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? – Nameless Jul 22 '19 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 Jul 22 '19 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... – Nameless Jul 23 '19 at 6:48