Module 1: Time vs Frequency Domains
Summary
TLDRThis script explains the difference between the time and frequency domains in signal analysis. It describes how the time domain reflects real-world experiences and is represented by plots over time, while the frequency domain is more abstract, focusing on a signal's frequency components. The Fourier transform is used to convert between the two. Practical tools such as oscilloscopes and spectrum analyzers are discussed, along with the importance of understanding low-power signals and how filters affect signals in both domains. Concepts like negative frequencies and real vs. imaginary components are also explored.
Takeaways
- 🕰️ Time domain represents the familiar way we perceive the world, where events occur sequentially over time.
- 📊 The time domain is typically represented by a plot with time on the x-axis.
- 🔄 The frequency domain is an abstract concept that represents signals as a sum of frequency components.
- 🎶 Using Fourier series, any time-domain signal can be broken into cosines and sines in the frequency domain.
- 🔢 Engineers often use frequency (F) in analysis, but sometimes work with Omega (ω) in analytical calculations.
- 📡 Time-domain signals are measured using an oscilloscope, while frequency-domain signals are measured using a spectrum analyzer.
- 📉 Advanced features of spectrum analyzers are necessary to detect very low-power signals like microwatts or picowatts.
- 🔄 The Fourier transform allows conversion between time and frequency domains, but changes in one domain affect the other differently.
- 📏 Filters in the frequency domain can simplify signal processing, smoothing signals in the time domain.
- 🌀 Real signals contain both positive and negative frequency components, which are necessary to represent fully measurable real-world signals.
Q & A
What is the time domain, and why is it familiar to us?
-The time domain is the way we perceive events as they occur in real life, where time moves forward, and events happen over time. It's familiar because it reflects our everyday experiences and is typically represented by a plot versus time.
How does the frequency domain differ from the time domain?
-The frequency domain is an abstract concept that helps us analyze signals by breaking them down into frequency components, such as cosines and sines. Unlike the time domain, which deals with signals over time, the frequency domain focuses on the magnitude and phase of these frequency components.
What is the role of the Fourier series in signal processing?
-The Fourier series allows us to break any time-domain signal into a series of cosines and sines. This decomposition makes it easier to work with signals by analyzing their frequency components, particularly in the frequency domain.
Why do engineers often use frequency (F) instead of angular frequency (Omega)?
-Engineers commonly use frequency (F) because most instrumentation measures in terms of frequency. However, angular frequency (Omega) is used in analytical calculations, often when results are expressed in radians. Engineers can switch between the two using a conversion formula.
What instruments are used to measure time and frequency domain signals?
-To measure time-domain signals, an oscilloscope is used. For frequency-domain signals, a spectrum analyzer is employed. Both instruments provide valuable insights into the respective domains of the signals.
How can you move between the time and frequency domains?
-The Fourier transform allows for the conversion of signals from the time domain to the frequency domain and vice versa. Changes in one domain affect the other, although the changes are not identical.
What happens to a signal when it passes through a low-pass filter?
-In the frequency domain, a low-pass filter removes high-frequency components, which correspond to sharp edges in the time domain. As a result, the signal becomes smoother in the time domain.
What does the Fourier transform of a square wave look like?
-The Fourier transform of a square wave produces a sinc function in the frequency domain. This transform illustrates how a square wave, with sharp edges in the time domain, contains a wide range of frequency components.
What are real and negative frequencies, and how are they related?
-In the frequency domain, real signals have both positive and negative frequency components. Negative frequencies appear due to the complex nature of signals, which can be expressed as a combination of e^(jωt) and e^(-jωt). These components help create a real signal by canceling out the imaginary parts.
Why do we need both positive and negative frequency components to form a real signal?
-To form a real signal, we combine both positive and negative frequency components because a single component would result in a complex (imaginary) signal. Combining them cancels the imaginary parts and ensures the signal is entirely real and measurable.
Outlines
⏳ Understanding the Time and Frequency Domains
This paragraph explains the concept of the time domain, which is how we perceive events in everyday life, represented as plots versus time. It introduces the frequency domain as a more abstract way to represent signals by breaking them into sine and cosine components, referencing the Fourier series. These components are plotted in terms of frequency or phase to analyze signals. The paragraph also discusses how engineers use both the frequency (F) and angular frequency (Ω), noting that these can be interchanged with proper conversion.
