (because it’s being bent and pushed in ways similar to those that originally encouraged it to pop onto the wheel) and scurries away toward an unsuspecting fence.
> This secrecy was maintained long after the end of the war. “Efforts were made to give the message that the uranium came from Canada, as a way of deflecting attention away from the Congo..."
> Under Belgian rule, Congolese workers toiled day and night in the open pit, sending hundreds of tonnes of uranium ore to the US every month.
It’s unlike language because words can fluidly be invented or fall out of use. Emoji is designed by a committee. If it expands to include everything it will just be unusable. Why not just use pictures at that point anyway?
The existence of a committee, even an influential one, for a language does not mean that that committee actually controls that language. In the end, it's always the people who use a language that control it, not just in theory, but also in practice, as evidenced by how many of the Académie Française's proposals for pure French equivalents to English loanwords have failed to supplant the latter. These committees also tend to only concern themselves with low hanging fruit that is easy to point at, such as vocabulary, grammar, and spelling (which is technically speaking external to a language rather than part of it), while barely if at all targeting anything that would require more than an elementary-school-level understanding of language, such as phonetics/phonology, so their commentary also rarely even covers all aspects of the language.
I input Chinese everyday and it’s fairly simple. You input a phonetic sequence and choose the character. I already find it hard to use emoji. It takes forever to scan through the faces to find the one I want, and search online sometimes helps because emoji often have a popular use that isn’t congruent with their “official” purpose.
I don't know if there's a default OS solution that's any good, but I have a file of emoji + their text names and I use bemenu (like dmenu) to filter through it by name and put the selection on my clipboard. I bind this script to super-z on my keyboard in my Sway config. I based it on something[0] I found online that used dmenu or rofi and input the text for you. I adapted it to work better on Wayland by using a Wayland-native menu instead.
Windows, macOS, iOS, and Android all have dedicated emoji keyboards with search readily accessible. All Apple keyboard (software or hardware) from the last… 5(?) years have a button to bring it up!
Though in Linux there isn't only one window manager. This guy/woman/? chose to use his specific window manager (probably a tiling one), but he/she/xe could also use GNOME if he/she/xe wanted to. GNOME does have an emoji keyboard.
btw, this is much cooler than GNOME or whatever...
> They chose to use their specific window manager (probably a tiling one), but they could also use GNOME if they wanted to. GNOME does have an emoji keyboard.
There you go! Made it much easier for you and the reader!
But really,
> I'm not sure which window manager they're using, but GNOME does have an emoji keyboard
> Absolutely no excuse why we haven't done it since then
Well there is the fact that Apollo had (at its peak) something like 400,000 people working on it and involved a spend of $164bn (in 2021 dollars) [0]. I think we could get back to the moon if we had that sort of commitment.
Happy May Day! Here's a Twitter thread [0] that has excerpts/paraphrases from an old Soviet pamphlet called "How Soviet Workers Spend Their Leisure." (rolled up version [1])
You can find a scanned version of the pamphlet here [2]. There were many of these pamphlets published. One day I will scan them and post them. There was one on Waterways & Transport in the USSR, one about Soviet theater, one about the food industry in the USSR, one about light industries of the USSR, and one about the countryside of past and present of the USSR. I think these were a series published in 1939.
Article 119 of the Soviet Constitution:
> ARTICLE 119. Citizens of the U.S.S.R. have the right to rest and leisure.
>
> The right to rest and leisure is ensured by the reduction of the working day to seven hours for the overwhelming majority of the workers, the institution of annual vacations with full pay for workers and employees and the provision of a wide network of sanatoria, rest homes and clubs for the accommodation of the working people.
'Sanatoria' being health/medical centers or hospitals.
Did the Soviet government actually follow the Constitution of The USSR? I believe it also mentioned freedom of expression, freedom of religion and freedom of assembly, all of which seemed to be ignored.
I seem to remember a similar thing happening under Mao for a brief period of time, followed by a near total crackdown on anyone who had decided to speak up.
