Miller Puckette 之
1.7 Periodic Signals
A signal x[n] is said to repeat at a period τ if
x[n + τ ] = x[n]
for all n. Such a signal would also repeat at periods 2τ and so on; the smallest τ if any at which a signal repeats is called the signal’s period. In discussing periods of digital audio signals, we quickly run into the difficulty of describing signals whose “period” isn’t an integer, so that the equation above doesn’t make sense. For now we’ll effectively ignore this difficulty by supposing that the signal x[n] may somehow be interpolated between the samples so that it’s well defined whether n is an integer or not.
A sinusoid has a period (in samples) of 2π/ω where ω is the angular frequency. More generally, any sum of sinusoids with frequencies 2πk/ω, for integers k, will repeat after 2π/ω samples. Such a sum is called a Fourier Series:
x[n] = a0 + a1 cos (ωn + φ1 ) + a2 cos (2ωn + φ2 ) + · · · + ap cos (pωn + φp )
Moreover, if we make certain technical assumptions (in effect that signals only contain frequencies up to a finite bound), we can represent any periodic signal as such a sum. This is the discrete-time variant of Fourier analysis which will reappear in Chapter 9.
The angular frequencies of the sinusoids above are all integer multiples of ω. They are called the harmonics of ω, which in turn is called the fundamental. In terms of pitch, the harmonics ω, 2ω, . . . are at intervals of 0, 1200, 1902, 2400, 2786, 3102, 3369, 3600, …, cents above the fundamental; this sequence of pitches is sometimes called the harmonic series. The first six of these are all quite close to multiples of 100; in other words, the first six harmonics of a pitch in the Western scale land close to (but not always exactly on) other pitches of the same scale; the third and sixth miss only by 2 cents and the fifth misses by 14.
………
文本，談點『周期』概念容易『誤解』的事。
維基百科的『周期函數』Periodic function 詞條講︰
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.
An illustration of a periodic function with period 
Definition
A function f is said to be periodic with period P (P being a nonzero constant) if we have
for all values of x in the domain. If there exists a least positive[1] constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.
Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.
A function that is not periodic is called aperiodic.
Properties
If a function f is periodic with period P, then for all x in the domain of f and all integers n,
- f(x + nP) = f(x).
If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is periodic with period P/|a|. For example, f(x)=sinx has period 2π, therefore sin(5x) will have period 2π/5.
Double-periodic functions
A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. (“Incommensurate” in this context means not real multiples of each other.)
Quotient spaces as domain
In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space:
.
That is, each element in 
is an equivalence class of real numbers that share the same fractional part. Thus a function like 
is a representation of a 1-periodic function.
也許足以澄清 Miller Puckette 文本中大多數的內容。首先一個以 
為『周期』的函數，自然也以 
為周期【※ 
非零正整數】，所以實數周期函數才會有最小周期值。
如果 
以 
為周期， 
以 
為周期，假使 
的比值是個『有理數』，可以用『最簡分數』表示成 
。也就是說 
此處的 
也就是 
的周期。然而周期不必是『有理數』，周期的『比值』當然也未必是『有理數』，因此
兩個周期函數的和，卻未必是個周期函數
！！？？
要是此時重讀《字詞網絡︰ WordNet 《一》 索引》系列文本︰
Natural Language Processing with Python
— Analyzing Text with the Natural Language Toolkit
Steven Bird, Ewan Klein, and Edward Loper
一書第二章第五節《 2.5 WordNet 》之『字詞網絡』概念階層片段
※或可參考【譯著】
Figure 2-8. Fragment of WordNet concept hierarchy: Nodes correspond to synsets; edges indicate the hypernym/hyponym relation, i.e., the relation between superordinate and subordinate concepts.
『WordNet』字詞網絡計畫啟始於一九八五年，在普林斯頓大學『認知科學實驗室』由心理學教授『喬治‧A‧米勒』 George Armitage Miller 的指導下建立和維護的英語『詞彙資料庫』 lexical database 字典。因為它包含了多種『字詞』間之『語義關係』，所以別於通常意義下的『字典』。『WordNet』是什麼？也許最好先讀讀『創造者』怎麼說︰
What is WordNet?
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the creators of WordNet and do not necessarily reflect the views of any funding agency or Princeton University.When writing a paper or producing a software application, tool, or interface based on WordNet, it is necessary to properly cite the source. Citation figures are critical to WordNet funding.
