Frequency response 

It is defined as the magnitude (gain), and phase differences between the input and the output sinusoids. To plot the frequency response, a vector of frequencies is created first (varying between zero or "DC" and infinity), and compute the value of the transfer function at those frequencies. If G(S) is the transfer function of a system and w is the frequency vector, the G(jw) vs w is then plotted. Since G(jw) is a complex number, both magnitude and phase responses  can be plotted (bode plots).

Bode Plots are the magnitude (gain) and phase response curves as functions of log(w) (frequency). Every system has a magnitude and phase response where they describe the system. Sketching Bode plots can be simplified because they can be approximated as a sequence of straight lines. Straight-line approximation simplify the evaluation of the magnitude and frequency response. Usually the gain in decibels, abbreviated dB, and the phase are plotted linearly along the y-axis on graph paper that has several cycles of a logarithmic scale on the x-axis. Each cycle represents a factor of 10 in frequency.

Cut-off frequency is the final point at which the filter response drops 3dB or to 0.707 of its peak value.

Decade is a logarithmic way of measuring the gain or loss. Decibels are defined as 20log10 of a voltage ratio (Vo/Vi), which is the division of the output voltage with the input voltage.

dB = 20 log (V o/Vi )        ,        (V o/Vi ) = 10  

figure E : magnitude response

figure F : phase response

The gain margin is found by using the phase plot to find the frequency WGM, where the phase angle is 180. This is shown in figure G below . By observing  the magnitude plot at this frequency, the Gain margin GM can be determined, which is the gain required to raise the magnitude curve to 0 dB, shown in yellow colour. 

The phase margin can be found by using the magnitude curve to find the frequency WPM, where the gain is 0 dB. By then looking on the phase curve at that frequency, the Phase margin PM, is the difference between the phase value and 180, shown in fuchsia colour. This way, every system characterised by bode plots as gain and phase plots, it's gain and phase margins can be found easily. 


figure G : gain & phase margins