It wasn't followed, of course. Indeed, the early dissident movement in 1960s started with demands for the government to actually follow the constitution, and then kinda evolved from there. But also, as usual with constitutions, the wording in many cases is vague enough to allow for some ... creative interpretations.
In preparation for the 1980 Olympics, Soviets published a pamphlet titled "USSR: 100 questions and answers", meant primarily for Soviet citizens as an agitprop manual - basically a list of precanned answers to the most common questions that Western tourists were expected to ask. It is a hilarious read, but also very informative if you want to know how it was all sold in official propaganda.
Ok, but without consequence? There were a thousands of arrests and people hurt/killed voicing their frustration. Even journalists were shot at with rubber bullets and arrested. Yes, that's right, journalists were shot at and arrested by police in the US.
I think more accurate would be to say it allowed freedom from expression, freedom from religion and freedom from assembly. Unless in the service of the communist party of course.
A former colleague of mine, born and raised in the USSR, liked to quip that there was only one minor difference in the way freedom of expression was implemented in the USSR and in the West.
Both places guaranteed the freedom to express oneself, but only the West guaranteed your freedom also after you'd expressed yourself.
Prolonged periods of time spent without obligation to work for pay is one of the greatest luxuries you can have in this life. People shouldn't have to wait until retirement to enjoy it. Sabbatical, basically, although even sabbatical often comes with obligations for self-improvement serving the company offering it. A lot of discourse on working hours revolves around decreasing the hours worked per week. I would also like to see people talk about a right to take 6-12 months of unstructured leave after 3-4 years of service. Seems unrealistic? What is all this wealth and automation and productivity growth for, then?
> People shouldn't have to wait until retirement to enjoy it.
Also people should not pursue a strategy that concentrates ALL their enjoyment until retirement. It's a high variance strategy with IMO low EROI . My bias is because my father died in his 40s and my step father in his late 50s.. Both never saw a day of retirement despite saving for it.
I highly recommend a patterned approach to life enjoyment, yes this is highly privileged and I hope it will be universal one day.
Here's my strategy now that I'm financially stable:
I do something special, scaled to the time available, where they overlap I intersect them (ie, use the one weekend to build the one week)
* One weekend a month have a special plans
* One week a half disconnect, could be travel or camping, or staycation where you just chill and do stuff you love only
* Two weeks once a year - take an adventure, or a deep disconnect according to needs
* One month every 5yrs - Do something big that stretches you. Backpack foreign countries, or disconnect and focus on intensively learning a skill you intended to also practice going forward
* One quarter every 10 - Sabbatical (Sabbath) . Let your body and mind reset. Only do healthy and restorative things, celebrate every win you can remember, the only work you do should be investing in others, and do it at < 50% your capacity.
I'm happy you've boiled your recharging needs down to a science but none of this compares with long contiguous periods of unstructured time you can choose to use as you wish. Three months off every decade? Really?
I have ~ 15y of experience and I have a grand total of 3 weeks PTO. My father, at my age had 5 whole weeks and doesn't understand why I don't take big trips more. Despite the trend in 'unlimited vacation', I feel it's only getting worse.
> with long contiguous periods of unstructured time
Can you tell me more concretely what you're thinking? How do you define "long"? And, what is it about my list (despite it's prescriptive structure) that makes the time usage not "unstructured" ?
This is an interesting idea — there could be a sabbatical accrual of sorts.
One way this could realistically work today (with a lot of kicking and screaming about how it’ll ruin the economy) is each year of work earns a month of sabbatical, and once you’ve earned at least 2 months you can exercise your sabbatical rights, given some notice.
Companies could cap it at 6 months at a time so you don’t completely lose track of changes at work.
We could even make up to 2 months transferable between companies, and if you’re ever fired you can choose how or if you’d like to be paid out. That would help discourage retaliation against using the time, in addition to making it illegal.
I don’t see a reason that we couldn’t pass a law tomorrow that does this. Sure the economy would slow a bit, but what a giant quality-of-life gain it would be. And maybe it would actually make the economy grow faster in the long term if it encourages more private innovation.
But we already accept this idea for parental leave, this would be the same thing without the need for having a baby.
