In the book Skolnik, Introduction to Radar Systems, ed. 2, pag. 52 (sec. Transmitter Power) the radar equation in its simplest form is initially cited:

$$R_{\max}=\left[\frac{P_t G A_e \sigma }{(4 \pi )^2 S_{\min}} \right]^{1/4}$$

where $$P_t$$ is the transmitted power (spatial integral of the poynting vector generated by the transmitting antenna in far field), $$G$$ is the receiving antenna gain, $$A_e$$ is the equivalent area of the receiving antenna, $$\sigma$$ is the radar cross section of the target, $$S_{\min}$$ is the minimum receiving power that make possibile to detect that particular target located at the maximum distance of $$R_{\max}$$.

Subsequently, the author says: The average radar power $$P_{av}$$ is also of interest in radar and is defined as the average transmitter power over the pulse-repetition period. If the transmitted waveform is a train of rectangular pulses of width $$\tau$$ and pulse-repetition period $$T_p=\frac{1}{f_p}$$, the average power is related to the peak power by:

$$P_{av}=\frac{P_t\cdot \tau}{T_p}$$

and then immediately he rewrites the radar equation replacing $$P_t$$ with the expression $$\frac{P_{av}\cdot T_p}{\tau}=\frac{E_t}{\tau}$$ (he calls $$E_t$$ the transmitted energy).

I can't understand why, certainly apparently (but unfortunately I don't see where this 'apparently' is), it seems that it has mixed time and frequency as if they were interchangeable. In fact, according to me, if we want to write explicitly the radar equation, even in its simple form like the one above, it becomes:

$$R_{\max}=\left[\frac{P_t(\omega) G(\theta, \phi,\omega) A_e(\theta, \phi,\omega) \sigma(\theta, \phi,\theta_b,\phi_b,\omega) }{(4 \pi )^2 S_{\min}(\theta, \phi,\theta_b,\phi_b,\omega)} \right]^{1/4}=R_{\max}(\theta, \phi,\theta_b,\phi_b,\omega)$$

where $$\theta,\phi$$ is the direction in which the target is, $$\theta_b,\phi_b$$ is the orientation of the target with respect to the antennas, $$\omega$$ is the angular frequency of the EM wave.

Following what was written by Skolnik I would therefore have that:

$$P_{av}(\omega):=\frac{P_t(\omega)\cdot \tau}{T_p}$$

which to me means nothing, no kind of 'average value'. The only way I know of linking the temporal average of instantaneous power with complex power is as follows:

$$<\mathcal{P}(t)>=\int_{\omega=-\infty}^{\infty}P(\omega)\mathrm{d}\omega$$

Does anyone know how to justify what the author said?

The radar equation assumes you are pointing the radar at the target, so you don't need any of the thetas or phis. Everything about the antenna pattern that differs from isotropic is captured in the directionality, and when that is combined with electrical efficiency (losses), it is call the gain, $$G$$.

There seems to be some confusion about power and frequency. You do not want to analyze this as a complex impedance in a circuit with oscillating current and voltage, rather, it's just how much power is the radar putting out when it is on (the peak power), and then what is the average power, which is reduced by the duty cycle. The duty cycle is the fraction of time the thing is transmitting: $$\tau/T_p$$.

Those are all just hardware concerns. There is a further step when the signal is "sophisticated". The measure of that is the pulse length times the bandwidth:

$$s = \tau \times \Delta f \ge 1$$

where unity is an unsophisticated signal, e.g. a square wave envelope at the carrier frequency. Since the hardware may limit the peak transmit power, you can chirp (frequency sweep) or phase-code the signal, and then de-chirp it or convolve it with inverse phase-code, thereby compressing the power into a peak in the time-domain.

• Pointing the radar at the target, we have: $$R_{\max}=\left[\frac{P_t(\omega) G(\theta) A_e(\omega) \sigma(\theta_b,\phi_b,\omega) }{(4 \pi )^2 S_{\min}(\theta_b,\phi_b,\omega)} \right]^{1/4}=R_{\max}(\theta_b,\phi_b,\omega)$$ so how do you pass from this equation in $P(\omega)$ to an equation in $P(t)$? About this... There seems to be some confusion about power and frequency: power is a function of frequency in the domain in which radar equation is defined. Do you agree? Commented Jul 22, 2019 at 6:33
• I do not agree. The radar equation is for designing radars, and with pulsed radars it is on or it is off, so there is a peak power and an average power, at the frequency of the radar. Of course there is signal bandwidth, but that is not considered in the radar equation. Time domain vs frequency domain considerations are for later, once you have radar data to analyze.
– JEB
Commented Jul 22, 2019 at 18:19
• Gain is not defined only when radar is on or when it's off. Gain is defined and is different for every frequency, and if the radar works with impulses and not in CW, the total signal in time (repetition of impulses) has not a single frequency, and so the gain in radar equation is not a single number. Same things for equivalent area, RCS, transmitted power, and so on... Commented Jul 23, 2019 at 6:48

You are right for the extreme cases of "wideband" radar. Normally a radar antenna and its receiver is a rather narrowband device in the sense that both its directivity and its noise figure is essentially independent of the frequency over its instantaneous bandwidth. This was especially the case the time Skolnik wrote his books. Having an airborne radar in the X-band with a 100MHz instantaneous bandwidth has a relative bandwidth of $$~0.1/10 =1\%$$ and that is nothing for modern electronics.

Regarding the antenna, that is a more complicated issue, especially if it is frequency hopped over say ~2GHz. While it will not matter much to a reflector antenna be it fixed or mechanically scanned, an electronically scanned phased array maybe subject to frequency scanning that will distort the antenna pattern. There was an enormous effort put in these electronically scanned phased arrays to make their antenna pattern to be nearly frequency independent. On the other hand, some of the very first phased arrays, already in the WWII took advantage of their natural frequency scanning but hat idea has gone out of practice as the science of it has improved.

An especially interesting issue is the frequency sensitivity of the scattering cross section. No real target is an isotropic point reflector. While during the detection stage any frequency sensitivity is a curse as it reduces the minimum return and increases its variance most of the time, some very sophisticated target recognition techniques can take advantage of that sensitivity, resonances and what not, characteristics of the specific target itself that otherwise would be rather confusing to the radar processor.

Further complication in a wideband radar is that the received noise can be strongly frequency dependent and by noise here I mean man-made noise, especially intentional jamming. None of that is captured in Skolnik's formula but your frequency dependence indicates its potential significance. Especially sensitive to frequency variation can be the antenna sidebands and nulls that are used to suppress such jamming. And I have not even touched the problem of a well-matched high power amplifier flat over a wide bandwidth...