📊 Measuring Time and Frequency Domains
This section describes the instruments used to measure signals in the time and frequency domains. It introduces the oscilloscope, which represents the time domain, and the spectrum analyzer, which handles frequency domain measurements. The paragraph highlights that while oscilloscopes are more familiar, spectrum analyzers are key for measuring low-power signals such as microwatts and picowatts in radio systems. The advanced features of spectrum analyzers are necessary for detecting these small signals.
🔄 Fourier Transform: Linking Time and Frequency Domains
The paragraph discusses how the Fourier transform allows transitions between the time and frequency domains, noting that changes in one domain reflect differently in the other. It introduces the idea of filtering signals in the frequency domain (e.g., using a low-pass filter) and the corresponding smoothing effect in the time domain. This provides an example of how certain tasks are easier to understand in the frequency domain, while there is always an equivalent change in the time domain to consider.
📐 Understanding Sinusoids and Delta Functions
This section focuses on the Fourier transform of sinusoids, explaining how a cosine wave corresponds to two delta functions in the frequency domain. The concept of negative frequency is introduced, which results from the Fourier transform of a cosine, yielding both positive and negative components. The paragraph clarifies that real signals have both positive and negative frequencies and discusses the mathematical identity (e^jωt) that helps illustrate why these dual components are needed to form a real signal.
Mindmap
Keywords
💡Time Domain
💡Frequency Domain
💡Fourier Transform
💡Oscilloscope
💡Spectrum Analyzer
💡Low-Pass Filter
💡Sinc Function
💡Cosine Wave
💡Negative Frequency
💡Real and Imaginary Components
Highlights
The time domain is what we experience in everyday life, where time moves forward, and it is typically represented as a plot versus time.
The frequency domain is an abstract concept used to represent signals by their frequency components, making signal analysis more robust.
The Fourier series allows any time waveform to be broken into a series of cosines and sines, which make up the signal in the frequency domain.
In engineering, we often represent frequency by 'F', though 'Omega' is commonly used in analytical calculations, as it represents radians.
We measure signals in both time and frequency domains using instruments like the oscilloscope for the time domain and spectrum analyzers for the frequency domain.
Oscilloscopes plot signals against time, while spectrum analyzers plot the magnitude of each frequency, helping analyze even low-power signals.
The Fourier transform converts signals between the time and frequency domains, showing the correlation between changes in one domain and the other.
A low-pass filter in the frequency domain filters out unwanted frequencies and smooths out the signal in the time domain.
Operating in the frequency domain can simplify complex signal processing tasks that are harder to visualize in the time domain.
A sinusoid is considered a 'single tone' because it contains a single delta function in the frequency domain, with both positive and negative frequency components.
Real signals have both positive and negative frequency components, often referred to as images in the frequency domain.
The negative frequency component comes from the Fourier transform of cosine, which introduces a -J Omega T term.
To obtain a completely real signal, two components, e^jΩt and e^-jΩt, must be combined to cancel out the imaginary part.
This combination of e^jΩt and e^-jΩt results in either a cosine or sine wave, creating a real signal without imaginary components.
Real signals in the lab can be measured using instruments, and the frequency domain representations allow engineers to filter and analyze these signals efficiently.