>>I don’t see a reason that we couldn’t pass a law tomorrow that does this.
Why do we need for a law to be passed? There are so many people seemingly in favor of this, why don't they just band together and start a company and then pay people to take 6 month vacations?
I know you're asking this question in bad faith because you posted the same thing elsewhere in the thread, but the answer here is because they will be outcompeted by companies that don't offer that policy. In much the same way, businesses that use child labor will outcompete those that do not. Of course a great many people are against child labor, but you don't see people telling them to go start their own company that doesn't use child labor if they're so against it.
It's fairly common for a majority—even a large majority—of people to want something, but to be unable to make it happen without a law. Coordination problems are real, and they are everywhere.
And that's just at a basic game-theory level—quirks or dysfunctions of political systems can make passing a law extremely difficult, too, even given super-majority voter support.
In Canada, there is the concept of sabbatical accrual (https://ett.ca/know-your-rights-self-funded-leave-plan/). When my wife & I were backpacking in Patagonia we met a Canadian teaching couple with their two children doing the same thing, except instead of one week (us) they were on a 6mo trip from the southern tip of South America up to Alaska.
Then, the same year, we were visiting family in Grenada and met a young family (teachers) with a 4yo on a 40' sailboat they were cruising around the Caribbean on.
This would not only provide a break from work but also offer opportunities for personal growth and exploration.
This could reduce the career growth penalty for mothers. Companies may be hesitant to hire or promote women to leadership positions, fearing that they will take significant time off for childbirth. However, if taking sabbaticals every few years becomes normalized, women wouldn't be viewed as a special liability for taking leave around childbirth.
Furthermore, during these sabbatical periods, there could be programs created for potential career switchers to try out new jobs. These internships could provide workers with opportunities to explore different fields and find work that aligns with their skills and interests. This could ultimately benefit the economy as a whole by increasing job satisfaction and promoting a better match between workers and their jobs.
If you don't want to work for a year then just quit your job and then get a new job later. But it's ridiculous and unrealistic to think that an employer should hold your job open for a year. How are they supposed to get work done during that period, hire a temporary employee and then lay them off when you decide to return?
If you want that lifestyle then just become an independent consultant rather than an employee. I know a married couple who does that. They work for a while to save up enough money, then take off and sail their boat around until they run low on money and have to find new consulting gigs.
I know all of these things sound nice. But in the real world a lot of businesses aren't making the kind of profit that allows workers to just take time off and not generate any revenue for the company. Sure there are some that could pull it off. But most small businesses aren't making that kind of profit. Unless you are assuming workers are supposed to take unpaid time off. But then how are they going to afford to take time off?
> But most small businesses aren't making that kind of profit.
It always feels that way in a small business, but then all of a sudden a key employee needs to take a three month medical leave and somehow the business survives.
You just have to plan well, and I'm sure there will be devoted employees who answer the odd emergency email during their sabbatical.
Businesses don't start flush with cash. It's a long process to become established. Your viewpoint is that businesses just have positive cash flow at all times and if they don't it's poor planning. It's not poor planning. It could be investments need to be made in certain areas in order to grow. or they haven't established a big product or client portfolio yet. Everyone starts somewhere.
Most of the world has statutory minimum of vacation days that also applies to small businesses, and somehow those businesses manage to survive. They just include it in their business model and market finds new equilibrium.
Moved from where to where and which industries exactly? Do you think this happened in the entire world? I do not see Germany as de-industrialized country, for example.
If it can't for any reason, it's failed. The problem is the more things you pile on, the harder it gets. The main point of a business, and the main good it does, is not employing people. It's what goods or services it provides its customers, at a lower price.
If you moralise and add more things for a business to do, the big ones will do some performative stuff and spend their vast marketing budgets on it, and the small ones will go under, and we all lose because we get less choice and higher prices.
Not really. It could be a new business that is only a few years old and is still trying to grow and become the type of business can afford these. Or an established business that lost a major client. Your viewpoint is really limited.
