Sitting at my desk on an otherwise normal day, I received a rather curious email. It asked if, fully funded by the British Council in Pakistan, I wished to give a talk at a nanoscience conference at LUMS University Lahore, sight-see, meet students and researchers, and travel with eight (or more?) other PhDs from different UK institutions with distance ranging from UCL to Glasgow and Strathclyde. There would of course be 5* hotels, top tier flights, amazing food, and armed police escorts, all while being a UK dignitary and taken round by the same people that hosted the Duke and Duchess of Cambridge a couple of weeks prior to the visit.
“Ahhh… hmmm… yes?”
One quick day trip to Gerry’s Visas in Birmingham later, I arrived at Manchester Airport not quite knowing what I had got myself in for. After a short layover in Qatar (plus a few hours to sleep off the jet lag!) I found myself in Lahore in the company of the other researchers and our front-facing representatives from the British Council; Mubbashir, Hajira and Namra.
Hosted at LUMS, the ICONN nanoscience and nanotechnology conference contained an absolutely packed two days of talks, and was attended by several hundred people, including a pleasingly large number of university students. I was invited to give a 12 minute talk on the PhD I do with the nanoscience group here in Nottingham, applying the latest in machine learning technologies to automatically run the millions of pounds of probes we use for our research. (Going back through this blog, you may find a post I wrote on this topic during a summer internship during my undergraduate here).
My expectations of the conference were for it to be like a nanoscience conference in the UK –far-out future technologies and new-found properties of surfaces, using eye-wateringly expensive equipment. Instead, local research was specific, and highly focused on practical solutions to the immediate issues facing Pakistan: pollution, healthcare, clean drinking water and reliable battery storage. LUMS even offers space and expertise to start-up companies working towards this end.
The whole experience gave us a real taste of Pakistani science and was a real joy. In particular, the conference, visits to other universities, and allowing us to judge the conference posters during the cultural night at LUMS gave us plenty of chances to speak to loads of students, PhDs and researchers from up and down the country. This in particular was quite eye-opening to me – some of them explained that low funding meant that they often have to travel and pay out of pocket to use equipment and do analysis that we could routinely do in the first year undergraduate lab in Nottingham. Science is held in incredibly high regard in Pakistan, and I can only admire people doing work for the immediate benefit of their nation, even when it comes at their own expense. I must also mention how touching it was to see how enthusiastic and just proud the students were to be involved with science in general.
This conference only made up two days of the week we had, though. Besides the university visits, we also had time to do some sightseeing. Lahore fort and the old walled city was a real gem, as well as having dinner by the largest Mosque in the country – capable of holding 100,000 worshippers. We also squeezed in a visit to the British Council HQ, where they support people taking UK educational qualifications. Their library is wonderful (and open to international e-members) – it is a bustling, cross-cultural community hub with a real international spirit, and the British Council should be proud of what they have created there.
I will, however, confess that the real highlight of the trip for me was playing cricket at COMSAT University Lahore. The first thing we saw at COMSAT was their cricket green and pavilion – the faculty and cricket team were more than happy to arrange a five-over game with us over lunch. Unfortunately, I (captaining) found myself run out trying to steal a quick single…
But after some long goodbyes and games of ping-pong in our hotel, I found myself back in Manchester and welcomed by the UK’s torrential rain. It had to end at some point!
I need to say an absolutely massive thank you to Walther Schwarzacher at the University of Bristol for arranging the trip (plus his great talk on clouds), as well as to Mubbashir, Hajira, Namra, and the rest of the British Council team for their incredible hospitality. This trip really left me feeling genuinely quite inspired about just how much of a privilege and pleasure research is, and looking forward to getting to my lab to do more.
"A memorable two days international conference on nanoscience and nanotechnology (ICONN-2019) at LUMS in collaboration with the British Council and NUST… Wonderful opportunity to present and awarded by Prof Walther."#photooftheday#OSALUMSpic.twitter.com/Cmer5fTxse
VROOMFONDEL: We demand that machine not be allowed to think about this problem!
DEEP THOUGHT: If I might make an observation…
MAJIKTHISE: We’ll go on strike!
VROOMFONDEL: That’s right. You’ll have a national philosophers’ strike on your hands.
DEEP THOUGHT: Who will that inconvenience?
MAJIKTHISE: Never you mind who it’ll inconvenience you box of black legging binary bits! It’ll hurt, buster! It’ll hurt!
