The Delilah was based on these two ideas from the beginning, while in the X-system they had arrived by a back-door route.

The point about sampling was that it removed the redundancy of the continuous sound wave.
所谓采样，就是去掉连续声波的冗余信息。
Any sound signal could be represented by a curve such as:
声音信号可以表示成这样的曲线：
The point was that it would be unnecessary to transmit the whole curve.
艾伦的想法是，没必要传输整条曲线，只需要传输这条线的某些点就足够了。
It would be sufficient to communicate the knowledge of certain points on the curve, provided that the recipient could thereupon perform the exercise of 'joining the dots' to reconsitute the curve.
从理论上说，接受者可以通过这些点，来拟合出一条曲线。
Since sharp wiggles would correspond to high frequencies, it followed that provided there was a limit on the frequencies contained in the signal, then a sequence of discrete points, or samples, of the curve, taken at regular intervals, would contain all the information of the signal.
因为高频信号的摆动过于剧烈，所以要在信号中限制高频，然后每隔一定的时间进行采样，得到一组离散的点，这些点就能够包含信号所需的全部信息。
Since telephone channels did in any case cut off high frequencies, the restriction on allowed 'wiggling' of the curve was no real restriction at all, and in fact a rather small number of samples could be shown to suffice to convey the signal.
因为电话频道本身就会削去高频，所以并不需要对曲线进行更多的限制，因此只需要很少量的样本，就能够传递声音信号。
The idea was well-known to communication engineers. In the X-system, it was the practice to sample each of the twelve 25 Hz channels fifty times a second.
通信工程师们对这种想法很熟悉，比如说X系统就是每秒对12个25赫兹的频道进行50次采样。
These figures were illustration of a general result, that it was necessary to sample at a rate of twice the maximum variation in frequency of the sound, or bandwidth.
一个普遍的规律是，采样率必须达到最高频率的2倍，
There was an exact mathematical result to this effect, proved as early as 1915 but which Shannon had re-stated11 and discussed with Alan at Bell Labs.
这是一个精确的数学结论，最早是由香农在1915年证明的，并在贝尔实验室与艾伦讨论过。
If, for instance, the sound signal were restricted to frequencies less than 2000 Hz, then a sample taken 4000 times a second would be exactly enough to reconstitute the signal. There would be precisely one curve of the stated frequency restriction that passed through all the sampled points.
例如，如果一个声音信号的频率不超过2000赫兹，那么每秒4000次采样就可以足够精确地表征该信号，并能得到一条通过所有采样点的曲线。
Alan described and proved this result to Don Bayley as the 'Bandwidth theorem'. His 'Why not?' had come in asking why this elegant fact could not be made the pivot on which to turn the whole encipherment process.
艾伦向唐·贝利描述并证明了这个结论，他的想法是，可以试着利用这个事实，来改变整个加密过程。