Personally in my career I would have much preferred sabbaticals to a reduced work week. I think most companies would find sabbaticals hard to retrofit though. You either screw the people who have a long tenure with the company or you suddenly have an unsustainable number of people who are now eligible for a sabbatical all at the same time.
Just a quick reminder that while paid time off was mandatory, working was also mandatory in the Soviet Union. One couldn’t just go on a sabbatical, that’d be a crime there.
> Those who refused to work, study or serve in another way risked being criminally charged with social parasitism (Russian: тунеядство)
Hey if they let people work less because the machines do it, they might figure out we're "dangerously" close to post-scarcity in a lot of fields. Gandhi figured that out a long time ago and talked about it, it's why the only machine he liked was the sewing machine.
The things that people really covet will always be scarce no matter how much machinery we have. There will never be "post-scarcity", it's just a silly concept.
As always, the gap between reality and propaganda is hilarious. Soviet workers and their colonized subjects spent their leisure time filling in for the massive failures of the state to provide anything resembling their promises. Some excerpts from my childhood:
* Family outing to the fields to harvest carrots to pickle for the winter. This would be bartered with other families who went to harvest other veggies, so that your turshia would not be too monotonous.
* Going to my friend's apartment to roasting and canning peppers, because no one has to money to buy the factory made stuff. And if you did, you still wouldn't want such poor quality.
* Helping my dad unload window frames for firewood. He worked at a window factory and was able to redirect a small truck full of finished wooden frames, which we could break down and burn for heat over the winter. Sure, it would have made more sense to buy actual firewood or coal rather than steal finished industrial output, but you can't afford that on a regular salary. And it would have made even more sense to offer district heating, and the government had such plans, but you'd freeze long before this became a reality. Eventually we did get district heating - about 6 years after communism fell.
Yeah. I remember my parents getting a permit to vacation by the Black Sea like once. My grandfather went to Egypt once. The vacations even within the soviets were super scarce experiences and going to the west was only possible for a choir trip or something like that after being vetted by the kgb for your flight risk.
There were trips to leningrad or moscow but even those had years between them.
So yea, most vacations were helping out at grandparents farm. And reading books during rainy days.
These examples highlight the deficiencies of the Soviet economy, and it's evident that such failures caused considerable suffering among the population. Nevertheless, given more time to engage in activities akin to the first two examples, and separated from the threat of malnourishment, I believe I would experience greater happiness and a stronger connection to both my food and community. Through shared struggle, bartering, and communal work, we can create a sense of connection that is absent from the isolation brought about by specialization, which, although more economically efficient, also contributes to the social problems present in western society.
I used to go foraging for mushrooms with a close friend. That friend developed depression due to his perceived failures to keep up in the rat race of career advancement, and committed suicide last year. I know this because of some pretty explicit conversations I had with him about feelings of inadequacy due to money and career path. So now I don’t have a friend to forage mushrooms with. I can do it by myself, but that doesn’t really address the alienation issue.
My parents and grandparents were forever traumautized by this experience leading them to leave objectively worse lives. Instead of actual leisure they'd spend their free time working on their car in the garage (because there are no car services), working in the garden (because food supplies aren't too diverse and you can't just go to the store to buy jam), or doing something else. They still feel weird about me ordering apartment cleaning once a month.
My grandma would continue working in her garden until she was extremely old because she was used to this, even though we could have bought any veggies or fruits to her. This likely affected her health in a negative way.
Living in a pre-industrial society isn't "good". It doesn't bring any advantages.
As someone who also lived in that society, albeit briefly, I can tell you that "experiencing a stronger connection with your community" is not necessarily a positive. Especially when you don't exactly have much choice when it comes to picking said community in the first place.
Certainly, I completely agree that it's not always a positive. Western capitalist culture is better suited to providing freedom to non-conformists, and it's true that life can be arduous in communal cultures if one deviates from the norm.
On the other hand, there are psychological advantages to collaborating with one's community to address shared issues. In western capitalist culture, many individuals are predisposed to feeling that their problems are their own responsibility, leading them to believe that any misfortune they experience is solely due to a lack of effort or poor decision-making on their part.
