混疊

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目前已翻译5%，原文在Aliasing。

混疊(aliasing)，在訊號頻譜上可稱作疊頻；在影像上可稱作疊影，主要來自於對連續時間訊號作取樣以數位化時，取樣頻率低於兩倍奈奎斯特頻率。

在統計、訊號處理和相關領域中，混疊是指取樣訊號被還原成連續訊號時產生彼此交疊而失真的現象。當混疊發生時，原始訊號無法從取樣訊號還原。而混疊可能發生在時域上，稱做時間混疊，或是發生在頻域上，被稱作空間混疊。

在視覺影像的類比-數位轉換或音樂訊號領域，混疊都是相當重要的議題。因為在做類比-數位轉換時若取樣頻率選取不當將造成高頻訊號和低頻訊號混疊在一起，因此無法完美地重建出原始的訊號。為了避免此情形發生，取樣前必須先做濾波的動作。

[编辑] 概要

兩個不同的正弦波卻有相同的樣本值。藍色正弦波的頻率 $f\,$ 較低；紅色正弦波的頻率 $f_s+f\,$ 較高。

[编辑] 週期現象上的混疊

太陽在天空由東往西移動，兩次的日出間隔了24小時。若某個人每23小時對天空拍張照片，太陽會好似由西向東移動，並且日出週期由24小時轉變成552小時(24×23=552)。相同的現象也會發生在高速旋轉的車輪鋼圈，視覺上看到的旋轉方向和實際上相反。這就是時間混疊。

若對一個穿有人字呢圖案外衣的人攝影，播放時會發現影片中的線條數目會少於真實圖案的線條數目，此現象被稱為moiré pattern。這就是空間混疊的例子，它的成因之後會闡明。

[编辑] 週期訊號的取樣

In the same way, when one measures a sinusoidal signal at regular intervals, one may obtain the same sequence of samples that one would get from a sinusoid with a different, possibly lower frequency. Specifically, if a sinusoid of frequency $f\,$ (in cycles per second for a time-varying signal, or in cycles per centimeter for space-varying signal) is sampled $f_s\,$ samples per second or per centimeter, the resulting samples will also be compatible with a sinusoid of frequency $Nf_s - f\,$ and one of frequency $Nf_s + f\,$ , for any integer $N\,$ . In the area's jargon, each sinusoid gets aliased to (becomes an alias for) the others. If $f_s > 2f\,$ , the lowest of these alias frequencies will be the original signal frequency, but otherwise it will not.

Therefore, if we sample at frequency $f_s\,$ a continuous signal that may contain several such sinusoids with identical samples, we will not be able to reconstruct the original signal from the samples, because it is mathematically impossible to tell how much of each component we should take.

[编辑] 奈奎斯特準則

Image:CriticalSamplingFrequency.png

Recreation of Black's Fig. 4-5 Minimum sampling frequency for band of width B

One way to avoid such aliasing is to make sure that the signal does not contain any sinusoidal component with a frequency equal to or greater than $f_s/2\,$ . More generally, this condition can be generalized to allow energy in some band or set of bands such that no frequencies that are aliases of each other with respect to the sample rate (according to the formulas above, for any and all values of $N\,$ ) are present in the signal.

This condition is sometimes called the Nyquist criterion, and is equivalent to saying that the sampling frequency $f_s\,$ must be high enough; either greater than twice the highest frequency or some other more complicated criterion.

In the case of a single band of width B with lower and upper frequency limits $f_1\,$ and $f_2\,$ , the criterion was spelled out by Harold Stephen Black in his 1953 book Modulation Theory. The criterion in that case is that the sampling rate must exceed $2f_2/m\,$ , where $m\,$ is the largest integer not exceeding $f_2/B\,$ . See the plot to the right, where the segments correspond to integer values of $m\,$ starting with 1.

[编辑] 名詞的起源

The term "aliasing" derives from the usage in radio engineering, where a radio signal could be picked up at two different positions on the radio dial in a superheterodyne radio: one where the local oscillator was above the radio frequency, and one where it was below. This is analogous to the frequency-space "wrapround" that is one way of understanding aliasing.

[编辑] 音訊的例子

The qualitative effects of aliasing can be heard in the following audio demonstration. Six sawtooth waves are played in succession, with the first two sawtooths having a fundamental frequency of 440 Hz (A4), the second two having fundamental frequency of 880 Hz (A5), and the final two at 1760 Hz (A6). The sawtooths alternate between bandlimited (non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22.05 kHz. The bandlimited sawtooths are synthesized from the sawtooth waveform's Fourier series such that no harmonics above the Nyquist frequency are present.

The aliasing distortion in the lower frequencies is increasingly obvious with higher fundamental frequencies, and while the bandlimited sawtooth is still clear at 1760 Hz, the aliased sawtooth is degraded and harsh with a buzzing audible at frequencies lower than the fundamental. Note that the audio file has been coded using Ogg's Vorbis codec, and as such the audio is somewhat degraded.

Sawtooth aliasing demo {440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased}

[编辑] 數學上的解釋

The preceding explanation and the Nyquist criterion are somewhat idealised, because they assume instantaneous sampling and other slightly unrealistic hypotheses, although useful approximations to these things do exist. The following is a more detailed explanation of the phenomenon in terms of function approximation theory.