About WordNet
WordNet® is a large lexical database of English. Nouns, verbs, adjectives and adverbs are grouped into sets of cognitive synonyms (synsets), each expressing a distinct concept. Synsets are interlinked by means of conceptual-semantic and lexical relations. The resulting network of meaningfully related words and concepts can be navigated with the browser. WordNet is also freely and publicly available for download. WordNet’s structure makes it a useful tool for computational linguistics and natural language processing.
WordNet superficially resembles a thesaurus, in that it groups words together based on their meanings. However, there are some important distinctions. First, WordNet interlinks not just word forms—strings of letters—but specific senses of words. As a result, words that are found in close proximity to one another in the network are semantically disambiguated. Second, WordNet labels the semantic relations among words, whereas the groupings of words in a thesaurus does not follow any explicit pattern other than meaning similarity.
Structure
The main relation among words in WordNet is synonymy, as between the words shut and close or car and automobile. Synonyms–words that denote the same concept and are interchangeable in many contexts–are grouped into unordered sets (synsets). Each of WordNet’s 117 000 synsets is linked to other synsets by means of a small number of “conceptual relations.” Additionally, a synset contains a brief definition (“gloss”) and, in most cases, one or more short sentences illustrating the use of the synset members. Word forms with several distinct meanings are represented in as many distinct synsets. Thus, each form-meaning pair in WordNet is unique.
Relations
The most frequently encoded relation among synsets is the super-subordinate relation (also called hyperonymy, hyponymy or ISA relation). It links more general synsets like {furniture, piece_of_furniture} to increasingly specific ones like {bed} and {bunkbed}. Thus, WordNet states that the category furniture includes bed, which in turn includes bunkbed; conversely, concepts like bed and bunkbed make up the category furniture. All noun hierarchies ultimately go up the root node {entity}. Hyponymy relation is transitive: if an armchair is a kind of chair, and if a chair is a kind of furniture, then an armchair is a kind of furniture. WordNet distinguishes among Types (common nouns) and Instances (specific persons, countries and geographic entities). Thus, armchair is a type of chair, Barack Obama is an instance of a president. Instances are always leaf (terminal) nodes in their hierarchies.
Meronymy, the part-whole relation holds between synsets like {chair} and {back, backrest}, {seat} and {leg}. Parts are inherited from their superordinates: if a chair has legs, then an armchair has legs as well. Parts are not inherited “upward” as they may be characteristic only of specific kinds of things rather than the class as a whole: chairs and kinds of chairs have legs, but not all kinds of furniture have legs.
Verb synsets are arranged into hierarchies as well; verbs towards the bottom of the trees (troponyms) express increasingly specific manners characterizing an event, as in {communicate}-{talk}-{whisper}. The specific manner expressed depends on the semantic field; volume (as in the example above) is just one dimension along which verbs can be elaborated. Others are speed (move-jog-run) or intensity of emotion (like-love-idolize). Verbs describing events that necessarily and unidirectionally entail one another are linked: {buy}-{pay}, {succeed}-{try}, {show}-{see}, etc.
Adjectives are organized in terms of antonymy. Pairs of “direct” antonyms like wet-dry and young-old reflect the strong semantic contract of their members. Each of these polar adjectives in turn is linked to a number of “semantically similar” ones: dry is linked to parched, arid, dessicated and bone-dry and wet to soggy, waterlogged, etc. Semantically similar adjectives are “indirect antonyms” of the contral member of the opposite pole. Relational adjectives (“pertainyms”) point to the nouns they are derived from (criminal-crime).
There are only few adverbs in WordNet (hardly, mostly, really, etc.) as the majority of English adverbs are straightforwardly derived from adjectives via morphological affixation (surprisingly, strangely, etc.)
Cross-POS relations
The majority of the WordNet’s relations connect words from the same part of speech (POS). Thus, WordNet really consists of four sub-nets, one each for nouns, verbs, adjectives and adverbs, with few cross-POS pointers. Cross-POS relations include the “morphosemantic” links that hold among semantically similar words sharing a stem with the same meaning: observe (verb), observant (adjective) observation, observatory (nouns). In many of the noun-verb pairs the semantic role of the noun with respect to the verb has been specified: {sleeper, sleeping_car} is the LOCATION for {sleep} and {painter}is the AGENT of {paint}, while {painting, picture} is its RESULT.
More Information
Fellbaum, Christiane (2005). WordNet and wordnets. In: Brown, Keith et al. (eds.), Encyclopedia of Language and Linguistics, Second Edition, Oxford: Elsevier, 665-670
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是否會有不同之體驗乎？？！！