Diễn đàn Hải Dương học

Giương buồm ra biển lớn !


Tần số Nyquist và hiện tượng giả danh (Aliasing)

Share

tieuminh2510

*****
*****

Tổng số bài gửi : 350
Danh dự : 5
Join date : 15/01/2008
Age : 30

Tần số Nyquist và hiện tượng giả danh (Aliasing)

Bài gửi by tieuminh2510 on Tue Jun 24, 2008 4:54 pm

A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth B.

A signal that is bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate.
To formalize these concepts, let x(t) represent a continuous-time signal and X(f) be the continuous Fourier transform of that signal (which exists if x(t) is square-integrable):


The signal x(t) is bandlimited to a one-sided baseband bandwidth B if X(f) for all | f | > B
Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate fs (in samples per unit time)

or equivalently

2B is called the Nyquist rate and is a property of the bandlimited signal, while fs is called the Nyquist frequency and is a property of this sampling system.

The time interval between successive samples is referred to as the sampling interval


and the samples of x(t) are denoted by



The sampling theorem leads to a procedure for reconstructing the original x(t) from the samples x[n] and states sufficient conditions for such a reconstruction to be exact.

Aliasing


Hypothetical spectrum of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "brick-wall" low-pass filter can remove the images and leave the original spectrum, thus recovering the original signal from the samples.

If the sampling condition is not satisfied, then frequencies will overlap; that is, frequencies above half the sampling rate will be reconstructed as, and appear as, frequencies below half the sampling rate. The resulting distortion is
called aliasing; the reconstructed signal is said to be an alias of the original signal,
in the sense that it has the same set of sample values.

Top: Hypothetical spectrum of an insufficiently sampled bandlimited signal (blue), X(f), where the images (green) overlap. These overlapping edges or "tails" of the images add creating a spectrum unlike the original.
Bottom: Hypothetical spectrum of a marginally sufficiently sampled bandlimited signal (blue), XA(f), where the images (green) narrowly do not overlap. But the overall sampled spectrum of XA(f) is identical to the overall inadequately sampled spectrum of X(f) (top) because the sum of baseband and images are the same in both cases. The discrete sampled signals xA[n] and x[n] are also identical. It is not possible, just from examining the spectra (or the sampled signals), to tell the two situations apart. If this
were an audio signal, xA[n] and x[n] would sound the same and the presumed "properly" sampled xA[n] would be the alias of x[n] since the spectrum XA(f) masquarades as the spectrum X(f).

For a sinusoidal component of exactly half the sampling frequency, the component will in general alias to another sinusoid of the same frequency, but with a different phase and amplitude. To prevent or reduce aliasing, two things can be done:

  1. Increase the sampling rate, to above twice some or all of the frequencies that are aliasing.
  2. Introduce an anti-aliasing filter or make the anti-aliasing filter more stringent.
The anti-aliasing filter is to restrict the bandwidth of the signal to satisfy the condition for proper sampling. Such a restriction works in theory, but is not precisely satisfiable in reality, because realizable filters will always allow some leakage of high frequencies. However, the leakage energy can be made small enough so that the aliasing effects are negligible.The reconstruction low-pass filter transition band is between B and fs-B and
the filter response need not be precisely defined in that region (since there is no non-zero spectrum in that region). However, the worst case is when the bandwidth B is virtually as large as the Nyquist frequency fs/2 and in that worst case, the reconstruction filter H(f) must be:


where rect[u] is the rectangular function.
With H(f) so defined, it is clear that


Spectrum, Xs(f), of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "brick-wall" low-pass filter, H(f), removes the images, leaves the original spectrum, X(f), and recovers the original signal from the samples and the spectrum of the original signal that was sampled, X(f), is recovered from the spectrum of the sampled signal, Xs(f). This means, in the time domain, that the original signal that was sampled, x(t), is recovered from the sampled signal, xs(t).

codaipm12

****-
****-

Tổng số bài gửi : 278
Danh dự : 1
Join date : 15/01/2008
Age : 30
Đến từ : 05HD

Re: Tần số Nyquist và hiện tượng giả danh (Aliasing)

Bài gửi by codaipm12 on Tue Jun 24, 2008 9:40 pm

bài này nằm trong môn KHai thác dữ liệu có nên để ở đây ko?

tieuminh2510

*****
*****

Tổng số bài gửi : 350
Danh dự : 5
Join date : 15/01/2008
Age : 30

Re: Tần số Nyquist và hiện tượng giả danh (Aliasing)

Bài gửi by tieuminh2510 on Tue Jun 24, 2008 11:18 pm

Ah mình lại nghĩ khác đó, môn Phân tích xử lý số liệu là một nhánh của Xác suất thống kê, và Xác xuất thống kê là một nhánh trong toán học.

vuieng

****-
****-

Tổng số bài gửi : 93
Danh dự : 1
Join date : 15/01/2008
Age : 29

Re: Tần số Nyquist và hiện tượng giả danh (Aliasing)

Bài gửi by vuieng on Sat Jun 28, 2008 12:00 am

xài cái này trong đề tài lun đi Minh mò


_________________
Đem yêu thương phủ lên trên cuộc đời

Sponsored content

Re: Tần số Nyquist và hiện tượng giả danh (Aliasing)

Bài gửi by Sponsored content Today at 1:08 am


    Hôm nay: Sat Dec 03, 2016 1:08 am