Doug Kerr
Well-known member
The color of light is determined by its spectrum, for which the more precise technical name is power spectral density (PSD). In the usual meaning in color science this is a function of wavelength (we can think of it in its graphical presentation as a curve plotted against wavelength) that describes how the power contained by the light is distributed over different wavelengths.
In electrical engineering, the concept of power spectral density is also important. There, we most commonly speak of a function of frequency that describes how the power contained by a "signal" is distributed over different frequencies.
We can plot the PSD of something either in terms of wavelength or frequency, as best fits our context. Either will describe the same situation.
Wavelength is of course directly related to frequency, thus:
L=C/f
where L is the wavelength, f is the frequency, and C is the speed of light.
Thus we might think that the two kinds of PSD (lets speak of them as their graphic plots) differ only in the meaning of the scale of the y-axis: whether that is in terms of frequency or wavelength.
But in fact there is a far greater difference.
The word "density" in the name power spectral density refers to the concept of a density function. Lets consider a PSD of the frequency flavor. There, the ordinate (vertical value) does not tell us how much power there is at the specific frequency. In fact, assuming a "continuous" spectrum, there is zero power at any specific frequency.
If this seems startling, let me give an analogy that may help. Suppose we have an aluminum bar of circular cross section but tapering diameter. Now, imagine a plane through the rod, perpendicular to its axis at some particularly place along its length, say exactly 2 inches from some reference point. How much aluminum (in volume) is there in the rod at that place? Zero - the volume of an area on a plane is zero, so in the part of the plane that is within the rod, there is zero aluminum.
Now while we have this analogy in front of us, let's proceed. Suppose that at the point of interest the cross-sectional area of the bar is 0.001 m². Consider a "slice" of the bar of a very tiny thickness - we will call the thickness dx. Actually, this slice is a cylinder of average cross-section area 0.001 m² and length dx. Thus, its volume is 0.001 dx m³.
Said another way, the volumetric density of the rod at this point is 0.001 m³ per inch of length (assuming a very tiny length - the famous "dx" of the calculus, so we need not be concerned with the cross-sectional area changing).
Now, back to our PSDs.
We often speak of a uniform PSD (a "flat" curve of the function). For such, there is a tendency to think that the abscissa can be frequency or wavelength, whichever best fits our context, with no effect on the "shape" of the curve, given that it is flat. But far from that.
If we start with a power spectral density, in terms of frequency, that is uniform, this means that the amount of power per unit frequency is constant over the range of frequency of relevance.
Now consider plotting the same power spectral density as a function of wavelength. Now, if the amount of power per unit frequency is constant, the amount of power per unit wavelength is not, because of the nonlinear relationship between wavelength and frequency (L=C/f).
Consider a radio spectrum running from 100 Mhz (L=3.00 m) through 200 Mhz (L=1.50 m). If the power spectral density on a frequency basis is uniform with frequency, the the power in the range from 99.5-100.5 Mhz (an interval of 1.0 Mhz) is the same as that in the range from 199.5-200.5 Hhz (also an interval of 1.0 Mhz).
But the power in the range of wavelength 2.995 - 3.005 m (an interval of +0.01 m, but an interval of frequency of -0.333 Mhz) is not the same as the amount of power in the range 1.495 - 1.505 m (also an interval of -0.01 m, but -1.333 Mhz, ). It is in fact essentially 1/4 as great.
Thus the plot, on a wavelength basis, of a power spectral density that (on a frequency basis) is uniform will be a curve that declines with increasing wavelength - it is in fact an "inverse square" curve, as can be readily demonstrated with a little work with (the) calculus.
Thus, we need to be very cautious when we begin to work mathematically with a power spectral density.
In complicated theoretical work on color spaces this distinction needs to be carefully attended to.
Best regards,
Doug
In electrical engineering, the concept of power spectral density is also important. There, we most commonly speak of a function of frequency that describes how the power contained by a "signal" is distributed over different frequencies.
We can plot the PSD of something either in terms of wavelength or frequency, as best fits our context. Either will describe the same situation.
Wavelength is of course directly related to frequency, thus:
L=C/f
where L is the wavelength, f is the frequency, and C is the speed of light.
Thus we might think that the two kinds of PSD (lets speak of them as their graphic plots) differ only in the meaning of the scale of the y-axis: whether that is in terms of frequency or wavelength.
But in fact there is a far greater difference.
The word "density" in the name power spectral density refers to the concept of a density function. Lets consider a PSD of the frequency flavor. There, the ordinate (vertical value) does not tell us how much power there is at the specific frequency. In fact, assuming a "continuous" spectrum, there is zero power at any specific frequency.
If this seems startling, let me give an analogy that may help. Suppose we have an aluminum bar of circular cross section but tapering diameter. Now, imagine a plane through the rod, perpendicular to its axis at some particularly place along its length, say exactly 2 inches from some reference point. How much aluminum (in volume) is there in the rod at that place? Zero - the volume of an area on a plane is zero, so in the part of the plane that is within the rod, there is zero aluminum.
Now while we have this analogy in front of us, let's proceed. Suppose that at the point of interest the cross-sectional area of the bar is 0.001 m². Consider a "slice" of the bar of a very tiny thickness - we will call the thickness dx. Actually, this slice is a cylinder of average cross-section area 0.001 m² and length dx. Thus, its volume is 0.001 dx m³.
Said another way, the volumetric density of the rod at this point is 0.001 m³ per inch of length (assuming a very tiny length - the famous "dx" of the calculus, so we need not be concerned with the cross-sectional area changing).
Now, back to our PSDs.
We often speak of a uniform PSD (a "flat" curve of the function). For such, there is a tendency to think that the abscissa can be frequency or wavelength, whichever best fits our context, with no effect on the "shape" of the curve, given that it is flat. But far from that.
If we start with a power spectral density, in terms of frequency, that is uniform, this means that the amount of power per unit frequency is constant over the range of frequency of relevance.
Now consider plotting the same power spectral density as a function of wavelength. Now, if the amount of power per unit frequency is constant, the amount of power per unit wavelength is not, because of the nonlinear relationship between wavelength and frequency (L=C/f).
Consider a radio spectrum running from 100 Mhz (L=3.00 m) through 200 Mhz (L=1.50 m). If the power spectral density on a frequency basis is uniform with frequency, the the power in the range from 99.5-100.5 Mhz (an interval of 1.0 Mhz) is the same as that in the range from 199.5-200.5 Hhz (also an interval of 1.0 Mhz).
But the power in the range of wavelength 2.995 - 3.005 m (an interval of +0.01 m, but an interval of frequency of -0.333 Mhz) is not the same as the amount of power in the range 1.495 - 1.505 m (also an interval of -0.01 m, but -1.333 Mhz, ). It is in fact essentially 1/4 as great.
Thus the plot, on a wavelength basis, of a power spectral density that (on a frequency basis) is uniform will be a curve that declines with increasing wavelength - it is in fact an "inverse square" curve, as can be readily demonstrated with a little work with (the) calculus.
Thus, we need to be very cautious when we begin to work mathematically with a power spectral density.
In complicated theoretical work on color spaces this distinction needs to be carefully attended to.
Best regards,
Doug