I wonder if it would be possible to build a society that gets the positive parts of both approaches, or if there is a better balance to be sought between individualism and collectivism.
I don't recall much "collaboration with one's community", at least not in urban Soviet landscape - perhaps things were different in the villages (but if they were,
I don't think the Soviets could claim credit for that). The society that they ended up building was actually very cutthroat in many ways once you got past the guaranteed basics that everyone had, precisely because anything above that was that much harder to get.
Thing is, you fundamentally can't enforce good socialization. You can enforce socialization in general, but the more it is forced, the less likely it is to actually be good. A good society is the one that gives opportunities for people to socialize if and when they want it, and accepts that some people really just don't and that's fine too.
Not exactly different from my parents experiences growing up in the US, where "summer vacation" was a trip to uncle's farm to detassel corn or pick cotton, hunting to get meat, etc.
Yep, all of my vacations growing up were either hunting or fishing trips. While getting out in nature is nice, when you take your kids out on a boat for 10h / day because you haven't caught enough fish it very quickly stops feeling like a vacation and more like work.
The information presented by the quote and JSTOR is accurate, at least insofar as it reveals that a significant number of individuals in USSR went on either tourist tours or sanatoria each year for rejuvenation.
In the 90s, state sanatoria maintained their popularity in post-Soviet countries. To this day, travel agencies in some of these nations primarily focus on two types of journeys: relaxation and "learning" holidays. This deeply ingrained cultural practice persists throughout the region.
I don't know how many and which groups of people went on annual holidays in the USSR, but I know it wasn't an insignificant number from what I've observed living there as a kid.
The down voting of the parent is wholly unjustified. This is a factual reality that has already taken place and continues on to this day.
“The Nixon campaign in 1968, and the Nixon White House after that, had two enemies: the antiwar left and black people. You understand what I’m saying? We knew we couldn’t make it illegal to be either against the war or black, but by getting the public to associate the hippies with marijuana and blacks with heroin, and then criminalizing both heavily, we could disrupt those communities. We could arrest their leaders, raid their homes, break up their meetings, and vilify them night after night on the evening news. Did we know we were lying about the drugs? Of course we did.”
There is nothing about the use of prison labor that benefits a free market. The opposite is true. By controlling a source of labor that can operate at below market prices it's easy to destroy any competion and gain total control. That is the antithesis of free market economics.
If anyone is a 'tankie' it's people who support government-supplied cheap labor...
There's a recent episode in the Citations Needed podcast [0] (episode 179) that discusses public libraries. There is a bit of history that is presented about Andrew Carnegie's contributions to the current system we have today: how the US library system started out with private funding before transitioning to public funding.
One misconception that many make regarding the Nyquist frequency is thinking that the sampling rate needs to be twice the highest frequency.
Your sampling should really really be twice the bandwidth.
e.g. your bandwidth is 100 MHz centered at 1 GHz (it needs to actually be bandlimited to 100 MHz**). You do not need to sample at 2.2 GHz. You sample at 200 MSPS (really, you should sample a little more than that, say 210 MSPS, so that the bandwidth of interest doesn't butt up against the Nyquist zone edges.)
The folks who are telling you you’re wrong don’t understand Nyquist’s criterion very well. Curse those undergrad courses for only effectively teaching about Nyquist at baseband frequencies.
You can sample 100MHz of bandwidth at 1GHz just as you describe at 210MSPS. You’ll get everything in the 950-1050MHz band.
Trouble is, without an antialiasing filter, you’ll get every other band that’s a multiple of that sampling rate. The Nyquist criterion works at every multiple of the sampling frequency.
Bandpass filter your analog input appropriately from 950-1050MHz and you’re golden.
This is the way nearly every commodity Wi-Fi chip downsamples 2.4/5GHz raw RF. Sigma-delta ADCs are cheap, fast, and space efficient for die area using this method.
The most fiendish application of this effect that I've seen is polyphase filtering. I can't remember the details, but at the time I can remember the wonder of understanding (in a lecture by fred harris) how most the logic was running at a low sampling rate yet the input was at a high rate. The mixing was done by aliasing.
