# What's a good technique to understand the differences between the time and frequency domains?

I understand that phonetic analysis can be performed in various domains, among which: time, frequency, cepstral, and spectral. In the time domain, one studies e.g. Rate, energy, and duration. In the frequency domain one studies e.g. f0. This makes logical sense -- but how am I to conceive of these two domains as being independent of one another? I'm looking for more than a mathematical explanation (such as the one that rightly distinguishes cepstral from spectral) - but rather an intellectual framing to scaffold my interpretation of "domain". Thanks!

• They are not independent... they just are some way to categorize things. Commented Jun 9, 2015 at 15:41
• But how can I conceptualize the difference between the domains? Commented Jun 10, 2015 at 7:08
• Please try to be more specific in your question. Right now it seems to me you are saying "I don't understand it, please explain it to me". But you already know something on the topic, so please edit your Q, add what you already know, and point out specific issues that are unclear to you. Commented Jun 10, 2015 at 11:06

There are two "domains", time and frequency. If you are using seconds as the units of measurement, the analysis is in the time domain. If the unit or measurement if Hz, the analysis is in the frequency domain. You are not supposed to analyze one domain as independent of another, indeed "frequency" can only be defined by reference to a unit of time (it recurs this number of times within a unit of time). Cepstral and spectral are not different domains.

To understand this in a non-mathematical way: Think of "sapmle", whether read or heard. You are likely to understand its meaning from context, because the order of the symbols doesn't matter much as long as the symbols are all there.

When talking about the frequency domain, you simply take a sample that spans a certain length of time, derive the frequency spectrum (by magic, if you will) and throw away the temporal information. The word may become difficult to comprehend without that information, because there's little difference between "my" and "I'm". Hence, you might want an infinitely small window, down to a point, so that you can look at the frequency domain for every point individually, in sequence. Such a sequence can be plotted over time. If you chose an alphabetic character representation, that looks more or less just like writing does. With sound, you care about the volume (attenuation, velocity) of each individual frequency (or frequency bands, lumping ranges of close frequencies into a single bucket).

Practically this becomes quite complicated, but the terms wavelength, frequency, sample-frequency (or sample-size) and band-width are need-to-know. frequency is the inverse of wavelength [Hz]=[1/s]. The thing with the bandwidth is rather fundamental, it comes up in the sampling-theorem. You need to watch a signal for twice the length of the wavelength you are interested in (that is, low pitch has a longer wavelength). That's an irrelevant technical detail perhaps. But consequently, you need to know your sampling frequency. You cannot record an infinite amount of sample points, so you need to look at points in certain intervals that are shorter than half the wavelength of the highest frequency that you are interested in. You need enough samples to cover the wavelength of the lowest frequency.

The beauty of it is that periodic signals (like a long "aaaaaaaaaaa") can be freely transformed between time and frequency domain. The two domains are not "independent of one another", as you put it. It's only a matter of representation. For non-periodic sounds, we can still say that the signal can be approximated by a sum of periodic waves across the spectrum. A bell's gong e.g. has a very low fundamental frequency that determines how long it takes to quiet down (I guess?). Obviously, a bigger bell has a darker tone and takes longer to quiet down because you have to put in more energy to get it to the same average velocity. We can't hear frequencies below or above a certain frequency (ca 20Hz to 20 kHz), but surely we can hear volume change with lower frequencies (hence called low-frequency-oscillation). This is easier to see in a 2D diagram plotting velocity over time and easier to think about in terms of multiplication rather than summation of wave forms (which gives an effect called enveloping see). To catch a glide (like in "eye"), as a beginner, you need a spectrogram across time, most likely.

Another thing to look for is phase shift, which is another way to describe shifts in frequency eg. known from the Doppler shift effect (e.g. the changing sound of a passing ambulance's siren).

You can test a lot of this for good fun with musical synthesizers, filters and a spectrograph. That's important when cleaning up speech recordings, not so much when analyzing, except if you want to filter for a certain phoneme, or synthesize speech.

You can think the frequency as the number of some same event happen in a time. So frequency is a number in the domain of time, generally this number is a qualitative quantity. Examples of frequency: few, rare, never, once, etc...

So it is not an independent domain of time, it is more like a binding of the domain of time and the domain of quantities.

The "reverse" to frequency is number of time at some event happen, i.e, how many time happen (not events, just time) at the point some event happen.

And generally this "number of time at some event happen" is what sometime we call just "time", but a "pure time" dont rely in any quantity, it is just a dimension of reality (e.g., the concepts past, future or present dont rely on quantities). Generally time and space are mixed in many domains, you never see it "pure" or "alone".

So generally when you talk about time, but not ever, you are counting time at some point where something happen (5 minutes, 2 days, etc.) And when you talk of frequency you are counting the number of times some same event happen in a period (period: some quantity of time, e.g. I take a shower 1-2 times per day, I drink about 5 coffees per day, etc...)

Im not linguistic, Idk anything about these kind of domains that you are talking about... but the general sense for frequency, or time, is this. Mathematically speaking frequency is just that: number of times some event happen in a fixed quantity of time.

Since human perception is based on frequency analysis, as an aid to the intuition, you can just forget about time domain analysis.