What are bode plots?

In this section, we're going to describe what a frequency response diagram is, as well as take a look at their primary characteristics. By definition, if we fit in a pure sinusoidal tone into a generic linear time-invariant system, the output will also be a pure sinusoidal tone. Notice that a linear time-invariant system can affect the amplitude of the signal and can introduce a shift in the phase of the sine wave. But it will not alter its fundamental frequency.

Comparing the input and the output sinusoidals, we can measure the change in amplitude and phase shift introduced by the system. If each of these individual measurements is plotted on its own axis against the corresponding pure tone frequency of the sine wave, it will give us a single point on a frequency response diagram.

By repeating this process for a number of different pure tone frequencies, we can construct a trace for the amplitude variation, also known as the gain of the system, as well as a trace for the phase shift introduced by the system for any given frequency range. This type of frequency response diagram is what is normally called a bode plot.

Notice that in order to capture all the relevant dynamics, we need to make sure that we excite our system at enough frequency points to get a good trace. This is why when estimating responses empirically or experimentally people usually use random white noise or sinusoidal chirps with enough frequency content as excitations.

Now, in general, if we need an analytic way to calculate this frequency responses directly from the dynamic equations of the system, in the Laplace domain, setting the operator s to a pure, imaginary number like jw is equivalent to exciting the system with a pure sinusoidal tone or frequency w.

The resulting expression for G will be a vector in the complex plane-- part real, part imaginary. The magnitude of that vector is the gain of the transfer function or, in other words, equivalent to the amplitude gain from the input to output sine waves. And the phase angle of that vector corresponds to the phase shift introduced by the system. Both of this will be functions of the frequency w.

As a quick note, frequency diagrams are normally drawn on logarithmic scales. The magnitude is usually displayed in decibels, which is defined as 20 times the log of the amplitude ratio. And the phase angle is usually displayed in degrees. To create a frequency response diagram, we need to evaluate both of these functions at different values of w in the frequency range of interest.

Of course, in a numerical engine like MATLAB, we can easily define our transfer functions directly on the s domain and call the desired built-in function to automatically perform this evaluation and create the appropriate frequency diagram, in this case, bode plot. Before I proceed, I want to define at least some of the standard terminology we will be using for describing the characteristics of our frequency response plot just to make sure that we are all on the same page.

First, when we talk about DC gain, we are referring to the magnitude gain at zero frequency, or DC. Next, roll-off rate refers to the slope at which the magnitude drops off at high frequency. This is usually expressed in dBs per decade, meaning a factor of 10 on the frequency scale. Any peak in the magnitude trace in general will be associated to some natural frequency of the system.

And any magnitude value above 0 dBs means that the system is amplifying the input signal. Remember that 0 dBs corresponds to the log of 1, which will happen when the amplitude of the output matches the amplitude of the input. So the magnitude ratio is 1. And any value below 0 dBs means that the system is attenuation the input signal. The point where that change happens is defined as the crossover frequency.

The last important characteristic I wanted to bring up is what is called bandwidth frequency. This is usually defined at the point where that magnitude crosses -3 dBs. -3 dBs means that the attenuation is going below the RMS value of the input signal where output to input amplitude ratio is (2^0.5)/2.

From our practical perspective, the bandwidth of a system is commonly used as a measure of the upper limit at which our system dynamics can be controlled. Any inputs past that frequency will be severely attenuated by those system dynamics and will barely be seen in the output.