From Fit The Fourth of The Hitch-Hiker’s Guide To The Galaxy, Douglas Adams. Broadcast on BBC Radio 4, March 29 1978.
I suspect that this is going to be a contentious post.
Having spent my time on the picket lines over the last eight (non-)working days…
Come on @UoN_Physics. We need as many of you as poss, out here fighting for our pensions, pay and casualisation. The more we have the stronger our fight and quicker this is sorted!!! The humanities are showing us up with their solidarity. #uss @ucu @UoNUCU pic.twitter.com/nSBO1jtTys
But what have these eight days on strike actually achieved?
Yes, I know that we’ve demonstrated a great deal of solidarity and that the time on the picket lines has been morale-boosting (and at least it wasn’t as sodding cold as last year). But still, pragmatically, what did we achieve?
Here at Nottingham, at least, the response from the “powers that be” has been a deafening silence. (And Nottingham’s hardly alone in this.) For many departments, including my own, it’s been business as usual; the car park has been full, lectures and lab sessions went ahead with nary a disturbance, and coursework was dutifully marked and returned to students. This is not to downplay in any way, I hasten to add, the heartening efforts of my UCU colleagues and our incredibly supportive students, including, in particular, those who occupied UoN’s iconic Trent Building…
And I’ve also got to highlight the incredible energy, charisma, and tenacity of Matt Green, the President of Nottingham’s UCU Committee, who has been as outstanding as ever.
But the upshot of our eight day strike is that …drum roll… the UCU is going to call for yet more strikes in January. The argument is that we’ve got to keep the pressure up. But who, exactly, are we pressuring? Or, as Deep Thought puts it in that salient quote that opens this post, who, exactly, are we inconveniencing? We’ve hardly brought senior university management to their knees, have we?
For those who, like me, were on the picket lines — and, indeed, for those who weren’t — ask yourself this: which of the options below hurts the university more? Which is more likely to cause some sleepless nights for the senior executive?
A.An empty seminar room or lecture theatre,
B. A five- or ten-strong picket line chanting at a university entrance,
C.A low score in the National Student Survey/ low league table ranking/ damaging media coverage for their university?
Not only did we have PVCs and other senior staff crossing picket lines with wild abandon, but quite a few union members — and, indeed, some erstwhile union reps — didn’t strike, let alone picket. University management will be well aware of this lack of engagement with the strike either now or when the figures for non-pay in January are returned. They save on the salary bill and they can rest easy that the impact on students’ progression is minimal, at best, and negligible, at worst.
Because what most matters to universities is their brand. If we want to have greater influence and bargaining power I would argue that we have to be a little more canny in our tactics and exploit exactly the corporatisation and marketisation culture we criticise and that underpins the behaviour of the 21st century university. (I’ve written before about the frustrating tendency of the left to not always be entirely cognisant of the value of “optics” and PR.)
Sceptical? Here are a few examples of brand management that might help to make my case…
Along with a number of APM colleagues, I spent six months chasing up a (very modest) honorarium payment for an invited speaker. Six months. The speaker eventually reached the point where, exasperated, she tweeted about the University’s lack of payment to her tens of thousands of followers (tagging in @UniOfNottingham). Within minutes she had a response from UoN, and within days the money was in her account.
Down the road, at Nottingham-Trent University (Guardian University Of The Year 2019),Liz Morrish was subject to disciplinary proceedings when a post hit 10,000 views on Liz’s own blog and trended at the Times Higher website, as described in the article linked in the tweet below.
And Warwick hardly covered itself in glory in this appalling case because they placed their brand management well ahead of students’ safety. That’s how engrained the importance of protecting the university brand can be.
“The top six universities are like the most beautiful cities in the world, reputable even if they have failing sewers, arrogant mayors and dodgy no-go areas…A folklore builds up around them, as do money and fans.”
I’d sit alone and watch your light My only friend through teenage nights And everything I had to know I heard it on my radio
From “Radio Ga Ga”, Track 1, Side 1 of Queen’s “The Works” (EMI Capitol 1984).
Written by Roger Taylor.
I’ve been delighted by the response to the “Joy Of Radio“, and accompanying “Resonant Frequency” , video that Brady uploaded to Sixty Symbols a couple of weeks ago. In addition to the very many kind, supportive, and heart-warming messages left under the video — which have done a lot to almost restore my faith in the humanity of YT’s comment sections — I’ve received a number of affecting e-mails telling me how much the video brought back happy memories.