Polyphase filtering is less crazy than it initially sounds. Conceptually, you can think of it as: I have this signal in frequency f. I want to resample it to frequency (b/a)*f, where a and b are integers. (You can also do polyphase filtering to resample of non-rational or varying ratios, by essentially approximating towards a rational, but let's ignore that for the moment.) a and b can be pretty large if you want, e.g. a=160,b=147 will downsample from 48 kHz to 44100 Hz.
So what you do to resample a signal (again conceptually), is: 1. Add <a> zeros between every input sample (which repeats the spectrum <a> times), 2. Apply a suitable (long!) FIR lowpass filter so that the signal is bandlimited, 3. Take every <b>-th sample (which doesn't cause any aliasing due to #2).
Now the core of the polyphase filtering idea: We don't need to actually calculate the FIR filter for the samples we don't want in #3. And most of the input values to the filter will be zero due to #1. So instead of storing all the zeros and stuff, we simply pick out every <a>-th tap of the FIR filter and use that on the input signal directly. But since a and b don't line up perfectly, this means we get a different subset of the FIR filter for every output sample; we have a time-varying filter (or a filterbank, if you want). You get <b> different such filters before you're back where you started.
Implemented a polyphase filter in Verilog once. I learned the hard way that it’s easy to mix in unwanted stuff into your polyphase chain if you’re not careful with your implementation.
I know what you're getting at, but your statement, as others have pointed out, is incorrect. Your sampling rate always always has to be twice the highest frequency of the signal you are sampling.
If you are sampling an RF-modulated signal with a center frequency of 1GHz and 100MHz of baseband bandwidth, then yes, you do need to sample at 2.2GHz+. And some applications do exactly that.
If you're taking the RF signal, mixing it down to baseband, and filtering it to bandlimit, then you have a signal with maximum frequency component of 100MHz, and in that case, yes, your sampling rate can be 200MHz+
From an information theoretic perspective (which is the perspective Nyquist was originally coming from, though it didn't yet have that name), you don't need to mix the signal down. Assuming it is truly band-limited, you can sample the signal directly at RF, and reproduce it from those samples. Additionally, you will need to modulate the reproduced signal into the original band, which means you need to know where that band is - perhaps this is the detail you're pointing out?
Another way of looking at it is that sampling inherently does the mixing down to baseband. Although it may not be exactly the baseband you want if the spectrum isn't cleanly symmetric about a multiple of the sample frequency.
I've worked on ultrasound systems that definitely worked this way, not just in theory but also in practice. Bandpass filter 20–40 kHz, sample directly at 40 kHz (giving 20 kHz bandwidth). No mixer step involved, but your spectrum becomes inverted (e.g. if you do an FFT, a 22 kHz tone will be in the 18 kHz bin, not the 2 kHz bin as you would perhaps expect).
Aliasing makes more sense (to me, anyway) if you think about the spectrum of complex signals, in which signals of real samples are modeled as the sum of positive and negative frequencies.
In the sampling operation, all sinusoids are shifted down to the "natural baseband" by adding or subtracting some multiple of the sampling frequency that places the resulting frequency within +/- half of the sampling frequency. So for your example of 22kHz, that real frequency has two components: +22kHz that gets shifted down to -18kHz=22kHz-40kHz, and -22kHz that gets shifted up to +18kHz=-22kHz+40kHz.
Note that this "natural baseband" is an abstraction of our own invention. You can just as easily think of the spectrum as ranging from 0Hz to the sampling frequency f_s, rather than -f_s/2 to f_s/2. The fact that some prefer one over the other is precisely why fftshift exists.
To clarify: "band-limited" usually means X(w) = 0 for abs(w) > B for some B, where X is the frequency spectrum. And that's the definition Shannon used in the original proof, which is where the idea of Nyquist Frequency comes from.
If you add the additional constraint of the signal being "bandpass-limited" where, X(w) = 0 for A > abs(w) > B for some A, B, then yes, you can under sample.
And that's where the information-theory idea comes in where the amount of information contained in the band only "needs" 2X sampling rate to reconstruct perfectly.
