[MUSIC] Welcome back to Linear Circuits, this is Dr. Ferri. This lesson is on the Frequency Spectrum. Before we can do analysis of the frequency responsive circuits, it helps to understand the frequency content of signals, and that's what the frequency spectrum is. For example, look at this signal right here. It has a number of different frequencies in it and this could be a voltage or a current signal. We're going to be building upon what we know about sine waves. In particular, if I have sinusoids defined this way, in terms of radiants per second or in terms of hertz, where f is equal to the frequency in hertz, and 2 pi f is equal to omega. We define the frequency spectrum this way. It's a plot of the Amplitude versus frequency. So if the Amplitude is in A, it is plotted at that given frequency. Now frequency spectrum could be done in terms of Hertz or radians per second. Now let's look at a specific example. I've got a sine wave with an amplitude of 1 and a frequency of f is equal to 2. The frequency spectrum looks like this. So at that frequency right here, I've got an amplitude of 1. Now, let's look at this frequency right here. This is obviously a higher frequency because it moves faster, the frequency here is 6. So if I plot that at 6, I have an amplitude of 0.2. So, that's the frequency spectrum of this signal and this signal. Now, what happens if I add them together? If I just, point by point, add these two signals together, I get this one here. It looks like this signal with the high frequency superimposed on it. So it ends up like this, the sum of them is x1 plus x2. Well it turns out I just have to sum the two frequency spectrum or spectra as we call it of the x1 and x2 and I get this. A couple things to notice about this is that, if I look at this signal in the time domain, this looks like it's dominated by a low-frequency, it looks mostly low-frequency. And that domination of the low-frequency is more evident in the frequency spectrum, because you can see that this amplitude is much larger than this amplitude. Let's look at another example. In this case, x1 is a low frequency signal, with an amplitude of 0.2 at a frequency of 2 hertz, so there's the 2 hertz, and an amplitude of 0.2. And the higher frequency, has an amplitude of 1 and frequency of 6. So I plot that right there. Now when I combine these, I add x1 x2, and I also give an offset to it. And that offset, we'll say it's a DC component. And that shows up right here. A DC component Corresponds to omega = 0, or f=0. Frequency of 0, and then I plot that offset value, or DC component. So, the DC component shows up here as, the average value of this. So if I plot this along here, the average value is 1. And then I add the other two components that I found before, the value at 2 and the value at 6Hz, and what I see here, is I see the DC component, and, other than that, I see the dominance of the high-frequency. And you can look at this combination here, and it looks a lot more high-frequency, it looks a lot more like this signal than it does this signal. So this particular example I'm giving you a time domain signal, v of t, and saying plot the frequency spectrum. And in this particular case, I'm telling you what the two frequency components are. And you can actually figure that out from the plot because if I look at this, I've got a low frequency that repeats at this. So this is, I can figure out what the time is, 1 over t, call it t sub 1. That's f1, that's the low frequency. The high frequency you can get by looking at the shorter peaks that are much faster. And 1 over t2, that distance is t2, 1 over t2 is f2. So from this plot, we can figure out what f1 and f2 are. Now we want to figure out what the amplitudes are of the f1 and f2 components. Well that's a little bit more tricky, but you have to realize that it's a super position of a low frequency and a high frequency signal. I can look at this signal and kind of figure out what the low frequency average, the slower moving average of this, and kind of sketch that in. And just to show, done a little bit more carefully, it looks like this. From this, I can pull out what the amplitudes are. So the low frequency amplitude I figure out from this slow moving average, and that is an amplitude of 1.5. And then the high frequency is the amplitude off of this low frequency average. So this amplitude is .5. So if I draw the spectra, I have a frequency of 200, and a frequency of 1000 frequency in hertz, and an amplitude at the low frequency of 1.5. So this is the amplitudes, the amplitude at the high frequency of 0.5. And you can see again this is dominated by low frequency, and you can see the frequency domain as well. Let's do another example very similar to that one. In this particular case, I do the same thing where I kind of sketch in the low frequency average value of this. Or done more carefully it looks like this. So the amplitude here, the amplitude, the peak to peak amplitude here, I have to look at this from here to here is 1. So, peak to peak. That's what we always call p to p, or peak to peak is 1. So that means the amplitude is half of that, is 0.5. Now, the value for the high frequency is the amplitude off of this. So from here To here, I get a value of about 1.5. Now the difference also here from the last example is that there's an offset here, and this particular value has an offset of 1 so the DC value is 1. So if I plot this now, at 0, that's my DC value. I have an offset of 1, at 200, that's the low frequency with an amplitude of 0.5. And then at high frequency of 1,000, I have the largest value which is 1.5. And if you look at this signal, this signal is more dominated by high frequency than the previous example that we looked at. And you can see it in the frequency domain by the strength of that mark right there. So in summary, we've introduced a frequency spectrum, which is a graphical way of showing the amplitudes of different sine waves. And we plot again the amplitude of a given sine wave at a given frequency. And the other thing that we looked at was time domain and frequency domain representations of the signal. So in the time domain, it looks something like this, but in the frequency domain it looks like this. And sometimes it's easier to see dominance of a signal by looking at the strength of it in the frequency domain versus the time domain. For example, this particular one is dominated by high frequency and you can see it in the frequency spectrum a little bit easier by the strength of this. All right, thank you very much. [MUSIC]