One of these came from Rocco Cuffaro, a third year Mathematical Physics student here at Nottingham. Rocco saw the video, immediately thought of his crystal-radio-loving granddad (who is eighty years young), and emailed him a link to the video. Rocco’s granddad’s wonderfully engaging response is below. Thank you both for sharing this with me (and for giving me permission to post it here). Over to you, Rocco’s granddad…
Where do I begin?
After you and I built the Meccano type of Rocket steam engine it was my intention for you and I to build a crystal set exactly as the simple circuit in Prof Moriarty’s book. For the coil we would have used the cardboard centre of a toilet roll as a former. For the crystal I would have used a crystal from a ex war time radar if I still had some or more likely buy a silicon diode from Maplins. I have or did have plenty of variable vane capacitors that we could have used. The box of capacitors that Prof Moriarty showed us included some wide vane capacitors that were probably I guess ex transmitter components.
I very much like his approach with the experiment set up live and using a corroded penny with a wire whisker very much as it was with crystals in the early days before my time. Just a hook up of components rather than a workshop prepared demonstration.The early sets used a crystal and a wire whisker and just as he said there were many swear words so it was when people tried to find the right spot on the crystal and were then afraid to jog the thing.
I attended an exhibition at the IET’s home in Savoy place in London( I am still a member but thinking of cancelling as I am in information overload with so many and varied things to read) which I believe marked the anniversary of the broadcast station 2LO which was the forerunner of the BBC. Many crystal sets from way back were on display and included some with more complex circuits that I was not aware of until then.
Before we became the affluent society in the 1960s I knew the elderly lady that occupied the old observatory tower in Falmouth. There was not much money around in those days so I built her a crystal set in an old cigar box and used an ex service radar diode for the crystal. The aerial was one to envy in that situation. A vertical wire 50 feet down the side of the building. The lady came from London and was thrilled to hear Big Ben for the first time in years. That happened not very long after the war.
When Nan and I lived in Ware we had a reel to reel tape recorder. I do not recall how it came about but I discovered that if I put a wire in the microphone socket the recorder thought it was a radio and picket up a strong radio signal of one programme only from obviously Brookmans Park not a million miles away. This can easily happen if all the real and stray capacitances and inductances etc allow this to happen with some non-linearity in a component giving that rectifier action that Prof Moriarty was demonstrating. I was able to record a complete set of the Gilbert and Sullivan operettas in this way.
The converse of this phenomena is modulation and also cross modulation. At Lands End Radio the receiving aerials were close to the building housing the receivers. The transmit aerials were not very far away by a couple of fields. A problem arose when unwanted radio stations were splashing across the entire spectrum of the receivers (around and either side of 2182khz) [Editor’s note — see footnote #2 below]. I traced this to a dustbin (!) in the grounds of the station. The lid was placed lightly over the bin. On removing and replacing the lid the problem cleared. This was in my view caused by rust around the edge acting like the penny and whisker in Prof Moriarty’s experiment in reverse. The bin would have been receiving a very strong signal from the transmitting aerials of course plus a multitude of other radio signals from all over the place. The bin was transmitting a cross modulation of the lot and dumping it on the receiver aerial system.
I will wind up there except to mention that there were a number of interesting cases of interference including a grandfather clock with no electrical connection interfering with television and the static from a lady combing her hair interfering with television through a party wall.(not mine but a colleagues) The frequencies in use then were much more susceptible to interference than now.
 I only realised this after reading the comments, but, remarkably, when I hit the desk to “kick start” the crystal radio at the start of the Resonant Frequencyvideo, the song that blasts out is…Queen’s Radio Ga Ga. I was entirely oblivious to this while making the video. (This is a good thing. I’m already as overly enthusiastic as a litter of labrador puppies in that video. (Those five coffees before Brady arrived probably didn’t help.) If I had realised at the time that we’d picked up a broadcast of Radio Ga Ga I may well have melted down with excitement…)
 I deliberately set up the simulation of the crystal radio described in this post to mimic this “bleeding” of signals across the frequency spectrum.
I’m writing this from the not-so-sunny climes of Maidstone West station, waiting for the 09:03 to take me to London St Pancras (via the wonderfully-named Strood), from which I’ll get the train to Nottingham. I’m travelling back from Maidstone after a Skeptics In The Pub talk last night in the Market House pub there. It’s been over three years since my last visit to Maidstone SITP, which was again at the invitation of Rob Millar, the local Skeptics… coordinator. Thank you, Rob, for the invitation, the hospitality, sorting out accommodation, locating a guitar amplifier at the eleventh hour , andyour careful chairing of what became a rather “robust” Q&A session at the end of the night.
