Related Experiment Video
Updated: Jul 12, 2026
07:45
Quasi-light Storage for Optical Data Packets
Published on: February 6, 2014
Signature quantization
Victor Guillemin1, Shlomo Sternberg, Jonathan Weitsman
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
Abstract:
We associate to the action of a compact Lie group G on a line bundle over a compact oriented even-dimensional manifold a virtual representation of G using a twisted version of the signature operator. We obtain analogues of various theorems in the more standard theory of geometric quantization.
Related Concept Videos
02:43Quantum Numbers
It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.
01:15Sampling Theorem
In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.
01:11Sampling Continuous Time Signal
In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
In the...
In the...
01:18Aliasing
Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original signal...
01:20Downsampling
When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
01:22Upsampling
Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...