Over at Quanta Magazine [Shalma Wegsman] asks What Is the Fourier Transform?

[Shalma] begins by telling you a little about Joseph Fourier, the French mathematician with an interest in heat propagation who founded the field of harmonic analysis in the early 1800s.

Fourier’s basic insight was that you can represent everything as a sum of very basic oscillations, where the basic oscillations are sine or cosine functions with certain parameters. [Shalma] explains that the biology of our ear can do a similar thing by picking the various notes out from a tune which is heard, but mathematicians and programmers work without the benefit of evolved resonant hairs and bone, they work with math and code.

[Shalma] explains how frequency components can be discovered by trial and error, multiplying candidate frequencies with the original function to see if there are large peaks, indicating the frequency is a component, or if the variations average to zero, indicating the frequency is not a component. [Shalma] tells how even square waves can be modeled with an infinite set of frequencies known as the Fourier series.

Taking a look at higher-dimensional problems [Shalma] mentions how Fourier transforms can be used for graphical compression by dropping the high frequency detail which our eyes can barely perceive anyway. [Shalma] gives us a fascinating look at the 64 graphical building blocks which can be combined to create any possible 8×8 image.

[Shalma] then mentions James Cooley and John Tukey and the development of the Fast Fourier Transform in the 1960s. This mathematical tool has been employed to study the tides, to detect gravitational waves, to develop radar and magnetic resonance imaging, and to support signal processing and data compression. Even quantum mechanics finds use for harmonic analysis, and [Shalma] explains how it relates to the uncertainty principle. The Fourier transform has spread through pure mathematics and into number theory, too.

[Shalma] closes with a quote from Charles Fefferman: “If people didn’t know about the Fourier transform, I don’t know what percent of math would then disappear, but it would be a big percent.”

If you’re interested in the Fourier transform and want to dive deeper we would encourage you to read The Fastest Fourier Transform In The West and Even Faster Fourier Transforms On The Raspbery Pi Zero.

Header image: Joseph Fourier, Attributed to Pierre-Claude Gautherot, Public domain.

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The PAX D177 PoS terminal helpfully tells you which tamper points got triggered. (Credit: Lucas Teske)
These days Points of Sale (PoS) usually include a digital payment terminal of some description, some of which are positively small, such as the Mini PoS terminals that PAX sells. Of course, since it has a CPU and a screen it must be hacked to run something else, and maybe discover something fun about the hardware in the process. Thus [Lucas Tuske] set out to do exactly this with a PAX D177 PoS, starting with purchasing three units: one to tear apart, one to bypass tamper protections on and one to keep as intact reference.

As expected, there are a few tamper protections in place, starting with pads that detect when the back cover is removed and a PCB that’s densely covered in fine traces to prevent sneaky drilling. Although tripping the tamper protections does not seem to affect the contents of the Flash, the firmware is signed. Furthermore the secrets like keys that are stored in NVRAM are purged, rendering the device effectively useless to any attacker.

The SoC that forms the brains of the whole operations is the relatively obscure MH1903, which is made by MegaHunt and comes in a dizzying number of variants that are found in applications like these PoS terminals. Fortunately the same SoC is also found on a development board with the AIR105 MCU that turns out to feature the same MH1903 core. These are ARM Cortex-M3 cores, which makes targeting them somewhat easier.

Rather than try to break the secure boot of the existing SoC, [Lucas] opted to replace the SoC package with a brand new one, which was its own adventure. Although one could say that this is cheating, it made getting a PoC of custom code running on one of these devices significantly easier. In a foll0w-up article [Lucas] expects to have Doom running on this device before long.

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Building a computer on a breadboard is a seminal project for many builders, but it can become complicated quite quickly, not to mention that all the parts needed for a computer are being placed on a medium which often lends itself to loose wires and other hardware bugs. [3DSage] has a working breadboard computer that is as simple as it can possibly be, putting it together piece by piece to show exactly what’s needed to get a computer which can count, access memory, and even perform basic mathematical operations.

The first step for any computer is to build a clock, and in this case it’s being provided by a 555 timer which is configured to provide an adjustable time standard and which steps through the clock pulses when a button is pressed. The next piece is a four-bit counter and a memory chip, which lets the computer read and write data. A set of DIP switches allows a user to write data to memory, and by using the last three bits of the data as opcodes, the computer can reset, halt, and jump to various points in a simple program.