I promised Rob and the other SITP regulars that I’d upload the slides I used, so here they are:
The majority of the talk was focused on “Uncertainty To 11…” themes, and I was delighted that Maidstone Skeptics asked very many perceptive, smart, and challenging questions about the nature of the quantum world (and much more). I just hope that my lengthy discourse on spatial frequencies through the medium of Stryper‘s sartorially-challenged stage attire was not the cause of too much indigestion last night. (Extra brownie points to the SITP regular who correctly identified both Stryper and Bad News  from the photos in my presentation. Clearly a man who, like myself, knows a little too much about eighties metal…)
Given that this was a Skeptics crowd, I felt obliged to include a couple of diversions from the quantum-meets-metal theme on the nature of skepticism, the devaluation of expertise, and the hysteria and hypocrisy of certain reactionary factions/fanatics in relation to university education. Those gender studies and lefty sociology courses mean that the nation is doomed, don’t you know…?
I think an ABSOLUTE PRIORITY for the next Government must be to root out leftist culture from our schools, universities, civil service, state-run services and Media (esp BBC).
If we don’t change the mindset soon we are all doomed.
Hat tip to Tony Padilla for making me aware of Lance’s over-excited tweet above. Mr. Forman doubles, triples, and quadruples down on his pearl-clutching in a series of increasingly hyperbolic responses, including this:
The sad thing the left have be one so blinded by their self-righteousness, they have become the fascists and deniers of free speech.
Lance’s call to root out opinions he doesn’t agree with in order to, ahem, protect free speech — as one uber-reactionary pundit would put it, you can’t make this up — isn’t, of course, an entirely original demand. Jordan Peterson, along with other members of the self-styled Intellectual Dark Web — stop sniggering at the back there — has been howling for academics’ heads on a plate for quite a number of years because they simply will not toe the line, do as they’re told, and goddamn teach his preferred doctrines.
Mention of Peterson’s self-help psychobabble last night (see Slide #17 above) led to quite a heated discussion in the Q&A session following the talk. Three years ago I spent quite a bit of time lampooning Deepak Chopra’s “quantum woo”to a receptive Skeptics audience in Maidstone. What I find so difficult to get my head around is that very many of those who would identify as “rational skeptics” (or similar), and who rightly dismiss Chopra’s fairy tales out of hand, also represent a significant proportion of Peterson’s core fanbase. And yet, as I discussed at length in a talk for Nottingham’s Agnostic, Secularist, and Humanist society last year, Peterson’s “12 Rules For Life” and “Maps of Meaning” push the bullshit meter just as far above 11 as anything Chopra has written. Peterson’s style-over-substance, read-into-it-whatever-you-like, self-help gobbledegook also, hilariously, has very much in common with the wilfully impenetrable junk that is produced by the worst of the postmodernists he so despises.
… if you are going to popularize the idea that leftist academics and human-rights organizations are poisoning the minds of children, and fomenting a subversive ideology antithetical to the health of your nation, then you simply cannot meet with an authoritarian prime minister who has used nearly identical arguments to justify state crackdowns on independent universities and NGOs — then issue no public explanation of why you took this meeting or objection to reports characterizing your conversation as convivial — and call yourself a principled defender of liberal values.
And to hammer it home:
Meanwhile, it isn’t hard to see how Peterson and Orbán might see eye to eye. The latter has effectively banned “gender studies” from his nation’s universities, while the former has called on his nation to do the same. What’s more, in a diatribe that Orbán’s speechwriters may wish to crib from, Peterson went so far as to suggest that left-wing instructors at a Canadian teachers college should be prosecuted for crimes against the state.
Lance Forman’s tweet above looks positively moderate in this context.
The central problem, however, is that the cult of personality surrounding Peterson (and, indeed, Chopra) is such that counter-arguments, data, and evidence are not going to sway those who feel that the great man has personally “spoken” to them via “12 Rules For Life” (or, in Chopra’s case, “The Seven Spiritual Laws for Success“) and changed them for the better. I asked the following question last night of the SITP regular who was a fan of Peterson:
“You say that Jordan Peterson’s “12 Rules For Life” spoke to you and made a difference in your life. How? Can you give me a specific example of something he wrote that had such an impact on you?”
