|Frequency Representation||Fourier Series / Transforms||Background Material|
Once we start thinking in terms of frequency space (frequency space and time space “look” exactly the same, they just have a different unit on the ‘x-axis’), we can do things in frequency space like we would do in time space. Certain “actions” in the frequency space will have their implications in the time space, and vice versa (since they are related by an integral).
Introduction to Bode plots
Graphs that plot frequency versus gain are commonly known as Bode magnitude plots. Usually these graphs are plotted on a semilog plot meaning one of the graph's axes is on a logarithmic scale and the other is not. Although the magnitude (or gain) is not plotted on a logarithmic scale, it is not plotted in its normal units either. The magnitude is plotted in units of decibels [dB]. Bode phase plots are also useful in giving us how much phase shift occurs at particular frequencies. Bode plots are particularly useful in allowing us to understand how a particular control system or filter reacts with a in a range of frequencies just like a time domain graph allows us to see values of our signal of interest in a range of time intervals.
Figure 1. Sample Bode plot graph (Acquired from http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Bode/BodePaper1.gif)
We can apply any of the filters discussed on this page to our signal simply by multiplying the frequency response of the filter to the Fourier transform of our signal, and taking the inverse transform.
How to plot Bode plots given the transfer function:
A transfer function is usually given in terms of where s = jω. The values of 's' that set the numerator to zero are known as the 'zeroes,' and the values of 's' that set the denominator to zero are known as the 'poles.' Follow the instructions in this pdf document to learn how to draw a Bode plot.
Here is a sample MATLAB program that has several examples of plotting transfer functions. Download it here (right click and "Save link as ...", should show up as Bode_plot.m), and type "Bode_plot" at the MATLAB prompt to run it. Make sure the directory you are in is the one where Bode_plot.m is stored. Email the TAs if you have any questions.
Low-Pass Filter (LPF)
A low-pass filter is a transfer function (H(s)) such that when it is convolved to an input signal, it removes high frequency content so that only low frequencies “pass” through. It a frequency response of:
This is an example of an active analog low pass filter. If we have an arbitrary signal that we would like to apply a low pass filter to, we would multiply the low pass filter frequency response (shown in the figure above) to the frequency response of the signal. Drawing back on our knowledge of the multiplication rule, what would this correspond to in the time domain? Convolution! Thus, if we only had our signal in time domain, we could convolve our arbitrary time signal to the inverse Fourier transform of the low pass filter transfer function to filter out the high frequencies. Lets see if we can analyze a simple resistor-capacitor (RC) network and understand the fundamental theory behind a low pass filter.
Impedance Review: Electrical Impedance is a measure of opposition to time-varying electric current. A circuit element with zero impedance (impedance = resistance) has nothing preventing current from passing through it. This is usually modeled in a circuit with a wire (because wires resistances on the order of a few ohms). A circuit element with infinite (or a very large) impedance can be modeled by an open circuit (a cut in the circuit). This means there is a very high barrier to passing current through that particular circuit element.
Let's think about this circuit qualitatively and contemplate why certain elements are connected the way they are. The capacitor here plays a big role. At high frequencies, the capacitor acts as a short (or a wire) and since current takes the path of least resistance, high frequencies are directly "tied" to ground (Remember: the impedance of a capacitor is , thus at high frequencies Zc=1/∞=0). At low frequencies, the capacitor (Zc=1/0=∞) acts as an open circuit and any voltage coming in from the input is passed to the output. Thus, this filter passes low frequencies and blocks high frequencies.
Mathematically, we can calculate the transfer function (H(s) or H(jω), which will tell us the amount of input signal which is "transferred" to the output.
Note that the cut-off frequency (the frequency past which the signal starts becoming attenuated) can be adjusted by tweaking the resistor and the capacitor.
For those of you who are interested in a challenge or have a background in electrical engineering, challenge yourself to derive the equation for the cut-off frequency!
High Pass Filter (HPF)
A high-pass filter is a transfer function that removes low frequency content so that only high frequencies "pass" through. It has a frequency response of:
We can also qualitatively think about what's going on here to understand why this arrangement of circuit components yields a high-pass filter. Since the capacitor is inversely related to frequency, low frequencies cause the capacitor to act as an open (see Low-Pass Filter for explanation) and high frequencies cause it to become a short. Thus, the low frequencies are blocked by the capacitor and only the high frequencies pass through.
For those of you interested in a challenge, try to derive the frequency response for this circuit. (Hint: Impedance of a capacitor = 1/jωC, where ω = 2πf)
Band Pass Filter
A band-pass filter is a transfer function that removes frequencies outside of a certain "band" but allows frequencies within the "band" to pass through. Its frequency response looks a little something like:
The bandpass filter may look more complicated, however, it is really only just a low-pass filter in series with a high-pass filter. The low-pass filter blocks the high frequencies and the high-pass filter blocks the low frequencies, yielding what looks like an inverse parabolic curve for its Bode plot. The cut-off frequencies can be adjusted by tweaking the circuit components.
Bandpass filter challenge problem:
A notch filter "notches" out a specific frequency that may have a particularly high amplitude interfering with a particular signal of interest. Notch filters are rated based on their Q-factor. Generally, the higher the Q-factor, the more exact the notch. A notch filter with a low Q-factor may effectively notch out a range of frequencies, whereas a high Q notch filter will only delete the frequency of interest.
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