Module 1: Time vs Frequency Domains

Summary

This 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

00:00

⏳ 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.

05:03

📊 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.

Keywords

💡Time Domain

The time domain refers to how signals are perceived in everyday life, where events are tracked over time. In the video, it is explained as the familiar way of plotting signals, typically as a function of time, and is measured using devices like oscilloscopes.

💡Frequency Domain

The frequency domain represents an abstract way of analyzing signals based on their frequency components rather than time. It breaks signals down into sines and cosines, allowing engineers to view the magnitude and phase of each frequency. In the video, it is compared to the time domain and measured using a spectrum analyzer.

💡Fourier Transform

The Fourier Transform is a mathematical operation that converts a time-domain signal into its frequency-domain equivalent. The video explains that using Fourier transforms, a signal's characteristics can be studied in both domains, showing how modifications in one domain affect the other.

💡Oscilloscope

An oscilloscope is a device used to measure and visualize signals in the time domain. In the video, the oscilloscope is introduced as a standard tool for examining how signals vary over time, with the time axis clearly represented on the screen.

💡Spectrum Analyzer

A spectrum analyzer is a device used to measure and display signals in the frequency domain. The video demonstrates how this tool is crucial for analyzing the frequency components of a signal, especially in fields like radio systems where low-power signals must be detected.

💡Low-Pass Filter

A low-pass filter allows low-frequency components of a signal to pass through while attenuating higher frequencies. The video uses this filter to illustrate how filtering a signal in the frequency domain has an equivalent effect of smoothing the signal in the time domain.

💡Sinc Function

The sinc function is the Fourier transform of a square wave. The video explains how certain time-domain waveforms, such as square waves, result in sinc functions when transformed into the frequency domain, demonstrating the relationship between shapes in the two domains.

💡Cosine Wave

A cosine wave is a fundamental periodic signal composed of a single frequency component. The video describes how a sinusoid or cosine wave appears as a single tone in the frequency domain, represented by a delta function.

💡Negative Frequency

Negative frequency refers to the mirrored frequency component of a real signal in the frequency domain. In the video, it is explained that all real signals possess both positive and negative frequency components, a necessary feature for generating a purely real signal.

💡Real and Imaginary Components

Real and imaginary components refer to the parts of a complex number or signal, with the imaginary part often representing a phase shift. The video differentiates between real, measurable signals and the use of imaginary components in mathematical representations like in-phase and quadrature signals.

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