This is now the third time this has happened during a Q&A session. I don’t like putting people on the spot but I find it fascinating that when asked to highlight just one instance of Peterson’s writings that made a difference, each time I get a blank response to that question. This is entirely in line with Peterson’s writing style. He resonates with so many because, as Nathan Robinson explains so well in his classic take-down, Peterson’s writing is so nebulous and unclear that the reader takes their own meaning from the text. To be fair to Peterson, that type of writing takes a particular kind of skill. I only wish I’d realised this long before now  but it’s eerily reminiscent of the boilerplate that drives the horoscope market, as described in a classic Physics World article by Iggy McGovern : Aspects of Low Resolution Horoscopy.
This leaves us with a conundrum. If even a hardened sceptic — an atheist/agnostic who rejects the likes of Deepak Chopra’s woo, for example — is taken in by Peterson’s guff, how can they be persuaded to be just a little more, um, sceptical (or, indeed, skeptical)? I am not at all suggesting that my approach last night — a rather full-on lampooning of Peterson — is any way to reach across the aisle, cathartic and fun though it was. Moreover, as an academic whose political leanings are left of centre, I will often be seen as one of the enemy. It is therefore going to be difficult, if you’ll excuse the understatement, to convince the Peterson faithful– whose numbers, I am willing to bet, include Lance up there — that I am not seeking to indoctrinate their children/ cause the collapse of Western civilisation/ establish a Cultural Marxist collective where it will be an ABSOLUTE PRIORITY to outlaw right wing views (delete/expand to taste).
So how do we connect? If we want to get beyond preaching to the converted, we obviously have to first find common ground with those who don’t share our political/ideological mindset. That’s tricky. But for the less evangelical of Peterson’s flock, science might be a way in. Or music. Or, indeed, both. There’s this book I could recommend…
 Thanks also to Ben for providing said amplifier.
 If you’re a Spinal Tap fan and you haven’t seen either of The Comic Strip Bad News specials, beg, borrow, or download the episodes asap. I guarantee that you won’t be disappointed. Sample quote, from Bad News’ lead guitarist and singer, Vim Fuego (aka Alan Metcalfe): “I could play “Stairway To Heaven” when I was 12. Jimmy Page didn’t actually write it until he was 22. I think that says quite a lot.”
And here’s the Bad News boys in action…
 Thanks to Rob for bringing my attention to Peterson’s meeting with Orbán.
 Thanks, Lori, for drawing this parallel with horoscopy.
 Iggy was the external examiner for my PhD. He’s a poet as well as a physicist so knows a thing or two about writing style. This video with Iggy was a lot of fun to make back in (gulp) 2011…
Woke up this morning to find this rather tired and emotional missive nestling in my inbox:
Hi, I read your post about Alessandro Strumia, and have to tell you to pray for a new brain. Yours is ideologically possessed and rotting. You make about as much sense as a demon possessed lunatic. His stuff is empirical. I’ve read that paper. You make me sick. That’s all.
I’ll spare my correspondent’s blushes and not reveal his name. As is usual in these cases, there’s absolutely no attempt at all to provide any type of counter argument to the scientific points raised in my posts about Strumia (here, here, and here. And elsewhere.) As I’ve said before…
…what I have always found striking throughout my time online is that those who claim vociferously to be solely driven by reason, logic, science, rational debate, and/or individualism — aka the Fuck Your Feelings (FYF) brigade — are often among the most hypersensitive, overwrought, tribal, and emotionally driven out there. The type of over-emotional response that the FYF tribe attribute to the big, bad bogeyman of “The Left” is equally, and often more, prevalent within their own ranks. (There are key parallels here with the deeply intolerant patriotic correctness of the right.)
That wonderful “demon-possessed lunatic” line has, however, put me in the mood for some metal to start the day…
With help and advice from my uncle Benny, and with a “recipe” from that wonderful Ladybird making a Transistor Radio book shown in the photo, as a science-and-sci-fi fixated ten year old back in 1978 I built a crystal radio from a diode, a variable capacitor, a coil wound around a ferrite rod, and a long length of wire (which was slung up in the attic of the family house, “threaded” out through a vent on the side of our house, and tied to a tree on the other side of the garden.)
Unfortunately, that’s not the original book — nor the original crystal radio — shown above. But it is a second-hand copy of roughly the same vintage (published 1972) as that we used. Strolling through Cambridge (MA, not the one ‘down south’) a few months ago, I spotted what I thought was that Ladybird book in a bookshop window. It turns out that it was just a postcard of the cover — grrr — but that was still enough to get me eagerly hunting online for a second-hand copy of making a Transistor Radio that I could order.