You can think of aliasing being somewhat orthogonal to that in the sense that you need 2X bandwidth so you don't corrupt the signal, but 2X max frequency so you don't alias anything else into the signal. (I say this realizing that aliasing is what would cause the former signal corruption, hence "somewhat")
"In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate (twice the upper cutoff frequency), but is still able to reconstruct the signal.
When one undersamples a bandpass signal, the samples are indistinguishable from the samples of a low-frequency alias of the high-frequency signal. Such sampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF-to-digital conversion."
Yes, but this only works if, as the page points out, the signal is bandpass filtered, which GP did not mention. It's not true in the general sense, nor is it practical for many (most?) RF systems, especially those with multiple channels.
I can see why you might think that but consider that in RF systems, while the wanted signal is bandlimited, you also have a lot of unwanted "blockers" all over the spectrum that need to be dealt with before sampling.
I'm afraid you're mistaken (source: worked as DSP engineer for 15 years). Often you apply your filter around the RF frequency you want and then sample at a lower rate. You're right that the signal will get aliased doing that, but the information is always preserved.
If you sample s.t. your folding frequencies are in an appropriate place, you can fold your desired region into the first nyquist region without needing to mix it down. This is especially desirable if you can avoid having to build an IQ mixer because they're hard to keep balanced.
The worst case doing this is that your signal spectrum is reversed in frequency, but you can correct that easily digitally.
I'm afraid I'm not mistaken (source: I design integrated RF transceivers) ;)
Yes, you can subsample if you have a suitably bandpass-limited signal. But that's not the general case, nor is it what the nyquist-shannon theorem proves, which is where "nyquist frequency" comes from.
Nyquist frequency by the original definition is 2X highest frequency, though some papers textbooks evidently have started using it to mean 2X bandwidth, enough so that wikipedia[1] actually mentions it.
In integrated circuits, IQ mixing isn't problematic as we can fairly easily do gain and phase calibration to correct for the mismatch.
You have to have a band limited signal to sample anyways, where it's at in the spectrum doesn't matter. The first thing you'll do before feeding anything to an ADC is running it through a filter to make _sure_ it's band limited. Whether that filter's at DC or some Rf doesn't matter.
Here's the result from his original paper where he specifically says that it doesn't have to be at DC:
My point is that practically speaking, it does matter where the signal is, depending on how you filter it. If you lowpass filter an RF- (or, more realistically, IF-) centered signal, you can't just sample it at 2X bandwidth because you'll get aliases from the unwanted content between DC and the bottom frequency edge of the signal.
It may not be a common scenario anymore, but it was very common in the early GSM days when the signal wasn't mixed to DC but near-DC.
Ah yes you're right that you have to be careful, it'll fold at multiples of the nyquist frequency and you want to make sure your SOI is entirely contained in one of those zones.
That's true, but there are a couple of things more. First, your DAC or ADC need to have such analog bandwidth. Working in a higher Nyquist zone also require higher amplification since the signal would be considerably weaker and more complex filtering to remove the signal from the other zones
Or use the traditional "lock-in" amplifier technique of mixing with a known reference at the frequency mid-point of the range you care about? (That's how NMR spectrometers / MRI scanners worked for decades
Isnt the lock-in amplifier technique used to improve the SNR ratio of a signal by filtering out noise at frequencies outside a specific range of interest? High-speed sampling would still be required to accurately measure transient signals.
Consider a signal whose value at x seconds is f(2x) - 2 f(3x) + f(4x), where f(x) = sin(2πx)/x. Considering that the absolute frequencies of f(x) are uniformly distributed from 0 to 1 Hz, the absolute frequencies of this total signal should be constrained to between 2 and 4 Hz. Thus, a bandwidth of 2 Hz. But if we sample at 6 Hz (three times the bandwidth!) including x = 0, we'll get all zeros.
Granted, we might say that from the perspective of the complex Fourier transform using signed frequencies, the frequencies of this signal actually range over [-4 Hz, -2 Hz] U [+2 Hz, +4 Hz]. But I'm not sure that's the interpretation you had in mind.
