How exactly is the SAR image resolution increased by the linear frequency modulation of the transmitted signal and further by the compression of range and azimuth?
In the comment on your main post you've specified that your question in particular is about range compression, so I'll address that here:
Here's a whole bunch of words that explains the motivation of why we might want to use pulse compression:
What it boils down to is that when the radar receives a target signal return of say a point target source (ideal case), the signal we measure at the receiver is a time delayed copy of the signal the radar transmitted. Neglecting noise and other disturbances, one could simply measure a change in amplitude, and mark the "start" of when the signal was seen by the receiver.
Now in the real world, and this is especially true in SAR imagery, the target returns are numerous, and the scattering points are complex. Within a single receive window, there might be many returns. In order for the receive to be able to "see" the target returns, long pulsewidths are often used as well to get more energy on target.
This creates an interesting problem, where in order to "see" a target better (assuming fixed transmitter gain), one could use any number of linear transforms to try to discern a signal. A very basic idea would be to transmit a sinusoid, and use a DFT to detect its presence. I say this is basic, because in using a DFT, we trade our knowledge of the time domain for knowledge in the frequency domain. Keep in mind that for radar, where the signal "starts" in the time domain correlates directly with its position in range, so that's information we want to preserve.
So, in light of that, we seek an alternative transform (preferably linear) that we can use such that we can easily tell when a signal "starts" in the time domain. Enter the concept of pulse compression. Wikipedia has a great long winded explanation of it, but here's some simple Octave/MATLAB code that might clue you in on why pulse compression is really important in SAR image formation:
% generate an LFM and a tone f1 = -10; f2 = 10; t = linspace(0,1,500); xc = exp(1i*pi*(f2-f1)*t.^2); % LFM xf = exp(1i*pi*f1*t); % tone % calculate the matched filter output for the LFM [pc_c,lags] = xcorr(xc) % plot the results figure(1); hold off; plot(lags,20*log10(abs((pc_c)))); hold on; plot(lags,20*log10(abs(xcorr(xf)))) ylabel('Power (dB)'); xlabel('Range Bin'); xlim([-500 500]); grid on; legend('Matched Filtered LFM','Matched Filtered Sinusoid'); title('Comparison of Pulse Compressed Waveforms')
This code produces the plot show below. Note that this is a simple example and I didn't time delay the signals at all (both signals will appear at the zero range bin). One can see that after pulse compression, the LFM produces a fairly sharp peak with these settings in comparison to just using a pure sinusoidal signal. If we just use a sinusoidal signal, we get a fairly amorphous blob as an output.
Now continuing on with this example, lets add in a second target (see the figure below this paragraph). In this example, lets say one target comes into the receiver at the first sample, and the next target comes into the receiver at the 100th sample. Using pulse compression with an LFM, we can easily observe the two targets; if we try to use a sinusoidal signal, we get our red blob again, and we cannot reliably measure the time of arrival of the signals using the sinusoidal signals.
Thus, pulse compression gets you a way to precisely measure time of arrival of signals, even if there are many time delayed signals that overlap in time/frequency by varying amounts. Now with your SAR image, this translates direct to resolution in the range domain: without pulse compression, you would have very little if not no resolution in the range domain. Hope this helps!