The wave of nostalgia that hit me when the book arrived a week or so later was, to say the least, palpable. Brady Haran had suggested many moons ago that a Sixty Symbols video where I built a crystal radio from the Ladybird “blueprint” might make for a neat project, and he was (as ever) dead right. We had a lot of fun making this…
I casually state in the video that the diode rectifies the electrical signal, only allowing current to flow one way. That’s of course true, and it accurately describes the role of the diode in the crystal radio. But just why is it that rectification allows us to recover audio information, with frequencies in the sub-20 kHz range, from a radio wave whose frequency is ~ 1 MHz? Like so many problems in physics (and elsewhere), the answer is beautifully simple if we consider the Fourier domain. However, to my mind there’s a great deal of unnecessary complicating detail in textbooks and online resources on the subject of AM radio — and modulation in general — that I feel needlessly obscures the core, and ever-so-elegant, principle.
But sometimes I really want to explain the physics in a more in-depth, tutorial style. And that is very definitely the case for the crystal radio. Not only was that humble receiver such an important factor in my getting hooked on physics but it’s an amazing demonstration of the power and elegance of a Fourier perspective on signal processing. So I spent a few hours over the weekend writing some Mathematica code  to simulate the process of amplitude modulation and demodulation that underpins just how the crystal radio works. I hope this will help demystify (de)modulation for those who, like me, struggled to get their head round the concept as a student.
Let’s start with the simplest possible wave — a pure sinusoidal tone. We’ll choose a 1 kHz frequency. It looks like this…
…and it sounds like this:
Yawn. An irritating, characterless whistle.
But this boring, characterless tone can teach us a lot.
A much more “compact” way of looking at any wave is to plot its frequency (aka Fourier) spectrum. In the case of that 1 kHz sine wave, it’s very straight-forward to interpret…
There is no new information in the figures above — they’re simply alternative representations of exactly the same waveform data. Instead of plotting how the wave varies in time, we are representing it in a much more compact, and therefore much more easily interpretable, form. The main figure is what’s known as the power spectrum — it shows the power associated with different frequencies.  Don’t worry about the numbers on the y-axis. (Not that you can in any case — I’ve not shown them so you won’t get distracted.) Focus instead on the x-axis. We have a peak at 1000 Hz. And nothing else. A single, solitary peak, the height of which is proportional to the “volume”  of the note.
The inset shows what’s known as a spectrogram. This tells us how the frequency spectrum varies in time, with the brightness on the plot now representing the “volume” of the waveform. In this case, the note is a constant 1 kHz frequency, so we have a simple flat line (at 1 kHz on the y-axis) across the 1 second duration of the note.
“Never mind the aggravation. Modulation. Gimme another key.”
To transmit this incredibly boring signal via conventional AM radio, we’d first need to imprint it on a carrier wave having a frequency in the 535 – 1605 kHz range. I didn’t use frequencies quite that high in the Mathematica simulation because that would be overkill (and would dramatically slow down processing speed due to the sheer volume of data; a 1 MHz signal, for example, would require a minimum of a 2 MHz digital sample rate for accurate representation, as Nyquist and Shannon (and others before them) showed us many moons ago.) But we can see exactly the same (mathematical) physics in action and understand AM modulation/ demodulation using much lower frequencies.
So let’s just pluck a carrier frequency out of the air. We’ll opt for 21.12 kHz, for no particular reason. We’re going to imprint our audio information — that single sine wave — onto our carrier wave; more technically, we’re going to modulate the carrier wave with the audio information. In this case, “modulate” just means multiply:
fmod(t)= (1 + mfsignal (t)) fcarrier(t)
where fmod(t) is our modulated wave, fsignal (t) is our signal (i.e. that boring sine whistle), and fcarrier(t) is our carrier wave (which is another sine wave but at a much higher frequency (i.e. 21.12 kHz)). That m variable in the equation above is the modulation depth — it controls the extent to which we imprint our signal on the carrier. (I’ve set m to a value of 1.0 for this example but tweaked it a little for the demos further down this post.)
Here’s what a couple of milliseconds of our carrier wave looks like…
(You really only need to see a couple of milliseconds — it just repeats this cycle over and over.)
And here’s what our modulated wave, fmod(t), looks like over the same period of time:
The much more slowly varying 1 kHz frequency modulates the carrier, causing its amplitude to vary up and down.
Now here’s the especially neat trick that becomes so much easier to understand in Fourier space. What does the frequency spectrum of the modulated waveform look like? You might initially think that it contains two frequencies — the 1 kHz signal and the 21.12 kHz carrier.