Although these three chips make it possible to perform basic programming, [3DSage] takes this a bit further in his video by demonstrating some other simple programs, such as one which can play music or behave as an alarm clock. He also shows how to use a fourth chip in the form of a binary adder to perform some basic math, and then packages it all into a retro-styled computer kit. Of course you can take these principles and build them out as far as they will go, like this full 8-bit computer built on a breadboard or even this breadboard computer that hosts a 486.

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Federal attacks on free speech are rightfully getting most of the headlines, but Georgia’s unprecedented attempt to prosecute dozens of “Stop Cop City” protesters as members of a racketeering enterprise was as frivolous as the worst of the Trump administration’s antics.

This week’s dismissal of conspiracy charges against these opponents of the Atlanta police training facility is welcome news. But it’s alarming that these charges lingered over 61 people’s heads for two years. And their ordeal likely isn’t over.

Ridiculous indictment

Rather than filing charges aimed at individual alleged acts of arson, vandalism, or other actual crimes, prosecutors tried to rope together dozens of activists into one sprawling case under the Racketeer Influenced and Corrupt Organizations Act. The indictment didn’t even attempt to connect the dots between most defendants’ constitutionally protected speech and a handful of defendants’ alleged bad acts.

Why would prosecutors want to do that instead of just charging the lawbreakers? Primarily, because the prosecution wasn’t intended to punish illegality but to kill a movement. Charging only those who broke actual laws wouldn’t achieve that aim, so they got to work making up new ones.

Secondarily, because they’re vindictive dirtbags. To prove it, they chose the date of George Floyd’s murder as the start date of their fabricated “conspiracy,” even though plans for Cop City hadn’t even been announced yet when Floyd was killed.

Prosecutors pointed to the defendants’ alleged anarchist political philosophy, their publications, their social media posts, their constitutionally protected recording of police, their use of encryption and VPNs to communicate securely, and even their holding news conferences and talking to reporters — all as “proof” that they were each part of a criminal enterprise.

The theory goes that, if one anarchist writes an essay about the environmental impact of Cop City and another anarchist sets a police car on fire, then both anarchists effectively struck the match together. Or something like that. It’s too incoherent to make any sense of.

Chilling effect on both journalists and sources

By the prosecution’s logic, journalists who covered the protests could be portrayed as helping to “advance” a RICO conspiracy simply by giving activists a platform. Write about the movement, and you (and your sources) might become part of the case file. This time, they only went after writers who participated in the Stop Cop City movement, but what about next time?

The harm extends beyond direct impact on reporters. Newsmakers and whistleblowers will surely be wary of talking to journalists if they know prosecutors view doing so — and particularly doing so securely via encrypted messaging — as an act in furtherance of a conspiracy.

This week’s win doesn’t undo the harm, either to free speech or to the defendants’ lives (many say they’ve been unable to find housing or jobs due to the long-pending case). The court’s forthcoming dismissal order seems likely to be limited to procedural grounds — prosecutors neglected to get required authorization from the governor to bring the charges. Deputy Attorney General John Fowler, the lead prosecutor on the case, said he’ll appeal.

Prosecutors must know they’re unlikely to ultimately secure a conviction, much less one that higher courts will uphold. Even if the appellate courts give them a path to get past this week’s procedural dismissal, they’ll still have to reckon with the Constitution and the fact that their legal and factual theories are as flimsy as they come.

But the longer they can stretch the case out, the longer uncertainty will linger about whether a law intended to counter organized crime can be distorted to stifle protest movements. And the more reluctant critics of Cop City will be to speak to journalists whose reporting might clarify the goals of their movement and contradict law enforcement narratives.

Plenty of experts have explained how RICO laws are rife with abuse, even when used for their intended purpose. The Cop City case makes clear that the need for reform isn’t just a criminal justice issue; it’s a free speech one as well.

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Dear Friend of Press Freedom,

For 171 days, Rümeysa Öztürk has faced deportation by the United States government for writing an op-ed it didn’t like, and for 90 days, Mario Guevara has been imprisoned for covering a protest.

Our newsletters are taking a short break next week, but we’ll be back. Explore our archive for more press freedom stories.