Transcripts
so the time domain is what we think
about and we perceive in our everyday
life so in our world time marches
forward and we think about things
happening over time and so the time
domain is very comfortable and familiar
to us and typically is represented by a
plot versus time not much to say in here
but if we think about the other domains
we work with namely the frequency domain
this is a little bit different the
frequency domain is an abstract concepts
that we've generated that really helps
us deal with signals in a more robust
way and these are representing the
signals by the frequency components that
make them up if you remember from the
Fourier series that you could take any
time waveform and break it up into a
series of cosines and sines and the
frequency domain just simply plots the
magnitude and phase of those cosines and
sines that when you add up create the
time domain signal that you are
interested in and we generally represent
this in F or we can also talk about
Omega typically as engineers we talk
about F because all of our
instrumentation is in frequency directly
Omega is used quite often because
oftentimes analytical calculations end
up in radians and a make is more useful
so we can go back and forth between the
two almost interchangeably the only
thing you need to make sure of is that
if you're doing a calculation it may
likely be in terms of Omega and if you
want to convert to F you need to use the
formula right here
all right so the question comes in if we
have time domain is what we see in real
life if you will in frequency domain is
this concept how do we measure these two
and so you should know that there's two
primary instruments that we're going to
use to measure time and frequency domain
so let's start with the time domain
and this is a picture of an old
oscilloscope but you probably already
know that we use the escola scope here
to represent the time domain and so the
time axis is actually right right here
and then of course if we want to do the
frequency domain we use a spectrum
analyzer where this axis right here is
in terms of F and these are simply plots
of the magnitude of each frequency then
we're going to learn to use both of
these it's expected that you probably
already know how to use an oscilloscope
and we're going to go over some of the
basic functionality of a spectrum
analyzer and also go into some of its
advanced features because one of the
things about radio systems is that we
deal with signals of very low magnitude
and our very low power so microwatts
maybe even pico watts and it turns out
that we have to use some of these
advanced features in order to be able to
see these really small signals so we'll
talk about that more as time comes along
alright now we've been talking about the
time versus frequency domain and as you
know we can go between those two using
the Fourier transform and so the Fourier
transform will take us from a time
domain signal into a frequency domain
signal as you know and just as a
reminder if you change a signal in one
domain it gets changed in the other
domain however the change is not the
same if we change this signal here X of
Omega does change but the change happens
to the Fourier transform so except for
linear multiplications by scalars
changing X of T for instance shifting in
time does not just simply shift that on
the axis but this should be review for
you and so here is a couple examples of
Fourier transforms and one of the things
we want to show you here is sort of how
things look in the time domain and how
things look in the frequency domain so
if you remember the Fourier transform of
a square ends up with a sinc function
here and you're probably familiar with
that if we were though to take a
low-pass filter and I'll draw that in
the frequency domain so imagine I put in
a filter
that has a passband like this so it's
pretty easy and intuitive from the
frequency domain to realize we're just
gonna basically filter out all these
sides and this is the signal we get but
you need to remember that in the time
domain it has an equivalent one and it
smooths it out and we could have
actually thought about this directly in
that a low-pass filter is going to
remove all the sharp edges and smooth
things out so here's a great example of
how operating in the frequency domain
here is very easy and quick but there is
an equivalent time domain change that we
just need to keep in mind all right so
I've shown on the left a a time domain
waveform and on the right its Fourier
transform and just to be accurate here I
know that this right here is a cosine
because it has two positive Delta
functions so I would need to set my zero
for instance right here to make this
exactly accurate and so we often say
that a sinusoid is a single tone because
it contains a single Delta function in
the frequency domain and we also need to
remember that all real signals have a
negative frequency component as well are
sometimes called an image and so
whenever you look at the frequency
domain you should always see the image
if it's a real signal now I should just
be real careful by real here we mean we
can measure it
in the lab
okay i specify that because we're gonna
be talking a little bit later about real
and imaginary components in terms of
in-phase and quadrature and that just
has to do with the fact that we're using
imaginary to represent a 90 degree phase
shift but those are both signals we can
measure in the lab so real here is more
of just realistic or stuff we can
measure in the lab as opposed to real
and imaginary so I want to talk a little
bit more about this negative frequency
if you actually did the Fourier
transform of cosine you would see that
one of the coefficients comes out and it
has a minus J Omega T component and
that's what gives us this delta function
another way that I think helps me
remember two things one is what this is
and also that we need both is this
identity here you probably remember if
you think about these is plotting e to
the J Omega T and this is e to the minus
J Omega T right we know that e to the J
Omega T is cosine Omega T plus J sine
Omega T so if I just had one of these
this is what I would get I'd get a real
part and I'd get an imaginary part so I
wouldn't have a completely real signal
if I want a completely real signal what
I need to do is combine two of these
either for a cosine or a sine and what
that does is it creates either a cosine
or a sine and removes the imaginary
component so if you think about this
plot simply as a
e to the J Omega plot it helps me out a
little bit and remembering why I need
both it's because I need this to be real
and that we can think about a cosine
here simply as these two components
right there
5.0 / 5 (0 votes)