Here’s what it looks like:
There are three frequency components! And there’s no sign at all of a peak at 1 kHz in the spectrum. Our carrier frequency is certainly still there, standing tall and proud at 21.12 kHz. But it’s accompanied by two neighbours, or sidebands as they’re more commonly known. So where has our audio signal gone?
Rest easy. All the audio information is still there. It’s in those sidebands. The lower frequency sideband is at 20.12 kHz (i.e. 21.12 – 1.00 kHz), while its “sibling” is at 22.12 kHz (i.e. 21.12 + 1.00 kHz). The origin of the sidebands can be understand on the basis of secondary/high school trigonometry: multiply two sine functions, sin(A) and sin(B) and we end up with two cosines, one with a frequency of A-B, the other with a frequency of A+B. [Don’t worry about the sine-cosine transformation. A sine is just a phase-shifted version of a cosine. They both sinusoids. See also Footnote 5]. Or, in slightly more mathematical terms:
sin(A) sin(B)=½ [cos(A-B)-cos(A+B)]
The audio information is embedded in the sidebands. We’ve just moved the 1 kHz audio information to a higher frequency (above the audible threshold of 20 kHz). A real world AM transmitter would transfer the audio it’s broadcasting (be it music, comedy, news, the abomination that is Country & Western, or reports of the latest Trump family gaffe) to sidebands of a carrier frequency over one hundred times higher than the 21.12 kHz used in the simulation. But that’s just a matter of scale. The fundamental maths and physics are exactly the same. 
Rectifying the situation.
So the audio information is still there. It’s just been transposed to a different frequency. How do we recover it?
Remarkably, in a real radio we can recover the audio, i.e. demodulate the signal, using a very basic electronic component: a diode. That’s what I used as a ten year old (because it’s what the Ladybird book recommended). But as that Sixty Symbols video up there shows, we can use an even more humble “component” — a ‘rusty’ penny! — to recover our audio. The metal-semiconductor junction — a Schottky barrier — formed by the tip of the wire touching on the oxidised surface of the penny is enough to allow electrons to flow one way but not the other.
And that rectifies the modulated waveform…
The signal is now only positive. (It’s very easy to simulate the action of a diode in code simply by removing all the negative parts of the waveform. )
Now usually at this point in explanations of AM radio and modulation/demodulation, there’s a long discussion about the role of low pass filtering, and RC time constants and the like on the signal. That is, of course, important in the context of filtering out frequencies we don’t want — and I’ve judiciously filtered out frequencies above 20 kHz in the demodulated examples in this post — but we don’t need all that to understand how the diode demodulates the audio information from the radio signal. We just need the wisdom of Fourier again. Here’s what the power spectrum of that rectified signal looks like:
And if you’d prefer to see the spectrogram, here it is:
The most important thing to realise here is that simply by rectifying the radio signal we’ve recovered our original audio signal! There it is, back at 1 kHz in both the power spectrum and the spectrogram. (There’s also a spike at 0 Hz and new components at, and around, twice and four times the carrier frequency.  But we can filter those out.) What is key is that the rectification process (i.e. the demodulation) has recovered our original signal from the modulated waveform.
Here’s what our demodulated waveform sounds like…
“…and echoes with the sounds”
Radio would be exceptionally boring if all that was broadcast were single sine waves. (Although it’s got to be said that BBC R4 around about 7:00 pm any weekday evening is even less interesting than that monotone whistle.) So let’s take a slightly more exciting audio sample…
This has a rather more complicated frequency spectrum, as one might imagine…
But if the modulation-demodulation process works for one frequency, it will work for all frequencies. Here’s that audio signal modulating a 42 kHz carrier wave…
And this is its spectrum:
It might not look like it at first glance but exactly the same transposition of the audio spectrum as we observed for that solitary sine wave has occurred. The audio spectrum has been shifted to upper and lower sidebands around the carrier frequency of 42 kHz.
Here’s what that modulated waveform sounds like:
No, there’s no error. If you didn’t hear anything when you clicked the “Play” button, that’s good. All of the frequency spectrum is outside the audible range.
Now we demodulate…
…and instead of the void in the 0 – 20 kHz frequency range that we had for the modulated spectrum, we’ve recovered the audio spectrum.
This is the demodulated signal:
Turn Up The Radio
So that’s the modulation-demodulation process captured in code. Now let’s add in a few more “radio stations” across our frequency spectrum…
You might be able to guess just how many stations there are.
