# Tag Archives: Fourier transform

## cepstrum, quefrency, rahmonic

“By applying a low-pass lifter to the cepstrum in Figure 2 to extract the low quefrency components below the first rahmonic peak, the slowly varying curve (in red, upper graph) results.”

I read that to my wife and her eyes turned into a pair of shirred eggs. She was, for a time, speechless – a condition that, incidentally, the process described in the quotation would have been helpless in the face of.

Make no mistake: what Al Oppenheim and Ron Schafer are describing in their article (From frequency to quefrency: a history of the cepstrum, Signal Processing Magazine, IEEE (September 2004), 21 (5), 95–106) is freakin’ hard for most people to wrap their minds around. But while it might seem as dry as dust to you, that passage actually evinces a fundamental fact of true nerds: a sense of humour and playfulness.

There are four words in there that you need to look at: lifter, cepstrum, quefrency, and rahmonic. They are terms that apply to this specific mathematical process. The process itself is a little quirky, and applies to things that themselves require a bit of explanation to have real meaning – a bit more than I have space for here. But here’s a very short run-down – if your eyes start to glaze, skip to the paragraph that starts “So anyways.”

Sounds such as human speech are actually very complex, made up of a lot of different harmonic resonances on top of a basic sound frequency. It’s these resonances that allow people to discern the difference between different speech sounds: the position of your tongue in your mouth (among other things) changes the shape of resonating chambers and makes certain bunches of harmonics, called formants, stronger – you might say the formants are the informants of what speech sound you’re hearing.

When linguists – acoustic phoneticians in particular – and engineers and physicists analyze sound waves, they use a wonderful mathematical function called a Fourier transform to identify the different resonance frequencies in the sound waves, what is called the spectrum, a perfectly appropriate term since the spectrum of light is also the different frequencies. (Think about if someone were tapping 9 beats a second and someone else 12 beats a second and someone else 36 beats a second. If you graphed the sound waves, you would have something looking like :,..;..,:.,.:,..;..,:.,.:,.. and on and on. A Fourier transform would just show a graph plotting frequencies with one mark at 9 per second, one at 12 per second, and one at 36 per second.)

Well, if you treat the Fourier transform graph as though it were a graph of sound waves and perform a Fourier transform on it (it’s just slightly twickier than that, but that’s the general concept), you are performing a curious but useful inversion. You can identify how close together the harmonics are, and how close together the formants are; it tells you how frequent the strong frequencies are on the graph, so to speak. Believe me when I say this is useful, and not just in speech analysis: it makes cleaning up the sound on old recordings a lot easier, for instance – you can filter out unwanted resonances from the original sound-capture device.

So anyways, when you do this process, you get something that looks like a spectrum but is really a spectrum looked at the other way around, and you get what looks like frequency but is really frequency looked at the other way, and harmonics that aren’t actually harmonics, and you can apply filtering processes on the data that aren’t filters like the normal data filters are. You’re treating frequency as though it were time and time as though it were frequency.

So what do you do? You come up with new words for what you’re talking about. And if you’re a nerd, you may take this opportunity to be a little playful. (Businessmen would use wanton sesquipedalianisms and initialisms to try to sound impressive. Nerds don’t feel a need to try to sound impressive because they actually know what they’re talking about.)

That playfulness actually tells us some interesting things about language, too: not the way we perceive sounds (which is what the data that all this analyzes help us to understand), but the way we think of and group sounds and how we perceive the structure of words. You see, the guys who came up with this – Bogert, Healy, and Tukey, three engineers back in the early 1960s – wanted to signify the inversion by inverting the words. But you will notice they only inverted parts of the words, in order to maintain comprehension I suppose – in the process producing pseudomorphemes (I’ll explain, hold on) – and they did it in some particular ways:

spectrum –> cepstrum
frequency –> quefrency
filter –> lifter
harmonic –> rahmonic

In all of the words, they only inverted the first part of the word, thereby treating the front end of the word as the significant part and the remainder as a sort of tail (a common enough things for people to do – go to SoHo and ask JLo), and also treating them as separate bits of the word, like tweet plus ed in tweeted – meaning-bearing building-blocks called morphemes. Except that trum, ency, ter, and monic actually are not morphemes; they have no meanings of their own – they’re just phonological divisions.

And the way they inverted the first half is notable: in three of the four, they just reversed the letters in the first syllable, which in all cases also reversed the sounds (you should know from this that the original pronunciation of cepstrum is with a /k/ at the start). It was always the syllable, not any other division: not rtcepsum or nomrahic, which would be morphologically appropriate but phonologically and orthographically problematic. As usual, the sound patterns of the words guide how they’re treated – when you turn it around, it’s the sound you’re turning around (this is standard in most playful things we do with words, and it’s how we treat helicopter as heli plus copter rather than the original helico plus pter, and why we say a whole nother thing, and also why people asked to say my backwards will probably say “I’m” – reversing the phonemes – rather than like “yam” – reversing the actual sounds).

In the other word, that wasn’t possible – /rf/ and /wk/ aren’t acceptable syllable onsets. So the syllable onsets, /fr/ and /kw/, were simply swapped to make quefrency. The vowel sounds were not swapped: it’s just not comfortable in English to say /’kwε frin si/. But when you look at that word on paper, do you want to pronounce it with a “long e” on the first syllable? To me, thanks to other words starting with que, it looks first like the que is said like the one in question, making both vowels /ε/ and conforming the word to expected sound patterns rather than to the original sounds.

But at least quefrency looks like a swapping-up of frequency. When I first saw cepstrum, I didn’t see spectrum in it at all (obviously I wasn’t swirling and sniffing it at that point). It looked more like it was just some other Latin word I hadn’t seen, joining the long list of neuter nouns like rostrum and plectrum. And rahmonic, aside from making me think of Rahm Emanuel (and maybe rah-rah-rah), had a taste of rampike and mnemonic but took a moment to show its harmonic resonance. (Lifter happens to be an English word in its own right, and thereby carries unbidden resonances. Ironically.)

However, the resemblance of cepstrum to spectrum is not lost on those who are expecting to see spectrum. And the hazards of such wordplay showed up in an early publication by Oppenheim and Schafer on the topic – and make for a cautionary tale for editors and authors alike. I’ll quote directly from the same article I started with:

throughout the various stages of proofreading of this book, we constantly had to maintain vigilance to be certain that this “strange” term cepstrum wasn’t inadvertently “corrected” to what seemed to be more appropriate. . . . We breathed a sigh of relief when the last page proofs were returned to the publisher. When the first printing of the book appeared, it was clear that a particularly diligent proofreader at the publisher had caught the “error” at the last instant and cepstrum had been reversed to spectrum throughout.

Well, not entirely reversed – but run through a transformation aimed at making the strange look normal again. Ah, but too late – and sometimes you want to see things strangely.

Thanks to Colleen Kavanagh (@CanuckWordNerd) for drawing my attention to this whole sandbox of words.