Sampling Signals (3/13) - Fourier Transform of an Impulse Sampled Signal

We investigate impulse sampling in the frequency domain, i.e. we derive an expression for the Fourier Transform (FT) of a signal that has been impulse sampled. If x(t) is the continuous-time signal with corresponding FT X(w), the impulse sampled version of x(t) has a FT that consists of an infinite collection of X(w) shifted up and down the frequency axis. Each shifted version of X(w) occurs at an integer multiple of the sampling frequency ws.

[kaursingh]

my lord= every lecture is classic = i am addicted to u r lecture =thank u amarjit india

[beatrixzhou]

Thanks so much for this video! I have been wondering for long time about why there is infinite replica of the signal at frequency domain; now I get it - it's because of the impulse train at the frequency domain! Please make more such good videos! Good explanations really save the world!

[정상원-kp]

I have been trying to understand the topic for a few days and now I understand

[kevinwong]

Thanks so much, great explanantion! Couldn't find that good of an explanation on other videos

[ozzyfromspace]

I'm enjoying your sampling lecture series so much, Adam! Phenomenal job 🏆🎊🙌🏽

[faroukelkiouas]

waw, this is so helpful..who is watching that on2020?

[tojorabemananjara]

How did you get the "Fourier series representation of the signal" at 6:20? Can you please explain how you got that representation from the regular FS formula?

[azharyousuf]

Query: While computing the Fourier coefficient, why \delta(t) is used inside the integral instead of \delta(t-nT_s) ? I think at first one should use \delta(t-nT_s) and then proceed with substitution t-nT_s=z. After that, one may obtain the Fourier series coefficients as P_k=1/T_s

[fabianwenzel]

oh my god, thank you! this was an awesome explanation.

[gulshanmustafayeva]

Thank you for such an amazing explanation
very helpful 👏🏻

[신민우-wf]

i hava a question with P(w)
pulse train p(t) is periodic function, so p(t) is can compute Fourier Series
here my question is why scale factor "2*pi" of P(w) is multiplied??
plz help me...

[anilgugulothu]

Thank you sir....!!
Why we multiply original signal with impluse train

[sonicyouth]

Forever grateful. Very well explained.

[vaishnavj]

Suppose x(t) is a sinusoid of the form e^(wx t), then its F.T would result in an impulse placed at wx. How do we convolve this single impulse with the other impulse train for P(w)?

[bocckoka]

this is excellent, just a small typo at 7:53 in the sum representing the impluse train: it should be omega_0 instead of omega_s. or maybe it's intentional, i don't know :)

[subramaniantr]

Hi Adam, thank you for the explanation. If I just follow about finding FT of delta train directely, I get a sum of infinite exponentials in omega which I believe is sum of deltas because the sum of exp at a particular omega cancels out for non multiples of 2pi. But this qualitatively tells that there is another delta train and its separation in x-axis, but doesn't give the magnitude or the area of the delta. Can you tell how to go about finding it?

[wenzhouli]

Hello, sir.
I am trying to prove this Fourier Transform of an Impulse Sampled Signal is mathematically equivalent to DTFT(Discrete-Time-Fourier -Transform) since they both get a result of periodic extension. Can you help me with that?
Thanks a lot.

[Kevn]

I don't understand how the Fourier series representation of p(t) turns to what P(w) is equal to. I thought it would be Pkexp(-ikώt)

[giant]

Funny how a paid uni class teached by a very bad prof, over months, can't even come close to being clear like this 10 minutes video.