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Old January 2nd, 2018, 12:45 PM
Doug Kerr Doug Kerr is offline
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Join Date: May 2006
Location: Alamogordo, New Mexico, USA
Posts: 8,398

Here are some graphs that should help illustrate the points that I made in my earlier note.

Here we see the plot of the PSD (power spectral density) function of an illustrative (and wholly hypothetical) instance of an electrical signal, presented on a "wavelength" basis.
I chose to cast this as an electrical signal and not electromagnetic radiation so as to dodge the matter of the different "potency" metrics of electromagnetic radiation, such as watts per steradian, watts per steradian per square meter, watts per square meter, and so forth.

The x-axis is wavelength (lower-case lambda), denominated in meters. The range over which our signal contains any power is 100-300 m. Amateur radio enthusiasts and students of the history of radio broadcasting will recognize the neighborhood here: 200 m (a frequency of 1500 kHz) is considered the boundary between the top end of the "broadcast band" (today, we would say the "AM broadcast band") and "short wave". The range 100-300 m then corresponds to the frequency range of 3000-1000 kHz (stating the limits in the matching order)

We see that the SPD (wavelength basis) of this arbitrary, hypothetical signal is constant at 20 mw/m from 100-200 m, and at 30 mw/m from 200-300 m.
If we take the area under the curve, we find that the total power in this signal would be 500 mW (5 W).
In the next figure, we have shown the PSD of this same signal on a frequency basis:

We now see that (as expected) the range of frequency within which the signal contains any power is 1000-3000 kHz (now stating the limits in normal order). The vertical axis now shows the PSD of the signal (on a frequency basis) denominated in mW/kHz.

We see two important differences from the wavelength-basis plot:

The point at which the PSD changes abruptly, which was exactly halfway in the range as shown in terms of wavelength, is at only one quarter through of the range in terms of frequency. (This is the same spot in terms of signal frequency, or wavelength.)

We see that the PSD curve, which was flat in each region when using the wavelength basis, now declines over each region as the frequency increases.

By the way, if we were to reckon the area under this curve, we would find that it of course also would indicate that the total power in the signal is 5000 mW.

Quod erat demonstrandum.

Best regards,

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