Fourier Transform (Solved Problem 2)

00:06:53
https://www.youtube.com/watch?v=W1MW1dkE0ro

概要

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.

収穫

  • 📐 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.

タイムライン

  • 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).

マインドマップ

ビデオQ&A

  • 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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オートスクロール:
  • 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
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    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
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    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
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    transform equal to X Omega and time
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    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
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    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
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    performing the time shifting and time
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    scaling operations on signal XT and with
  • 00:01:28
    every operation we will perform an XT
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    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
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    minus 3 so let's perform the operation
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    known as time shifting operation and we
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    will perform the right shifting by 3 so
  • 00:02:01
    we will have XT minus 3 we are
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    performing the time shifting operation
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    with respect to T because T is the
  • 00:02:09
    independent variable you can see signal
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    XT is the function
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    of time T therefore all the operations
  • 00:02:15
    on time will be performed with respect
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    to T you cannot perform the shifting
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    operation with respect to 2 T because 2
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    T is not the independent variable so
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    this is one important point and this
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    point we have discussed a lot in the
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    basic lectures after this we will
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    perform the next operation which is time
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    scaling operation and we will scale the
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    time by 2 so we will have X 2 T minus 3
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    and again we have performed the scaling
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    operation with respect to time T only
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    now we are done with the operations and
  • 00:02:54
    now we will modify our Fourier transform
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    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
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    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
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    2 X Omega by 2 multiplied 2 e power
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    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
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    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
  • 00:05:13
    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
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    operation by 1.5 epower
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    minus J Omega 1.5 will be multiplied to
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    the Fourier transform we had after time
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    scaling operation and this Fourier
  • 00:05:46
    transform is same as this Fourier
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    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
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    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
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    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
  • 00:06:39
    I will end this lecture here soon the
  • 00:06:41
    next one
  • 00:06:42
    [Applause]
  • 00:06:45
    [Music]
タグ
  • Fourier Transform
  • Signal Processing
  • Time Shifting
  • Time Scaling
  • Method 1
  • Method 2
  • Frequency Domain
  • Homework Problem
  • Properties of Fourier Transform
  • Independent Variable