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!
Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam
Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (540+ videos) you might find helpful. Thanks, Adam
Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam
Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam
Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam
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?
Sure. In general, the complex Fourier series is: x(t) = sum_{-infinity}^{+infinity}Xn exp(j*n*omega0*t), Each term of this summation is a the Fourier Series coefficient Xn times the complex exponential exp(j*n*omega0*t). The Fourier Transform of this term is Xn delta(omega - n*omega0). Recall, the Fourier Transform of a complex exponential in time is a frequency shifted-impulse in the time domain. I'm just taking the Fourier transform of each term of the Fourier Series. Hope that helps. If you found the video useful make sure to check out my website adampanagos.org where I have a variety of other resources available that you might find helpful. Thanks, Adam.
@@AdamPanagos hey, im very new to this, and from a computer science background but trying to study digital image processing. I'm confused as to why you are taking the fourier transform of a fourier series. Dont we get the fourier series BY the fourier transform? that is don't we get this equation: x(t) = sum_{-infinity}^{+infinity}Xn exp(j*n*omega0*t) by taking the fourier transform of x(t)? more specifically, don't we get 'Xn' in this equation by the fourier transform? therefore from what I am understanding, we are taking fourier transform twice? once to find x(t) in terms of Xn, and then again taking fourier transform of x(t).. can you please let me know where I am going wrong?
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
The equation for computing the Fourier Series coefficients requires one to compute an integral on one full period of the signal. You're free to choose any time interval that consists of one full period. The period I selected was from -Ts/2 to Ts/2. On that interval, the signal is equal to delta(t). You could have selected a more general interval to work on, but this one centered about t = 0 is the easiest to work with I think. Hope that helps, Adam
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...
oh i find it p(t) is expressed by FS and p(t) can transformed by FT and parameter is expressed t=2*pi*tau dt=2*pi*d(tau) and bla bla.. so last question is.. why X_delta(w) has a scale factor 1/2*pi??
+신민우 Since w = 2*pi*f, then dw = 2*pi*df. The 2*pi is just a scale factor that accounts for the difference in the differentials df and dw. Hope that helps.
You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.
We are trying to create a signal that has content at only discrete points in time (i.e. we are trying to sample it). Multiplying by the impulse train creates exactly this type of signal.
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.
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?
That's really just the definition of the Fourier Transform. If we have a period continuous-time signal, the Fourier Transform will always be a collection of pairs of impulses. For example, if we had a single continuous-time sinusoid with frequency wc, the Fourier Transform is a pair of impulses (one at +wc and one at -wc). In this problem we've found all the frequencies present in the original signal (as quantified by the coefficients Pk), so the Fourier Transform is a pair of impulses for each of these terms.
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)?
Convolving with an impulse is "easy". Whatever gets convolved with the impulse just gets placed at the location of the impulse. So, just take the origin of the impulse train and shift it to the location of the impulse wx. There's probably a scale factor to apply as well, but that's not too bad to figure out. Hope that helps. Adam
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 :)
Most of the video derives what P(w) is equal to. We compute the FS representation of the time-domain impulse train to see that the we get a frequency-domain impulse train.
Most of the equations I work with are in the frequency domain using the radial frequency omega (w). Linear frequency (f) and radial frequency are related by the scale factor w = 2*pi*f. This means that dw = 2*pi*df. So, when working with integrals we'll often have a 1/2pi scale factor that's needed to have the "same integral" with respect to f instead of w. Does that make sense?
Yes, it makes perfect sense. I like to work with "f" and all sources write "using w" so I must constantly translating it over to "f" and i am not expert at math, so nice to get a confirmation.
