What is the original signal's Fourier transform?

The original signal XT has the Fourier transform X(ω).

What operations are performed to derive YT from XT?

The operations are time shifting and time scaling.

How is the time shifting operation applied?

The signal is shifted to get YT = X(2t-3), involving a right shift by 3 on XT.

What does the time scaling operation involve?

The signal is scaled by 2, so X(t-3) becomes X(2t-3).

What changes occur in Fourier transform due to time shifting?

After time shifting, the Fourier transform is multiplied by e^{-jω3}.

How does time scaling affect the Fourier transform?

Time scaling leads to a factor of 1/|2| and changes the frequency argument to X(ω/2).

What is the final Fourier transform of YT?

Y(ω) = [1/2] X(ω/2) e^{-jω1.5}.

What are the two methods discussed for the transformations?

Method 1 performs time shifting then scaling, while Method 2 performs scaling first.

What is the homework problem about?

It asks to find the Fourier transform of YT = X(-3t+9) using the described methods.

Is there a preferred method between the two?

Both methods yield the same result, so either can be used depending on preference.

Fourier Transform (Solved Problem 2)

00:06:53

https://www.youtube.com/watch?v=W1MW1dkE0ro

Ringkasan

TLDRThe video discusses how to find the Fourier transform of a signal YT = X(2t - 3) derived from an original signal XT, which has a Fourier transform X(ω). Two methods are explored: the first applies time shifting followed by time scaling, while the second applies scaling first then shifting. For Method 1, time shifting results in shifting XT by 3 units, while scaling by 2 transforms XT to X(2t - 3). The corresponding Fourier transform Y(ω) is computed as [1/2] X(ω/2) e^{-jω1.5}. Method 2 confirms this result through an alternative process. The topic underscores understanding Fourier transform properties for multiple operations.

Takeaways

📐 YT = X(2t - 3) involves both time shifting and scaling of XT.
🔄 Method 1: First shift, then scale the original signal.
🌀 Method 2: Scale first, then shift the signal.
✖ Time shifting changes the Fourier transform by multiplying with e^{-jω3}.
➗ Time scaling changes frequency as X(ω/2) and adds 1/2 factor.
✅ Both methods confirm Y(ω) = [1/2] X(ω/2) e^{-jω1.5}.
📚 Understanding these properties allows accurate transformation.
🔍 Two different methods yield the same Fourier transform result.
🛠 Homework involves finding transform for YT = X(-3t + 9).
🔑 Key: Knowing basic operations and properties of Fourier transforms.

Garis waktu

00:00:00 - 00:06:53
In this explanation, the speaker is tasked with finding the Fourier transform of a signal Y(t) which is derived from another signal X(t) by applying operations on X(t). Specifically, Y(t) equals X(2t - 3). Knowing that the Fourier transform of X(t) is X(ω), the speaker seeks to express the Fourier transform of Y(t), denoted as VY(ω), in terms of X(ω) by utilizing the properties of Fourier transforms. Method 1 involves first applying a time-shifting operation followed by a time-scaling operation. The operations are carefully chosen to ensure that every transformation conforms to the rules associated with the Fourier transform, resulting in the final expression for VY(ω).

Peta Pikiran

Video Tanya Jawab

What is the original signal's Fourier transform?
The original signal XT has the Fourier transform X(ω).
What operations are performed to derive YT from XT?
The operations are time shifting and time scaling.
How is the time shifting operation applied?
The signal is shifted to get YT = X(2t-3), involving a right shift by 3 on XT.
What does the time scaling operation involve?
The signal is scaled by 2, so X(t-3) becomes X(2t-3).
What changes occur in Fourier transform due to time shifting?
After time shifting, the Fourier transform is multiplied by e^{-jω3}.
How does time scaling affect the Fourier transform?
Time scaling leads to a factor of 1/|2| and changes the frequency argument to X(ω/2).
What is the final Fourier transform of YT?
Y(ω) = [1/2] X(ω/2) e^{-jω1.5}.
What are the two methods discussed for the transformations?
Method 1 performs time shifting then scaling, while Method 2 performs scaling first.
What is the homework problem about?
It asks to find the Fourier transform of YT = X(-3t+9) using the described methods.
Is there a preferred method between the two?
Both methods yield the same result, so either can be used depending on preference.

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Teks

Gulir Otomatis:

00:00:00
in this question it is given that
00:00:02
Fourier transform of signal XT is equal
00:00:05
to X Omega and we need to find the
00:00:08
Fourier transform of signal YT which is
00:00:11
equal to X 2 t minus 3 so it is clear
00:00:15
that we are performing multiple
00:00:17
operations on signal XT to get signal YT
00:00:20
and we know Fourier transform of XT is
00:00:23
equal to X Omega so we can obtain the
00:00:26
Fourier transform of Y T in terms of X
00:00:29
Omega by using the properties of Fourier
00:00:32
transform so let's see how we can solve
00:00:37
this question it is given that time
00:00:41
domain signal XT is having the Fourier
00:00:44
transform equal to X Omega and time
00:00:48
domain signal YT is equal to X 2 t minus
00:00:53
3 and let's say it is having the Fourier
00:00:58
transform equal to VY Omega so we are
00:01:01
required to calculate Y Omega we will
00:01:04
understand the method number one to get
00:01:10
the answer in this method number one we
00:01:12
will first perform the time-shifting
00:01:14
operation and then we will perform the
00:01:17
time scaling operation because here you
00:01:19
can see we are getting signal YT after
00:01:23
performing the time shifting and time
00:01:25
scaling operations on signal XT and with
00:01:28
every operation we will perform an XT
00:01:31
next Omega will change accordingly for
00:01:34
this we will use the properties I have
00:01:36
explained in the previous lectures so
00:01:39
let's move to the method number one we
00:01:42
will start from this scratch we are
00:01:45
having signal XT and we want X to t
00:01:48
minus 3 so let's perform the operation
00:01:53
known as time shifting operation and we
00:01:57
will perform the right shifting by 3 so
00:02:01
we will have XT minus 3 we are
00:02:04
performing the time shifting operation
00:02:07
with respect to T because T is the
00:02:09
independent variable you can see signal
00:02:11
XT is the function
00:02:13
of time T therefore all the operations
00:02:15
on time will be performed with respect
00:02:18
to T you cannot perform the shifting
00:02:20
operation with respect to 2 T because 2
00:02:23
T is not the independent variable so
00:02:25
this is one important point and this
00:02:28
point we have discussed a lot in the
00:02:30
basic lectures after this we will
00:02:32
perform the next operation which is time
00:02:38
scaling operation and we will scale the
00:02:41
time by 2 so we will have X 2 T minus 3
00:02:46
and again we have performed the scaling
00:02:49
operation with respect to time T only
00:02:51
now we are done with the operations and
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now we will modify our Fourier transform
00:02:57
accordingly initially for signal XT we
00:03:00
are having the Fourier transform X Omega
00:03:03
and we know after performing the time
00:03:05
shifting the Fourier transform X Omega
00:03:08
will get multiplied by E power minus J
00:03:12
Omega 3 after this time scaling
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operation will take place and the
00:03:19
Fourier transform will become 1 over mod
00:03:22
2 X Omega by 2 multiplied 2 e power
00:03:28
minus J Omega by 2 multiplied to 3 or we
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can write the Fourier transform Y Omega
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is equal to 1 over 2 multiplied to X
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Omega by 2 multiplied to e power minus J
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Omega 3 divided by 2 is equal to 1 point
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5 so this is the answer this is the
00:03:52
Fourier transform of signal YT and now
00:03:55
we will move forward to the method
00:03:57
number 2 in method number 2 we will
00:04:01
perform the time scaling operation first
00:04:04
and then we will perform the time
00:04:06
shifting operation so we will start from
00:04:08
this scratch we are having signal XT and
00:04:11
we want signal x2 t minus 3 so let's
00:04:15
perform the time scaling operation we
00:04:20
will scale the time by 2 so we have X 2
00:04:24
T now we will perform
00:04:27
the next operation which is time
00:04:29
shifting operation now here you have to
00:04:32
be little bit cautious I told you every
00:04:35
operation we perform is with respect to
00:04:37
the independent variable and we are
00:04:39
having the independent variable equal to
00:04:41
t naught to t so we will separate t and
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then we will perform the shifting by 1.5
00:04:52
so that when you open the bracket you
00:04:54
are having X 2 t minus 3 now we will
00:04:58
write down the Fourier transforms for
00:05:02
all the three signals last signal is
00:05:06
signal YT initially we are having XT
00:05:09
with Fourier transform x Omega and after
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time scaling we are having 1 over mod 2
00:05:16
mod 2 is equal to 2 signal X Omega by 2
00:05:22
now we will write down the Fourier
00:05:25
transform after time shifting operation
00:05:27
and we know after time shifting
00:05:31
operation by 1.5 epower
00:05:36
minus J Omega 1.5 will be multiplied to
00:05:41
the Fourier transform we had after time
00:05:44
scaling operation and this Fourier
00:05:46
transform is same as this Fourier
00:05:49
transform so this is equal to VY Omega
00:05:53
and this is our answer so you can see it
00:05:56
is very easy to get the Fourier
00:05:58
transform if you know the basics if you
00:06:00
know how to perform the multiple
00:06:02
operations and the properties of Fourier
00:06:04
transform now it is time for the
00:06:08
homework problem in this homework
00:06:10
problem signal XT is having the Fourier
00:06:12
transform X Omega and there is signal YT
00:06:15
which is equal to X minus 3 T plus 9 and
00:06:19
you need to find the Fourier transform
00:06:22
of Y T which is equal to Y Omega again
00:06:26
you are having the multiple
00:06:27
transformations and you have to proceed
00:06:29
in the same way you can follow either
00:06:31
method 1 or method 2 you will get the
00:06:34
same answer so once you have your answer
00:06:36
don't forget to post in comment section
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I will end this lecture here soon the
00:06:41
next one
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[Applause]
00:06:45
[Music]