Friis Transmission Equation for Wideband Transmission

Question

it's known the following Friis transmission equation:

$$P_{rx} = P_{tx} \cdot G_{tx}(f) \cdot G_{rx}(f) \cdot \left(\frac{c}{4 \pi R f}\right)^2$$

It's nice but, as far as I'm concerned, it works only for a sine wave of frequency f or, with a still good approximation, for narrowband transmission (bandwidth much lower than the carrier frequency).

Now, I'd want to find a similar equation for wideband transmissions, where the transmitted signal spectrum, here called BW (Bandwidth) is not much lower than the carrier frequency. If I were to evaluate it, I'd apply the Friis transmission equation for just the little amount of power around the generic frequency f of my spectrum BW:

$$S_{rx}(f)\cdot df = S_{tx}(f) \cdot G_{tx}(f) \cdot G_{rx}(f) \cdot \left(\frac{c}{4 \pi R f}\right)^2\cdot df$$

where S denotes the Power Spectral Density. Now, I'd integrate inside my domain BW:

$$P_{rx}=\int_{f_{carrier}-BW/2}^{f_{carrier}+BW/2} S_{tx}(f) \cdot G_{tx}(f) \cdot G_{rx}(f) \cdot \left(\frac{c}{4 \pi R f}\right)^2\cdot df$$

Now, my questions are:

1. Is my approach correct? Do you agree with this result?

2. Now let's prove why for a narrowband spectrum such an equation should turn to the common Friis Transmission Equation. If BW is small as compared to the carrier frequency, the integrand can be assumed to be constant. That's a consequence of the first order Taylor Approximation of the Antiderivative (here the proof). This means that:

$$P_{rx}= S_{tx}(f_{carrier}) \cdot G_{tx}(f_{carrier}) \cdot G_{rx}(f_{carrier}) \cdot \left(\frac{c}{4 \pi R f_{carrier}}\right)^2\cdot \int_{f_{carrier}-BW/2}^{f_{carrier}+BW/2}df$$

which becomes:

$$P_{rx}= S_{tx}(f_{carrier}) \cdot G_{tx}(f_{carrier}) \cdot G_{rx}(f_{carrier}) \cdot \left(\frac{c}{4 \pi R f_{carrier}}\right)^2\cdot BW$$

If we put:

$$S_{tx}(f_{carrier})\cdot BW = P_{tx}(f_{carrier})$$

we get:

$$P_{rx}= P_{tx}(f_{carrier}) \cdot G_{tx}(f_{carrier}) \cdot G_{rx}(f_{carrier}) \cdot \left(\frac{c}{4 \pi R f_{carrier}}\right)^2$$

Do you still agree with the last result?

Answer 1

The Friis equation, like most everything else in engineering, is a linear approximation. I think what you're trying to do here is meaningless from an engineering point of view, because this is a non-real situation. However, consider the following issues

1. The Transmit Antenna gain will vary in both frequency and directionality, so you need to factor in the exact vector direction to the receive antenna.
2. Same for the Receive Antenna.
3. The input impedance of your antenna will vary across the infinite bandwidth you're proposing, which will effect the antenna gain, so that will have to be factored in.
4. Your transmitter output power will vary across the bandwidth into a perfect load, and will vary even more when the nonideal antenna input impedance is taken into account.
5. The same effect of the Receiver input impedance varying across the band, effecting the received signal.
6. The Friis equation also neglects atmospheric effects (humidity, rain, molecular absorption) so there's an atmospheric attenuation that should be taken into account (The ITU has a nice standard for that)

There's probably a couple of other issues I've neglected.

Trying to reduce all those variations to a single number is pointless from a link budget point of view, there's too many variables.