The four sharp lines running horizontally across the spectrum above arise from the four different carrier frequencies I’ve chosen: 42 kHz, 55 kHz, 65 kHz, and 75 kHz. Now, ideally, I’d separate those carrier frequencies from each other by at least 20 kHz, to ensure no overlap when it comes to the ~ 20 kHz audio bandwidth, but I had to draw a compromise between processing time, avoiding artefacts due to aliasing , modulation/demodulation quality, and “fidelity” of sound reproduction. It turns out that the ~ 10 kHz separation of the “radio station” carrier frequencies is more than good enough to demonstrate the remaining principle we need to cover in order to understand how an AM radio works : tuning.
If we don’t have a mechanism for selecting a particular carrier frequency, all of the audio signal from each of the stations will overlap when we demodulate and all we’ll hear will be a chorus — or, more accurately, a cacophony — of sounds. In the crystal radio we use a tuned circuit, formed by a capacitor and inductor, to select different frequencies. As I describe in that Sixty Symbols video above (and the accompanying “extras” clip), the capacitor-inductor circuit has a particular resonance frequency, f0 , defined by a very simple equation:
f0 = 1/(2𝜋(√LC))
So, by changing the value of the capacitance, we can tune the resonant frequency of the circuit and thus select different parts of the spectrum. For example, let’s say we select a resonant frequency of 42 kHz. Our tuned circuit would select a band of frequencies centred on that 42 kHz resonance. (The width of that band depends, in turn, on something called the quality factor (Q-factor) of the resonance.) Something like this:
In the land of Mathematica we can use the simple BandpassFilter function to mimic the action of the LC circuit in the crystal radio. This video sketches out what happens when we vary the value of the centre of the bandpass filter:
Now we have a mechanism for selecting one radio station from another! If the stations are a little too close together in frequency (and/or our bandpass filter is not selective enough) then we can sometimes hear two overlapping signals from different stations.
And at this point we have all the elements in place to modulate, demodulate, tune in, and tune out. I’ll leave you with the video below, which shows the Mathematica radio simulation being put through its paces as the frequency is tuned across the various stations. As Autograph — those purveyors of quality 80s hair metal — put it so aptly, things go better with rock…
 This was my first foray into Mathematica. I’ve got to say a big thank you to Noah (Hardwicke) for his invaluable advice and expertise in all things Mathematica. I’m also immensely grateful to my colleague Paul Glover for a couple of very helpful conversations on modulation and demodulation (and the loan of Paul’s treasured Introduction To Communication Systems (FG Stremler).)
 For those who’d like a little more technical detail, the power spectrum is the modulus squared of the Fourier transform of the signal. (The Fourier transform is complex-valued — it has real and imaginary parts; the modulus squared is real-valued.) By plotting this, we’re throwing away all phase information. For example, the power spectrum of a 1 kHz sine wave is identical to the power spectrum of a 1 kHz cosine wave. Phase is a whole other can of worms in Fourier space that we can fortunately avoid when it comes to understanding amplitude modulation and demodulation.
I’m also only plotting a single-sided spectrum. No negative frequencies. For now.
 Very, very loosely speaking. Settle down, Fourier enthusiasts and sound engineers.
 Lyric from NOFX’s “Thank God, It’s Monday” .
 See  above. The power spectrum doesn’t distinguish between sines and cosines (or any other phase difference for that matter).
 An even more elegant way of thinking about the modulation process, for those of you with some background in Fourier analysis, is as follows. A multiplication in the time domain is the same as a convolution in Fourier space (and vice versa). I’ve not included negative frequencies in the graphs above but they’re there (because we can represent sines and cosines in terms of sums and differences of complex exponential functions). Here’s what we have for the spectra of our signal and carrier wave (excuse the scribble)…
That * symbol in the scribble above represents a convolution. When we convolve the frequency spectra for our carrier and signal waveforms (and include the carrier wave), this is the result:
Lo and behold, we have sidebands. Multiplication in time domain = convolution in frequency domain.
 In Mathematica, I used the Clip function to do this.
 If you’ve studied Fourier analysis, ask yourself why this is. (Note that the sample frequency I was using is 192 kHz and that means the highest frequency I can reliably observe in the spectra is 96 kHz.)
 As we’ll hear a little further down the post, it also leads to some authentic “bleeding” of the sound from one station into another — an effect that many crystal radio enthusiasts had to endure in days gone by.