That is, it's not quite as simple as saying you just need to sample at any frequency at least twice the bandwidth. Rather, it's the more complicated behavior described by this graph: https://en.wikipedia.org/wiki/Undersampling#/media/File:Samp.... That is, the general rule is that the ratio of the highest frequency in the signal to half the sample rate, and the ratio of the lowest frequency in the signal to half the sample rate, have to lie within an interval of consecutive natural numbers.
When the lowest frequency is zero, this is the familiar rule that the sample rate has to be at least twice the highest frequency in the signal. But more generally, it's more complicated.
Whoops, I should've pulled the division by x out of the definition of f. The example I had in mind was [sin(4πx) - 2 sin(6πx) + sin(8πx)]/x. [Another good example is [sin(6πx) - 2 sin(8πx) + sin(10πx)]/x, whose frequencies are between 3 Hz and 5 Hz, thus a bandwidth of 2 Hz, but sampling at 4 Hz or even 8Hz gets all zeroes.]
Anyway, the details on that example don't matter, the Wikipedia graph and article makes things more clear.
Is this assuming you have some analog hardware that's demodulating the signal in front of your ADC? How do you demodulate a signal from a 1GHz carrier with 200 MSPS?
As the sibling comment mentioned, you don’t need to demodulate first, because that is actually what the sampling process of your ADC does.
You can think of it as multiplying the original signal by a comb (in the time domain) of delta functions, which folds everything (in the frequency domain) back into the nyquist frequency of your ADC. Each delta function corresponds to one sample. If your original signal was truly band-limited to 100MHz, then what comes out is a replica of the band limited signal.
One catch (which is actually fairly easy to do in practice) is that the sampling window needs to correspond to around 1/f of the carrier frequency. This is what YakBizzaro is talking about (ADC analog bandwidth) in their sibling post.
Thanks for the explanation! Between your comment and the Undersampling wiki page diydsp linked to I think I am on the path to enlightenment.
> If your original signal was truly band-limited to 100MHz
In practice, this means you need to band pass before the ADC, right? i.e. "signal" in this case is the entire input to the ADC and not just the particular modulated signal you care about
> In practice, this means you need to band pass before the ADC, right? i.e. "signal" in this case is the entire input to the ADC and not just the particular modulated signal you care about
Right and right.
And, you’d normally want that to be a contiguous 100 MHz band of frequencies (you could in principle have multiple discontiguous bands that add up to 100 MHz if they are spaced right (they don’t fold down to the same base frequencies), but that would be quite an unusual application).
To quote a meme: “That’s the neat part. You don’t.” If you bandlimit your input, aliasing effectively strips out the carrier tone and leaves the modulated signal.
In a way, you’re relying on aliasing / frequency folding to do it for you.
You can even improve information transfer in these scenarios by using a synchronizer, which allows you to phase shift your sampling to be at the ideal transition point in your information stream.
No, this assumption is incorrect. You can ADC first and then demodulate afterwards. The spectrum of your high-frequency (near 1 GHz) signal will be aliased at frequencies below the Nyquist frequency, but it’s easy to calculate the original frequency, if you know that the signal is band-limited.
You do not want to "correct" the wiki because the wiki is not wrong. The person you are replying to is clearly thinking about some sort of RF system (given the frequencies mentioned) where it's important to have a baseband filter to eliminate aliasing, and that filter will have some sort of roll off region, resulting in a higher sample rate than available bandwidth. That's all great, but the Nyquist theorem isn't talking about an RF system. It's referring to sampling. When the wiki uses the word "bandwidth", they mean the frequencies that don't alias given a specific sample rate.
Is the wikipedia page really wrong though? Highest frequency is what the mathematicians care about. EEs care about bandwidth because they're always modulating stuff and thinking in terms of carrier and baseband. Strictly speaking, what the EE grandparent suggested is using aliasing to mix the signal down to baseband.
Yeah but you also need the bandwidth of the sampler to exceed the highest frequency of the sample. Most samplers are limited by some kind of RC time and not their sinc envelope. Most.
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