The concept of density functions

[Doug Kerr]

We learn that the color of light is determined by a property called the "spectral density function (SDF), which we hear described as telling us the distribution of the power in the light beam by wavelength. (It is, by the way, actually the "power spectral density function".)

The basic notion seems clear enough, but when we get into the details there are some wrinkles that can be baffling. For example, if we actually plot the SDF of come hypothetical light beam we see that the horizontal axis, labeled "wavelength", is calibrated in (typically) nanometers (nm). The vertical axis, perhaps labeled "spectral density", is calibrated in watts per nanometer. What does that mean?

This situation, spoken of as the concept of density functions, occurs in many fields, including colorimetry, electrical engineering, and statistics. I'll explain it with an example from electrical engineering.

We have, for some ongoing electrical "signal", a plot of its power spectral density (PSD) function (PSD), which we hear described as showing the distribution of power in the signal by frequency.

It is only an accident of history that this and the corresponding function for light have different common names and abbreviations. Both are really "power spectral density functions" (PSDF's, if you will).​

The horizontal axis, labeled "frequency", might be calibrated in Hertz. The vertical axis, labeled "power spectral density. might be calibrated in watts per hertz.

Looking at the curve at a frequency of 1000 Hz, we are tempted to think that the value of the curve there tells us how much power (or perhaps even what fraction of the total power of the signal) has a frequency of 1000 Hz. But in fact, in the usual case, the amount of power at (exactly) 1000 Hz is zero!

We can perhaps best come to grips with this startling fact by moving for a moment into a different field, statistics. Our concern is with the distribution of the height of the World's human population. We might ask, after all the data is in and analyzed, "what fraction of the Worlds' human population has a height of 6 feet (72")? Stated that way, we have no choice but to interpret the question as meaning, "what fraction of the Worlds' human population has a height of exactly 6 feet (72")"?

The answer is that no human has a height of exactly 72". Putting aside the obvious impracticality of making precise measurements of stature (this is after all only a blackboard exercise), suppose one of our data gatherers says, "Yes, there is one fellow in Boston. We measure his height, and find it is 72.000000000000000000001". Indeed, no human will have a height of exactly 72".

More rigorously, we say that the probability of anyone having a height of exactly 72" is zero; that is not exactly the same as saying it could never happen. The difference is too subtle to bother with here.​

Now moving back to our electrical example, suppose, using a spectrum analyzer of very high resolution we determine that over the range of frequencies from:

999.000 Hz through 1000.001 Hz

(a range of 0.002 Hz), there is 5.0 mW of power.

Next, we determine that over the range of frequencies from:

999.9995 Hz through 1000.0005 Hz

(a range of 0.001 Hz), there is 2.5 mW of power.

Thus we see that, in this locality of frequency, at least, the power in any (very small) range of frequencies is 2.5 times the frequency span; that is, the power spectral density (in the vicinity of 1000 Hz) is 2.5 watts/hertz. If we use smaller and smaller ranges, this ratio holds (we say it "holds in the limit as the range of frequencies approaches zero").

Thus, at 1000 Hz, we can say unequivocally that the power spectral density is 2.5 W/Hz. And that is what is plotted on the curve at 1000 Hz.

But note that as we allow out "test" range to decrease, the actual amount of power in the range decreases, approaching zero in the limit (when the range becomes zero). Only the ratio of power to frequency range approaches a non-zero limit.

Now, we determine that over the range of frequencies from:

1000.9995 Hz through 1001.0005 Hz

(a range of 0.001 Hz), there is 2.51 mW of power.

Proceeding as before, we see that around a frequency of 1000.1 Hz, the spectral power density is 2.51 W/Hz (and that ratio holds as we shrink the range, keeping it centered about 1001 Hz). Thus, in our graph, for a frequency of 1001 Hz, we plot on the curve the value 2.51 W/Hz.

But how much power is there at a frequency of 1000 Hz? Zero, because "at 1000 Hz" implies a region of zero width, and the amount of power that can embrace is zero.

So, considering a plot of the spectral density function (SDF) of a certain light, how much of the total power of the light is at, for example, a wavelength of 550 nm? None.

Best regards,

Doug

 

[Doug Kerr]

In a density function, the area under the curve between two values of the independent variable (e.g., wavelength or frequency) tells us the total amount of "stuff" that lies between those two values. In the case of power spectral density functions (both the SDF in colorimetry and the PSD in electrical engineering - no different other than an arbitrary choice of a different abbreviation!), the "stuff" is power.

Thus, if we have the PSD curve for an audio signal, the area under that curve between the limits of 300 Hz and 15,000 Hz is the total amount of power of the components of the signal whose frequencies lie within that range.

But, in electrical engineering, we encounter one important density function in which the area cannot be related to any physical kind of "stuff" - and that can be troublesome.

Suppose we knew the waveform of an electrical signal (its "time-domain" representation, such as we would see on an oscilloscope), and we wanted to know its "frequency spectrum" (its frequency-domain representation). If we take the Fourier transform of the waveform, we get a function known as the amplitude spectral density function (and I will continue to use its full name here).

Amplitude, in electrical engineering, refers to the "height" of a waveform, as for example quantified in units of volts.​

The horizontal scale of a plot of this function is frequency, typically denominated in hertz. The vertical scale is amplitude spectral density, and is typically denominated in volts per hertz.

Now, what does this mean in "graphical" terms? If we take the area under this curve from, say, 20 Hz to 15,000 Hz, does that tell us the total voltage of the components of the signal within that frequency range? It seems that it should, but it does not - that "total voltage" turns out to not be a meaningful concept.

In fact, before we can do any meaningful graphic manipulation of our amplitude spectral density function, we must square it (that is, take the square of its value at each frequency), which gives us the power spectral density function.

Power is proportional to the square of voltage, or amplitude, for any given impedance of the circuit. Since in these exercises we are typically concerned only with relative values (as they vary with frequency), it is common to say that power is the square of voltage (thus arbitrarily assuming an impedance of one ohm, as good as any other in this context).​

So what is the meaning of the amplitude spectral density function? It has these meanings:

• It is the Fourier transform of a waveform.
• It is the square root of the power spectral density function of the waveform.

Much anguish is caused in electrical engineering by attempts to interpret areas under the curve of the amplitude spectral density function.

In photometry and colorimetry, we usually do not have the opportunity to get into this difficulty.

Best regards,

Doug

 

[Jeremy Waller]

Hello Doug,

This forum, via your tremendous efforts, really gets down to the nitty gritty !!

Yes the psd's do cause concern for some in some areas of communications. Normally one would define the psd as the fourier transform of the auto-correlation function for using this definition it is easier to apply statistical analysis. Where problems arise is in interpreting the performance (BER) of comms systems where one defines the term (Eb / N0) where N0 is the noise psd in unit bandwidth and Eb is the energy (per bit in joules).

Interestingly (See: posting in LL on diffraction referenced in Barts experiment) the diamerer of the Airy disc was given as a Bessel function (first kind) now this may be well approximated by a sinc squared function (ie. ( sine(t) / t ) **2 ) which is much kinder to deal with. Now the PSD of a binary phase shift keyed wave form is also a sinc squared function and these "dashed" bessel functions appear in FM waveforms - Just need to escape from this stuff . LoL.

Now you know why I took up nature photography.

Kind Regards,

Jeremy.