Welcome back to linear circuits. This is Dr Ferri.
Today's lesson is on the frequency spectrum.
So we are most of the way through this module, and in the previous class, we
introduced the transfer function as a way of
computing the circuit response to sinusoids of different frequencies.
Today's lesson is to introduce frequency spectrum as a way of showing the frequency
content of signals.
We're going to be introducing both linear and
log scales for displaying the frequency content and
there, they have different advantages and disadvantages showing
something on a linear scale versus a log scale.
So the purpose is to look at more complicated signals.
Ones that have multiple, frequencies to them.
And let's start out with something very simple.
A summation of sines.
In this particular case, we've got a sine wave
that has a frequency of omega in radians per second.
That is equal to 2 pi times 2, but we've seen from other lessons that
omega is equal to 2 pi f, where f is in hertz and in that case
f is the value of two in hertz. So this is a single sine wave.
With an amplitude of one, and a frequency in hertz of two.
I can plot that. It's going
to be called the frequency spectrum
where we've got an amplitude of one and a frequency of two.
And we look at another frequency.
This is a higher frequency, smaller amplitude.
Frequency of six and amplitude of two.
So I can look at the spectrum that way. Amplitude of 0.2, freqency of six.
Now, the point here is to look at. Multiples are
summations, super positions of frequencies.
If I look at this one, I add those two together.
I get something that is primarily low frequency
because the dominant signal is low frequency.
This has a much higher amplitude to it than the high frequency.
So this looks like a low frequency signal with
a little bit of high frequency added on to it.
But this is a lot more clear in the frequency domain, looking at the spectrum.
It's clear that it dominates a low frequency, the, by the low frequency.
So the frequency spectrum, by showing the
different amplitudes, it tells us what dominates.
This has got a much larger low frequency componnet
to it. Than it does high frequency.
Let me look at a different sign wave summation.
This case, the same frequencies, but I've switched around the amplitudes.
So now the low frequency the low frequency component has a much smaller amplitude.
So 0.2 at a frequency of two. We've got 0.2 amplitude frequency of two.
Here we've got an amplitude of one at a frequency of six.
We see that over here.
Putting them together, I see something very different then what I saw before.
In the previous case it was dominated by the low frequency.
This is dominated by the high frequency.
So we see that in the time domain, but it, it appears more.
More directly in the frequency domain. We have an amplitude of one,
at a frequency six.
It's clear that we're dominated by the
high frequency for the purpose of the frequency
spectrum is to very clearly see frequency content
and signals, and more particularly see what dominates.
So, let's look at a particular type of signal, one that
has a lot of sine waves in it, not just two.
And as sine waves are such that they're multiples of each other.
So if I look at this particular signal, I've got
what I'm going to call a DC component, that's the constant component.
And then I've got Frequency starting out when k is equal
to 1, I've got a omega naught frequency, and I'll call that
the fundamental frequency.
And then I've got these values when k is equal to two and higher.
And those what are called the harmonics.
For various values of k, their multiples.
If I plot this in this vector it becomes a lot more clear.
When DC component, I look at the amplitude and I plot that.
That would be a zero.
At the fundamental frequency of omega naught.
I plot a sub one, at the second, I'm going to label this a second harmonic,
right there I would plot a sub two.
A sub three would be the third harmonic and so on.
So in this case, we can see the harmonics are clearly multiples of the fundamental.
And then you would also want to be able to see
what the relative amplitudes are of these to see what dominates.
Oftentimes, we plot the frequency spectrum on a log scale, not a linear scale.
So far, my plots have been on a linear scale, but on a log scale, we oftentimes
plot signals both in the vertical axis as well as a horizontal axis on a log scale.
On the horizontal axis, equal divisions here,
so that division and that division being equal
vary by a order of magnitude, order times ten in
other words, so this frequency is ten times that one.
This frequency is going to be ten times this one.
And, but they're have the same distance here.
So on the vertical scale we get the log scale by taking
out the amplitude, taking the log of it, log base ten multiplying
it by 20.
And the resulting is displayed in what we'll call decibels,
or the units being db for short. Why do we do this?
It just seems so much easier to plot things on a linear scale than a log scale.
Well, it turns out that some frequency
components are better viewed in the log scale.
They just show up better.
We also have a larger dynamic range using a Log Scale while maintaining resolution
at low amplitude range and the other reason is just for historical purposes.
We often times use, we started using Log Scales a
long time ago, when grass had to be drawn by hand.
And hand calculations were done.
So some people still use log scales because that's just what we're used to.
What I want to do is a specific example to show
why log scales are really helpful. This is a particular signal, x of t.
And you could see that it has an average value here.
And that average value will be our DC component.
When we plot the spectrum, we'll see DC component with a value of one.
In addition we've got some noise on here.
The noise looks sort of periodic but not exactly.
And it's hard to tell, in the time domain.
What that frequency content is. So then we can do the spectrum.
This is a frequency spectrum, and in particular,
this is a linear scale, so when we see an amplitude
of one right there at our dc value, it's a value
of frequency of zero, and everything else
is really kind of small, it's really hard
to see where the content of this noise is. There is a little bitty blip right there.
At a value of 20 radians per second.
Hard to tell and we never really do see much else.
You know we're surroundiveness here.
Always see is a single frequency right there.
So in other words the linear scale doesn't show us that much.
But if instead all I did was
take my magnitude.
And take 20 times the log of it and plot it this way, then it's very clear at that
same frequency of 20, it's very clear at that same
frequency of 20 I get a big peak right here.
And then I also see a lot of
other frequency content there, and that gives us the
frequency content of this noise, which it turned
out to be random noise, so I can see
the frequency content a little bit better.
On the log scale than I can on the linear scale.
So in summary, a frequency spectrum is a plot of the frequency content of signals.
We look at harmonics, they include the fundamental frequency and multiples of it.
We often times use a log scale because it shows us content that is hard to see
in a linear scale and also because of historical reasons.
When we use the log scale, the units are decibels or dB.
In our next lesson, we will do a lab demo of a guitar string application
where you take an actual signal and look at the frequency content of that signal.
Please visit the forums to ask questions and I'll see you online.
Thank You.