The General Framework Of Signal Processing&OTFS Modulation Scheme(信号处理的一般框架&OTFS调制)（4）

阅前说明：这两个部分我感觉paper的着墨不多，在说核心思想。但是既无例子，也无推导过程，尤其是OTFS Modulation这个部分更是只说了核心的部分，比起前一篇论文OTFS的调制过程显得很空。看看后面会不会详细补充吧

2.3 The General Framework Of Signal Processing

The general framework of signal processing consists of three signal representations – (1)time, (2) frequency, and (3) delay-Doppler, interchangeable by means of canonical transforms. The setting can be neatly organized in a form of a triangle, as shown in Figure 5. The nodes of the triangle represent the three representations and the edges represent the canonical transformation rules converting between them.

信号处理的基本框架由三个信号表示组成：1.时间，2.频率，3.时延多普勒，通过经典的变换可以相互转换。

(p.s.感觉变换的方向标反了，FT难道不应该是从Time到Frequency吗？)
An important property of this diagram is that the composition of any pair of transforms is equal to the remaining third one. In other words, traversing along the edges of the triangle results in the same answer no matter of which path is chosen. In particular, one can write the Fourier transform as a composition of two Zak transforms:

这个图重要的性质是任何一对变换的符合等于剩下的那个变换，这个可以很明显的看出来，类似于矢量的加法，第三条边相当于加完之后的结果。傅里叶变换可以由两个Zak变换组成：$FT=Z_t\cdot Z_f^{-1}$
This means that instead of transforming from frequency to time using the Fourier transform one can alternatively transform from frequency to delay-Doppler using the inverse Zak transform $Z_f^{-1}$ and then from delay-Doppler to time using the Zak transform $Z_t$. The above decomposition yields an alternative algorithm for computing the Fourier transform which turns out to coincide with the fast Fourier transform algorithm discovered by Cooley-Tukey3. This striking fact is an evidence that the delay-Dopplerrepresentation silently plays an important role in classical signal processing.

这意味着从频率到时间的变换可以不使用傅里叶变换，从频域使用逆Zak变换$Z_f^{-1}$变到时延多普勒域，再在时延多普勒域使用Zak变换$Z_t$变到时域，这与FFT的想法一致，表明时延多普勒表示在经典的信号处理中扮演者重要的角色？？？

Going up one level of abstraction, we note that the delay-Doppler representation is not unique but depends on a choice of a pair of periods ($\tau_r,\nu_r$) satisfying the relation $\tau_r\cdot \nu_r=1$. This implies that there is a continuous family of delay-Doppler representations, corresponding to points on the hyperbola $\nu_r=1/\tau_r$, as shown in Figure 6. It is interesting to study what happens in the limits when the variable $\tau_r$ → ∞ and when the variable $\nu_r$ → ∞. In the first limit the delay period is extended at the expense of the Doppler period contracting, thus converging in the limit to a one-dimensional representation coinciding with the time representation. Reciprocally, in the second limit, the Doppler period is extended at the expense of the delay period contracting, thus converging in the limit to a one-dimensional representation coinciding with the frequency representation. Hence, the time and frequency representations can be viewed as limiting cases of the more general family of delay-Doppler representations.

• 因为$\tau_r\cdot \nu_r=1$,我们可以取无数个满足条件的$\tau_r,\nu_r$
  
• 时间表示和频率表示是时延多普勒的两种极限情况：当$\tau_r$ → ∞时，$\nu_r\rightarrow 0$,意味着时延多普勒收缩到了时域，对偶的，当$\nu_r$ → ∞时，$\tau_r\rightarrow 0$,意味着时延多普勒收缩到了频域

All delay-Doppler representations are interchangeable by means of appropriately defined Zak transforms which satisfy commutativity relations generalizing the triangle relation discussed beforehand. This means that the conversion between any pair of representations along the curve is independent of which polygonal path is chosen to connect between them. On a philosophical note, the delay-Doppler representations and the associated Zak transforms constitute the primitive building blocks of signal processing giving rise, in particular, to the classical notions of time and frequency and the associated Fourier transformation rule.？？？
不同的时延多普勒表示可以互相变换，标黄的不太能理解什么意思。

2.4 OTFS Modulation

Communication theory is concerned with transferring information through various physical media such as wired and wireless. The vehicle that couples a sequence of information-carrying QAM symbols with the communication channel is referred to as a modulation scheme. The channel-symbol coupling thus depends both on the physics of the channel and on the modulation carrier waveforms. Consequently, every modulation scheme gives rise to a unique coupling pattern which reflects the way the modulation waveforms interact with the channel.

Classical communication theory revolves around two basic modulation schemes which are naturally associated with the time and frequency signal representations. The first scheme multiplexes QAM symbols over localized pulses in the time representation and it
is called TDM (Time Division Multiplexing). The second scheme multiplexes QAM symbols over localized pulses in the frequency representation (and transmits them using the Fourier transform) and it is called FDM (Frequency Division Multiplexing).

It is interesting to convert the TDM and FDM carrier pulses to the delay-Doppler representation using the respective inverse Zak transforms. Converting the TDM pulse reveals a quasi-periodic function that is localized in delay but non localized in Doppler. Converting the FDM pulse reveals a quasi-periodic signal that is localized in Doppler but non localized in delay. The polarized non-symmetric delay-Doppler representation of the TDM and FDM pulses suggests a superior modulation based on symmetrically localized signals in the delay-Doppler representation, as shown in Figure 7. This new modulation scheme is referred to as OTFS, which stands for Orthogonal Time Frequency and Space.

There is an infinite family of OTFS modulation schemes corresponding to different delay-Doppler representations parameterized by points of the delay-Doppler curve (as shown in Figure 6). The classical time and frequency modulation schemes, TDM and FDM, appear as limiting cases of the OTFS family, when the delay and Doppler periods approach infinity, respectively. The OTFS family of modulation schemes smoothly interpolate between time and frequency division multiplexing.