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.