GeekBleep: Q’s Quandaries #2: What is 20dB/decade?
Andrew Pullin
· 3 years ago
First, you need to know what a decibel is. The Bell is a unit named after Alexander garahm bell. It is a relative intensity, like if I said "A plane is twice as loud as a car" ; we are talking about how many times louder something is. The factor of 2 is unitless. A decibell is 1/10 of a Bell, thats all. The Bell scale is as easy as 2x, 3x, 4x though. Instead, it follows whats called a "Logarithmic" scale, so 2x is equivalent to 3db, 4x to 6db, 8x to 9db, etc. Its also neat because it exchanges multiplication for addition. That is, if our base signal that we are comparing to is at 10db, then a signal that is 2x as big is 10 + 3 = 13 decibels. The "per decade" part is where it gets a little confusing. Usually it is in reference to another input. "Per decade" means that if the input gets 10 times as big, you gain 20 decibels on your output signal, or it becomes roughly 100x as big. So, say we are thinking about our guitar amplifier. If it was indeed 20 db per decade, that means if we turn it from a volume setting of 1 up to 10, then our output volume will be 100 times louder.
Maybe thats a little complicated still.
ridiculous_fish
· 3 years ago
Here's my understanding:
First, note that the little 'd' - the prefix 'deci' - means "tenths," just like...well, ok, there aren't any other units in wide use that have tenths. But that's what it means. So 20 tenths-of-a-bel per decade: multiply the 20 by the 1/10 and you get 2 bels per decade.
Now, decade can mean 10 years, but it can also mean powers of 10 - to go up a decade is to multiply the previous value by 10. So if the value is, say, 42, then that's log_10 42 decades.
Bel is also a base 10 logarithm! So 2 bels / decade means 2 * (log x / log y). Use the magical properties of logarithms, and you get log x^2 / log y. Exponentiating - changing your coordinate system to linear - gives you x^2 / y.
So saying something goes up "20 dB/dec," just means that y = x^2: the value you're calculating goes as the square of the value you're varying.
Presbywhoscwhatsit? Hey Davak, I'm an engineering geek, not a medical geek. Oh, okay, I'm a medical geek, too. But still no. And you already have a shirt!
No simple answer yet. I'm going to give it some time before I give any hints. Oh, okay, maybe some analysis. Andrew, breaking each part down is a great way to understand the whole. Rediculous_fish, you are also close, but something happened in the next to the last paragraph..
kftgr
· 3 years ago
It's basically a slope. If you have decibels on the y-axis and a powers of 10 logarithmic scale (decades) on the x-axis, then you have db/dec.
Now for the specific 20 dB/decade, you have a 100 fold increase in y (e.g., power) for every 10 fold increase in x (e.g., frequency).
qmchenry
· 3 years ago
Interesting.. with two answers relating to a factor of 100 for 20dB, I did some investigation. According to the Wikipedia entry on Decibels, they are calculated in two ways depending on the use. For power or intensity, it is 10*log(x/x0) but for amplitudes, it is 20*log(x/x0). The entry doesn't elaborate on the why and I learned long ago not to ask that.
In my training (electrical and biomedical engineering), we always used the 20*log(x/x0) version. Anyone who has studied linear systems and has had fun with high and low pass filters, etc., has seen 20dB/dec over and over again. For example, here's a graph of a low-pass filter frequency response which would have been a great thing to add to the initial post of the quandary.
kftgr, you are correct that it is a slope. We're using different definitions of decibel, that's all. I'm thinking your answer is shirt-worthy. Based on what I've said, does anyone have the 3 or 4 word answer I'm looking for?
F
· 3 years ago
The slope of f(x)=x.
George
· 3 years ago
Oh, man! I totally just got it. I don't know how long I've been doing transfer functions, but it never clicked for me.
"slope of 1"
Totally genius. Although I feel so stupid that I've been using this for so long.
qmchenry
· 3 years ago
Those are the exact words I was looking for: "a slope of 1" -- very well done. Actually, F has the correct answer, too (although it took me a minute to realize it). The slope of f(x) = x is 1. Three shirts go out again! kftgr, F, and George! Thanks for playing!
PhysicsGeek
· 2 years ago
I was starting to get the whole explanation, but then got lost when "F" said f(x)=x and slope is one. I don't see that???? I feel like its right there, but I can't relate the 20 dB/dec to a slope of 1???
All Comments
A decibell is 1/10 of a Bell, thats all. The Bell scale is as easy as 2x, 3x, 4x though. Instead, it follows whats called a "Logarithmic" scale, so 2x is equivalent to 3db, 4x to 6db, 8x to 9db, etc. Its also neat because it exchanges multiplication for addition. That is, if our base signal that we are comparing to is at 10db, then a signal that is 2x as big is 10 + 3 = 13 decibels.
The "per decade" part is where it gets a little confusing.
Usually it is in reference to another input. "Per decade" means that if the input gets 10 times as big, you gain 20 decibels on your output signal, or it becomes roughly 100x as big.
So, say we are thinking about our guitar amplifier. If it was indeed 20 db per decade, that means if we turn it from a volume setting of 1 up to 10, then our output volume will be 100 times louder.
Maybe thats a little complicated still.
First, note that the little 'd' - the prefix 'deci' - means "tenths," just like...well, ok, there aren't any other units in wide use that have tenths. But that's what it means. So 20 tenths-of-a-bel per decade: multiply the 20 by the 1/10 and you get 2 bels per decade.
Now, decade can mean 10 years, but it can also mean powers of 10 - to go up a decade is to multiply the previous value by 10. So if the value is, say, 42, then that's log_10 42 decades.
Bel is also a base 10 logarithm! So 2 bels / decade means 2 * (log x / log y). Use the magical properties of logarithms, and you get log x^2 / log y. Exponentiating - changing your coordinate system to linear - gives you x^2 / y.
So saying something goes up "20 dB/dec," just means that y = x^2: the value you're calculating goes as the square of the value you're varying.
:)
No simple answer yet. I'm going to give it some time before I give any hints. Oh, okay, maybe some analysis. Andrew, breaking each part down is a great way to understand the whole. Rediculous_fish, you are also close, but something happened in the next to the last paragraph..
Now for the specific 20 dB/decade, you have a 100 fold increase in y (e.g., power) for every 10 fold increase in x (e.g., frequency).
In my training (electrical and biomedical engineering), we always used the 20*log(x/x0) version. Anyone who has studied linear systems and has had fun with high and low pass filters, etc., has seen 20dB/dec over and over again. For example, here's a graph of a low-pass filter frequency response which would have been a great thing to add to the initial post of the quandary.
kftgr, you are correct that it is a slope. We're using different definitions of decibel, that's all. I'm thinking your answer is shirt-worthy. Based on what I've said, does anyone have the 3 or 4 word answer I'm looking for?
"slope of 1"
Totally genius. Although I feel so stupid that I've been using this for so long.