our topic today is frequency response and this is a common topic in signals and systems so if you remember from that class this will be very familiar so the idea is that we have a system G of s we're calling it and we want to see how the system reacts to specific frequencies before when we do it steps response you know we have an input it's just a step we're looking at all the frequencies at once but when we we want to actually isolate each frequency and see how it reacts individually so we're going to set up a problem so say we have so Valerie we all know Valerie she's actually a really great singer and you wouldn't know it but she's tiny but she can sing opera it's really amazing so say she can sing a perfect pitch a perfect frequency and we're going to put that into some system so it's some sound modulation systems and she's going to put that input just gonna sing a frequency into it and we're going to get some sort of signal out so first let's look at her input signal so the sound that she's singing so there's a we're going to put on a time domain so she's singing some frequency and we're going to say that it maybe it starts we're going to start here and it's some amplitude here I'm not very good at drawing these sine waves but go with it okay so we have this input signal and we're going to use the idea of a phaser so we're going to look at the magnitude of our system so we can break this as a sign or a sum of a sine a cosine but we're going to change it into a magnitude and a phase shift so the magnitude is simply going to be this amplitude from the peak to zero so that would be we call that M and we call mi for the input and so this is going to be mi is our frequency sorry is our magnitude and then in phaser notation you have a magnitude and an angle and here the angle is going to be so it's going to be this phase shift so say we're going to start off the stay our start starting point is from here we have some phase shift of that so this is our input so this is V I and Phi B Phi one of those so this is our phase shift our input and then this is our magnitude okay so we put that into our system and we will get out some system so say so here's our I'm going to put on the time domain again this is time this is the amplitude of the signal and this time it's going to this is going to amplify our signal so we're going to say it's amplified I'm not very good at drawing these okay let's do one more you can you can go ahead use your imagination and pretend it's a nice sine wave but the point I want a thing I want to show here is that we have should be the same amplitude we have some amplitude out we call it M 0 and o output and then we have some phase shift as well so here it's a little bit different phase shift it's changed a little bit and we're going to call that V out so here's M I signal here is mo vo so this is our input phasor this is our output phasor and in each of these when we do justice just individual frequencies we can model how this system will affect our input versus output so we can essentially model this only in this situation as a magnitude of this transfer function and a phase shift that it will cause such that when you multiply these two together you will get this output so essentially I'll write it out so yeah I see I multiplied by M G V G is equal to mo 0 so this is essentially our transfer function but only for an input of frequency and an output frequency so we call this a frequency response and without going through all of the math I can tell you that actually so I'll rewrite this let's just move it over here so if we want to think about this like frequency response remember it's the output of the input so we move this over will it'll get this alright so we can now get rid of that essentially this is our frequency response so the next question is how do we find this without going through all of the math I'm just going to tell you the answer so what we can do is we actually know that we can look at how the system reacts when you put in a frequency signal so what we do is we make s equal J Omega in this case so and if you think about this on the imaginary and real axis that is starting from so this is real and imaginary we are starting from if anything essentially on this this area it says on the imaginary axis which if we remember that's where our marginally stable occurrences can happen and we can associate these points with a frequency so in short we can put if we make s equals J Omega and put that into our transfer function here and it turns out that so G of J Omega we put that into our transfer function we get the magnitude of that that is going to be equal to our G mg here so the magnitude of our transfer function and we can do the same thing so now instead if we take the angle of G Omega J we will get this angle this shift so this would be fee G oh hi so and one more thing that this Omega remember is 2pi frequency this frequency is in Hertz so you can put in whatever frequency your input is here and you can figure out how this system will affect so how the effects on the magnitude and the phase at that given frequency okay so we're going to look at how to use this and draw it out in a bode plot and hopefully that will be a little bit clearer but to find the frequency response you have to take your actual G of s here you put in J Omega and you can take the magnitude of that and the angle of that to figure out how the input versus output will be affected okay so we'll stop here for now
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