Pulse compression is a signal processing technique commonly used by radar, sonar and echography to either increase the range resolution when pulse length is constrained or increase the signal to noise ratio when the peak power and the bandwidth (or equivalently range resolution) of the transmitted signal are constrained. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse.
Simple pulse
Signal description The ideal model for the simplest, and historically first type of signals a pulse radar or sonar can transmit is a truncated sinusoidal pulse (also called a CW—carrier wave—pulse), of amplitude A {\displaystyle A} and carrier frequency, f 0 {\displaystyle f_{0}} , truncated by a rectangular function of width, T {\displaystyle T} . The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse, s {\displaystyle s} . If we assume the pulse to start at time t = 0 {\displaystyle t=0} , the signal can be written the following way, using the complex notation:
s ( t ) = { e 2 i π f 0 t if 0 ≤ t < T 0 otherwise {\displaystyle s(t)={\begin{cases}e^{2i\pi f_{0}t}&{\text{if}}\;0\leq t<T\\0&{\text{otherwise}}\end{cases}}}
Range resolution Let us determine the range resolution which can be obtained with such a signal. The return signal, written r ( t ) {\displaystyle r(t)} , is an attenuated and time-shifted copy of the original transmitted signal (in reality, Doppler effect can play a role too, but this is not important here). There is also noise in the incoming signal, both on the imaginary and the real channel. The noise is assumed to be band-limited, that is to have frequencies only in [ f 0 − Δ f / 2 , f 0 + Δ f / 2 ] {\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]} (this generally holds in reality, where a bandpass filter is generally used as one of the first stages in the reception chain); we write N ( t ) {\displaystyle N(t)} to denote that noise. To detect the incoming signal, a matched filter is commonly used. This method is optimal when a known signal is to be detected among additive noise having a normal distribution. In other words, the cross-correlation of the received signal with the transmitted signal is computed. This is achieved by convolving the incoming signal with a conjugated and time-reversed version of the transmitted signal. This operation can be done either in software or with hardware. We write ⟨ s , r ⟩ ( t ) {\displaystyle \langle s,r\rangle (t)} for this cross-correlation. We have:
⟨ s , r ⟩ ( t ) = ∫ t ′ = 0 + ∞ s ⋆ ( t ′ ) r ( t + t ′ ) d t ′ {\displaystyle \langle s,r\rangle (t)=\int _{t'\,=\,0}^{+\infty }s^{\star }(t')r(t+t')dt'}
If the reflected signal comes back to the receiver at time t r {\displaystyle t_{r}} and is attenuated by factor A {\displaystyle A} , this yields:
… excerpt ends here. Continue reading the full article.
