Two echoes (in blue) come back with delays of 3 and 5 seconds and amplitudes equal to 0.5 and 0.3 times the amplitude of the transmitted pulse, respectively; these are just random values for the sake of the example. Since the signal is real, the intercorrelation is weighted by an additional factor. If two pulses come back (nearly) at the same time, the intercorrelation is equal to the sum of the intercorrelations of the two elementary signals. To distinguish one "triangular" envelope from that of the other pulse, it is clearly visible that the times of arrival of the two pulses must be separated by at least formula_3 so that the maxima of both pulses can be separated. If this condition is not met, both triangles will be mixed together and impossible to separate. Since the distance travelled by a wave during formula_3 is formula_24 (where "c" is the speed of the wave in the medium), and since this distance corresponds to a round-trip time, we get:  The instantaneous power of the transmitted pulse is formula_25. The energy put into that signal is:  Similarly, the energy in the received pulse is formula_27. If formula_28 is the standard deviation of the noise, the signal-to-noise ratio (SNR) at the receiver is:  The SNR is proportional to pulse duration formula_30, if other parameters are held constant. This introduces a tradeoff: increasing formula_30 improves the SNR, but reduces the resolution, and vice versa. How can one have a large enough pulse (to still have a good SNR at the receiver) without poor resolution? This is where pulse compression enters the picture. The basic principle is the following:  In radar or sonar applications, linear chirps are the most typically used signals to achieve pulse compression. The pulse being of finite length, the amplitude is a rectangle function. If the transmitted signal has a duration formula_3, begins at formula_33 and linearly sweeps the frequency band formula_34 centered on carrier formula_35, it can be written:  The chirp definition above means that the phase of the chirped signal (that is, the argument of the complex exponential), is the quadratic:  thus the instantaneous frequency is (by definition):  which is the intended linear ramp going from formula_39 at formula_33 to formula_41 at formula_42. The relation of phase to frequency is often used in the other direction, starting with the desired formula_43 and writing the chirp phase via the integration of frequency:  As for the "simple" pulse, let us compute the cross-correlation between the transmitted and the received signal.