@WMXZ
Hehehe - completely off-topic
, but I accept the challenge, we will meet at dawn tomorrow behind the barn and settle this thing once and for all! You've got this ancient academic on one of his favorite topics.
First, the units of the FFT are categorically not of spectral density! The units are simply amplitude. Sorry about that chief, but that is just the way it is - the DFT/FFT is simply a sum of numbers (amplitudes) multiplied by sines and cosines (which are dimensionless). As I point out below, however, the units of the continuous Fourier Transform are amplitude density...
The lines in the FFT have zero width - you couldn't squeeze anything into a bin there if you tried, and there is a complete nothingness between them. They are simply a set of complex numerical coefficients in a complex polynomial. Its like me taking a polynomial like
y = 3 +1x +5x^2 + 7x^3 -2x^4
and describing the set of coefficients that describe the polynomial, ie [3 1 5 7 -2], as "bins". In the DFT/FFT case the lines values are the numerical coefficients of terms of the form exp(-j.2pi.m.n/N). Same argument.
I have attached a handout from an old course I taught for many years, to provide a more mathematical basis:
View attachment SamplingDFT.pdf
and I will just outline my argument here:
1) We start in the continuous time domain, with the Fourier Series representation of periodic functions as a summation of discrete frequency (harmonically related) sinusoids, producing a line spectrum where each line has no width - it is purely a discrete spectrum, and nothing exists between the lines. Its units are also amplitude (again, as I will point out below, there is a very
strong relationship between the Fourier Series and the DFT/FFT)
2) The Fourier Transform is a spectral representation of a finite duration (aperiodic, or one-shot) continuous function f(t), and is derived by considering the behavior in the limit of the Fourier Series of a periodic-extension of f(t) as the period goes to infinity. In other words we assume that f(t) is in fact periodic, but the period is so long that we have not yet come to the end of the first period! The lines become more and more closely packed together, and in the limit the line spacing becomes zero and we end up with a continuous function of frequency.
The units of the Fourier Transform are of an "amplitude-density", such as volts/Hz.
2) Now we move to the discrete-time domain and the Discrete-Fourier-Transform DFT (of which the FFT is just an algorithmic implementation).
A sampled waveform is modeled mathematically by assuming that the continuous waveform f(t) has been multiplied by a periodic train of Dirac delta functions (which are infinitely narrow pulses of infinite amplitude, but with an area of unity, and integrated (see the attachment). This process eliminates ALL knowledge of the waveform f(t) except AT the sampling instants. Once we have the set of samples, the concept of continuous time doesn't exist any more, all we have is a bunch of numbers.
The DFTs (forward and inverse) are mathematical relationships between two sets of discrete complex numbers; continuous time and frequency do not enter into the transforms. But wait - there's more - there's an implicit assumption that each of the two data sets each are a single period of a PERIODIC pair of data sets, and when we say that {F_m} (M = 0...N)
F_m = sum(n = 0 to N-1) f_n exp(-j.2pi.n.m/N)
is the DFT of the sequence {f_n}, we are saying that it can be considered as a sum of complex exponentials - just as we imply when we use the Fourier Series representation of a periodic continuous f(t).
Indeed, another way of interpreting the DFT is as the Fourier Series of a periodic-extension of f(t). As I said at the outset the line spectrum of a Fourier Series has no interpretation of bins and/or line width. You must be very careful when you relate the DFT back and forth to the continuous world that (most of us) live in.
So where does the concept of "bins" in the DFT come from? Maybe somehow related to the concept of reconstruction of a bandlimited waveform between samples from its sample set using the sinc function (Whitakker or cardinal, see the attachment) interpolation formula?
My concept of "binning" is the grouping of nearby values into a common container, such as when constructing a histogram, or estimating a probalility-density-function (pdf). In my 40+ years of teaching this stuff I saw many, many outright errors and misunderstandings created by attempting to "bin" results from DFT analysis. I prefer to steer away from the term as far as I can!
See you out behind the barn at dawn tomorrow