Take action to stop attacks on journalists and transparency

People often ask how to support the causes we fight for other than donating. Freedom of the Press Foundation (FPF) has a new action center to allow you to do just that by writing or calling members of Congress to tell them that the public cares about press freedom.

This week we launched the first two actions — one to tell lawmakers to stop the decimation of the Freedom of Information Act and the other to encourage them to do something (besides the same old empty expressions of concern) about the massacre of journalists in Gaza. We plan to expand the action center, including by taking on locally targeted initiatives. Please use this easy, new tool to help us make a difference — and still donate if you can, of course.

CBS’s real bias monitor is Donald Trump

CBS announced this week that it’s hiring Kenneth Weinstein — a career partisan — as its “bias ombudsman.” FPF Director of Advocacy Seth Stern wrote for The Guardian that while CBS has a constitutional right to monitor bias as it sees fit, it’s a different story when the government gets involved.

The creation of the ombudsman role was one of many capitulations CBS’s owners made to the Trump administration to persuade the Federal Communications Commission to approve the Paramount-Skydance merger. And based on comments from the administration, it fully intends to use its foothold inside CBS’s newsroom to bend its reporting to its liking. Read more here.

‘Cop City’ case’s threat to press freedom persists despite win

Federal attacks on free speech are rightfully getting most of the headlines, but Georgia’s unprecedented attempt to prosecute dozens of “Stop Cop City” protesters as members of a racketeering enterprise was as frivolous as the worst of the Trump administration’s antics.

This week’s dismissal of conspiracy charges against these opponents of the Atlanta police training facility is welcome news. But it’s alarming that these charges lingered over 61 people’s heads for two years. Their ordeal likely isn’t over, and neither is the chilling effect on journalism and whistleblowing that the case has caused. Read more here.

Will 9/11 records be declassified?

Next year marks the 25th anniversary of the Sept. 11 attacks, and “a substantial body of records” about the event remains classified.

In theory, these records should automatically be declassified when they turn 25 in 2026. But this won’t happen with the 9/11 records, just as it hasn’t happened with other historically significant records, like the John F. Kennedy assassination records, which the public had to wait over 60 years to read.

It doesn’t have to be this way. Read more in The Classifieds.

Judge isn’t buying excuses for attacks on LA reporters

A group of journalists and others won a preliminary injunction in federal court on Sept. 11, 2025, placing new restrictions on the Los Angeles Police Department’s violent tactics while policing protests. The day before, they won a similar order against the Department of Homeland Security.

U.S. District Judge Hernán Vera outlined the decades-long history of press freedom abuses at protests in LA and said that “the First Amendment demands better.” We couldn’t agree more. Read more from FPF’s U.S. Press Freedom Tracker.

What we’re reading

Investigative reporter told to stop contacting police in California city (U.S. Press Freedom Tracker). You’d think police would’ve learned to stop messing with journalist Ben Camacho after the City of Los Angeles had to cut him a check for wasting his time with a frivolous lawsuit for publishing public records that they gave him.

China didn’t want you to see this video of Xi and Putin. So Reuters deleted it (The Intercept). When news outlets don’t resist censorship, “journalism’s independence “sinks to the lowest common denominator whenever news of global importance breaks in a country governed by a repressive regime,” Stern told The Intercept.

A tipping point at CBS News (Columbia Journalism Review). An alarming account of corporate meddling in CBS News’ editorial decisions, under new and old ownership alike, to appease the Trump administration and avoid criticizing Israel. And with CBS’s ownership now looking to buy CNN’s parent company, it’s unlikely to stop.

Cindy Cohn is leaving EFF, but not the fight for digital rights (WIRED). A giant in the civil liberties field! We all owe a debt of gratitude to Cindy Cohn for her indelible leadership at the Electronic Frontier Foundation for so many years (and for being our legal counsel at FPF since our founding in 2012).

Kansas prisons reject newspaper subscriptions, blindsiding publishers and cutting off information (Kansas Reflector). We can’t wait to hear Kansas prison officials explain how newspaper subscriptions threaten incarcerated people’s safety.

Boos of Donald Trump heard on ABC’s broadcast of US Open. Good (USA Today). Shame on the U.S. Tennis Association for demanding broadcasters to censor crowd reactions to Trump’s attendance of the U.S. Open men’s tennis final in New York on Sunday.

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