The General Framework Of Signal Processing&OTFS Modulation Scheme(信號處理的一般框架&OTFS調製)（4）

閱前說明：這兩個部分我感覺paper的着墨不多，在說核心思想。但是既無例子，也無推導過程，尤其是OTFS Modulation這個部分更是隻說了核心的部分，比起前一篇論文OTFS的調製過程顯得很空。看看後面會不會詳細補充吧

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.

信號處理的基本框架由三個信號表示組成：1.時間，2.頻率，3.時延多普勒，通過經典的變換可以相互轉換。

(p.s.感覺變換的方向標反了，FT難道不應該是從Time到Frequency嗎？)
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:

這個圖重要的性質是任何一對變換的符合等於剩下的那個變換，這個可以很明顯的看出來，類似於矢量的加法，第三條邊相當於加完之後的結果。傅里葉變換可以由兩個Zak變換組成：$FT=Z_t\cdot Z_f^{-1}$
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.

這意味着從頻率到時間的變換可以不使用傅里葉變換，從頻域使用逆Zak變換$Z_f^{-1}$變到時延多普勒域，再在時延多普勒域使用Zak變換$Z_t$變到時域，這與FFT的想法一致，表明時延多普勒表示在經典的信號處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.

• 因爲$\tau_r\cdot \nu_r=1$,我們可以取無數個滿足條件的$\tau_r,\nu_r$
  
• 時間表示和頻率表示是時延多普勒的兩種極限情況：當$\tau_r$ → ∞時，$\nu_r\rightarrow 0$,意味着時延多普勒收縮到了時域，對偶的，當$\nu_r$ → ∞時，$\tau_r\rightarrow 0$,意味着時延多普勒收縮到了頻域

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.？？？
不同的時延多普勒表示可以互相變換，標黃的不太能理解什麼意思。

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.