[编辑] 連續訊號

For the purposes of this analysis, we define a continuous-time signal as a real or complex valued function whose domain is the interval [0,1]. To quantify the "magnitude" of a signal (and, in particular, to measure the difference between two signals), we will use the root mean square norm (see L_p spaces for some details), namely

$||f||^2 := \int_0^1|f(t)|^2\,dt.$

Accordingly, we will consider only signals that have finite norm, i.e. the square-integrable functions

$L^2=L^2([0,1]):=\left\{ f:[0,1] \rightarrow \Bbb C : ||f||<\infin\right\}.\,$

Note that these signals need not be continuous as functions; the adjective "continuous" refers only to the domain.

To be precise, we do not distinguish between functions that differ only on sets of zero measure. This technicality turns || || into a norm, and explains some of the difficulties (see the S₀ sampling method, below.) For details, see Lp spaces.

[编辑] 取樣

The conversion of a continuous signal f to an n-dimensional vector of equally spaced samples (a sampled signal) can be modeled as a point sampling operator $S 0$ , defined by $S_0 f := (f(t_1), f(t_2), \dots, f(t_n))$ , where $t i = i / n$ . That is, the function is sampled at the points $1/n, 2/n, \dots, 1.\,$

Note that $S 0$ is a linear map: for any two signals f and g, and any scalar a, then $S_0(af+g)=aS_0(f)+S_0(g).\,$

Unfortunately, while $S 0 (f)$ is well-defined if f is continuous (say), it is not well defined on the space $L 2$ defined above. A symptom of this is, even if we restrict our attention to functions f that are continuous, function $S 0$ of f is not continuous in the $L 2$ norm.

In many physically significant settings, the $L 2$ norm, or a similar norm, is an appropriate measure of similarity between signals. What will then happen is that two signals f and g that are deemed very similar to begin with will sample to two signals $S 0 f$ and $S 0 g$ which are very dissimilar.

[编辑] 更好的取樣方式(濾波)

In order to preserve closeness of signals after sampling (in other words, to get a sampling method which varies smoothly as a function of the signal f) we need to modify our sampling strategy $S 0$ . An improved method is as follows:

$S_1f(k)=n \int_{(k-1)/n}^{k/n} f(t)dt, k=1,...,n$

This is a better filtering method, as $S_1\,$ is now a continuous linear map from $L^2\,$ to $\Bbb C^n$ .

This sampling method is also a better model of how an actual machine might sample a signal. For instance, telescopes sample light signals by accumulating photons on a film or CCD receptor. The resulting image is therefore approximately the integral of all the electrons received over a period of time and over a rectangular region of the image plane.

This rectangle function filter is just one of many possible filters for sampling. In the frequency domain, it is a lowpass sinc-shaped filter, with the first zero at a frequecy of one cycle per sample, which is twice the Nyquist frequency. This zero removes all the signal energy that would alias to DC (zero frquency), and greatly attentuates all frequencies that would alias to very low frequencies. The filter does not have a sharp cutoff at the Nyquist frequency, however, so doe little prevent energy just above the Nyquist frequency aliasing to just below it. The rectangle function filter is popular in computer-generated image anti-aliasing, where it is "good enough".

[编辑] 重建

Given a sampled signal $f(k)\in \Bbb C^n$ one would like to reconstruct the original signal $f(x)\in L^2$ . This is obviously impossible in general, as $L 2$ is an infinite dimensional vector space, while $\Bbb C^n$ is a finite dimensional vector space (of dimension n.)

In practice, one picks a subspace $H \subset L^2$ of dimension n and a reconstruction linear map R from $\Bbb C^n$ to H. The purpose of R is to turn a sampled signal into a continuous one in a way that makes sense to us.

An example reconstruction map would be

$R_1s=\sum_{k=1}^n s(k) 1_{[(k-1)/n,k/n)}$

where $1 E (x)$ is 1 if $x\in E$ and 0 otherwise.

Ideally, we would have $S (R (s)) = s$ for all $s\in \Bbb C^n$ . If this occurs, then R and S both have the same picture of how signals in $L 2$ and in $\Bbb C^n$ behave, we might say that S and R are coherent. Here, $R 1$ and $S 1$ are in fact coherent, but $R 1$ and $S 0$ aren't.

Another way of saying that R and S are coherent is that R is a right-inverse for S (or S is a left-inverse for R.)

[编辑] 混疊

For any sampled signal $v\in \Bbb C^n$ the set of continuous signals $f\in L^2$ which sample to the same $v$ are called aliases of one another. The fact that there are many aliases for any one given sampled signal is called aliasing. As previously mentioned, the large quantity of aliasing is caused by $L 2$ being infinite dimensional while $\Bbb C^n$ is finite dimensional.

[编辑] 最佳化濾波

In certain physical situations, the choice of R, H or S are somehow constrained. For instance, it is usual to choose H to be the linear span of low-degree trigonometric polynomials:

$H=\left\{\sum_{k=-n}^n \alpha_k \exp 2\pi ikx\,;\, \alpha_k \in \Bbb C \right\}.\,$

Further restrictions are that, for instance, S should coincide with $S_0 \$ on H. If sufficiently many of these demands are put forward, we eventually conclude that the sampling algorithm must take a very special shape:

$S_\mathrm{opt}f=S_0(\mathrm{sinc}*f) \$

where $\mathrm{sinc}*f \$ is some sort of sinc filter or sinc function.

The reconstruction formula R is chosen so that R and S are coherent.

[编辑] 提醒

It is important to keep in mind what is much repeated in the above discussion: the Nyquist theorem, the optimality of the sinc filter, the choice of the error norm (we chose $L 2$ ) and so on are all assumptions we are making about the underlying physical problem.

In many problems, these assumptions are unsuitable, and in these cases, the Nyquist theorem might need to be modified to be more relevant to the situation at hand.

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