"This spike you're looking at here above the winding frequency of five is the Fourier transforms way of telling us that the dominant frequency of the signal is five beats per second."
"If by some twist of fate the Fourier transform hadn't been studied prior to the beginnings of quantum mechanics in the early 1900s, it still would have inevitably been discovered once we attempted to understand the strange and fascinating laws of the quantum world."
"The wave function for a particle's momentum is the Fourier transform of the wave function for the particle's position."
"Circulant matrices are connected with the discrete Fourier transform."
"The quantum Fourier transform of that is exactly the Gaussian distribution supported over the dual lattice."
"So the idea then is that we will use these relationships to define an analysis and synthesis of aperiodic signals, and we'll refer to that as a Fourier transform."
"If I stretched time, I compress frequency."
"The Fourier transform of the constant one seems to be a delta function at zero of area 2pi."
"By representing it in the frequency domain with the help of the Fourier transform, it becomes trivial."
"The visibility function is effectively the two-dimensional Fourier transform of what the emission is on the sky."
"The Fourier transform at its heart is just a mathematical tool that decomposes a signal, any signal, into a sinusoidal component."
"The Fourier transform basically says that any periodic function can be written as a weighted sum of infinite sinusoids of different frequencies."
"The Fourier transform represents a signal f(x) in terms of amplitudes and phases of its constituent sinusoids."
"Whether you go forward with the Fourier transform, or come back from frequency representation back to spatial representation using the inverse Fourier transform, there is no loss of information."
"The Fourier transform of a constant is a delta function."
"For real functions, the square magnitude of the Fourier transform is symmetric."
"The easiest way to think about diffraction is Fourier transform."
"The Fourier transform makes our differential equation simpler."
"...the transition from Fourier series to the Fourier transform."
"Sampling is very important, and by thinking about in the Fourier domain, we get a lot of insights that we wouldn't have got otherwise."
"A Fourier transform takes a complex pattern and decomposes it into all the individual sine waves that are present."
"The Fourier transform takes any function and decomposes it to its set of frequencies."
"Later on in the course, when we've developed the concept of the Fourier transform after that, the Laplace transform, we'll see some very efficient and useful ways of generating solutions, both for differential and difference equations."
"The convolution of two time functions is the product of their Fourier transforms."
"The Fourier transform of the output is the Fourier transform of the input times the Fourier transform of the impulse response of the system."
"The Fourier transform of the convolution is the product of the Fourier transforms."
"We can implement the quantum Fourier transform on M qubits using M squared gates."
"That turns out to be an incredibly important computational problem, and there's a highly efficient algorithm for performing that computation known as the fasts Fourier transform."
"The Fourier transform, essentially by definition, is what we get as the Fourier series coefficients, as we focus on one period, and then let the period go off to infinity."
"Over the last series of lectures, we've seen how powerful and useful the Fourier transform is."
"The Laplace transform is the Fourier transform of an exponentially weighted time function."
"The Laplace transform is very closely associated with the Fourier transform."
"This particular expression, in fact, will correspond to what we'll refer to as the Fourier transform, the discrete-time Fourier transform of the system impulse response."
"To take a Fourier transform means to decompose a complicated function into a series of simple sine waves."
"What Fourier said, which was essentially his brilliant insight, is that, if I have a very general periodic signal, I can represent it as a linear combination of these harmonically-related complex exponentials."
"If the region of convergence includes the unit circle, then the Fourier transform exists."
"Adopting this point of view to give a rigorous foundation for Delta and then actually to also develop the Fourier transform was no less revolutionary."
"We need a more robust definition of the Fourier transform that will allow us verily to work with the signals that society needs to function."
"The Fourier transform of two functions that are convolved is equal to their separate Fourier transforms multiplied."
"If we take the Fourier transform of our image and we divide that in reciprocal space by the contrast transfer function and then we take the result and do an inverse Fourier transform, we will recover a more true version of the structure of the object."
"Through an inverse Fourier transform, you recover a CTF corrected image which is closer to the object than the image that you first recorded."
"What do we get when we take a pulse in the spatial domain and convert it into the frequency domain? A sync function."
"The Fourier transform is really just a linear map from one basis to another."
"To extract frequencies, you just run a Fourier transform that'll tell you what frequencies are present in your signal."
"The discrete Fourier transform has some important computational realizations and computational implications that will be one of the important things that we'll want to capitalize on in applying digital signal processing to real problems."
"The Poisson summation formula relates the Fourier coefficients of a periodic function to its Fourier transform."
"We have sort of looked at one derivation for the Poisson summation formula which relates the Fourier coefficients of a periodic function to its Fourier transform."
"It's what we call the Discrete Fourier Transform."
"The Fourier transform is right at the heart of quantum computation."
"The Fourier transform of the Delta function is one, the constant function one."
"Fourier transform generally applies to aperiodic functions, whereas Fourier series applies to periodic functions."