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下面的解析解怎么继续化简：\begin{align} |r_k|^2 &= \sum_{n=1}^N...

創建於：2025年12月21日

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問題

下面的解析解怎么继续化简：\begin{align}
|r_k|^2 &= \sum_{n=1}^N\big|\mathbf{f}n\mathbf{x}\big|^2\exp!\left(j\frac{2\pi k (n-1)}{N}\right)\sum{m=1}^N\big|\mathbf{f}m\mathbf{x}\big|^2\exp!\left(-j\frac{2\pi k (m-1)}{N}\right)\notag\
& = \sum{n=1}^N\sum_{m=1}^N \big|\mathbf{f}_n\mathbf{x}\big|^2\big|\mathbf{f}_m\mathbf{x}\big|^2\exp!\left(j\frac{2\pi k (n-m)}{N}\right)
\end{align}

where
\begin{align}
&\big|\mathbf{f}n\mathbf{x}\big|^2\big|\mathbf{f}m\mathbf{x}\big|^2 \notag\
&= \sum{p=1}^N\sum{q=1}^N\exp\bigg[j(\mathbf{f}p^H-\mathbf{f}q^H)\mathbf{v})\bigg]\exp\left(-j\frac{2\pi(n-1)(p-q)}{N}\right)\sum{p'=1}^N\sum{q'=1}^N\exp\bigg[j(\mathbf{f}{p'}^H-\mathbf{f}{q'}^H)\mathbf{v})\bigg]\exp\left(-j\frac{2\pi(m-1)(p'-q')}{N}\right)\notag\
&= \sum_{p,q,p'q'}\exp(j\mathbf{a}^H\mathbf{v})\exp\left(-j\frac{2\pi(n-1)(p-q)}{N}\right)\exp\left(-j\frac{2\pi(m-1)(p'-q')}{N}\right)
\end{align}
with
\begin{equation}
\mathbf{a}^H = \mathbf{f}p^H-\mathbf{f}q^H+\mathbf{f}{p'}^H-\mathbf{f}{q'}^H
\end{equation}

思考

Simplifying algebraic expression with DFT concepts

The user is asking how to further simplify an algebraic expression relating to exponentials and DFT. I’m looking at how to express ∣rk∣2|r_k|^2 in terms of quadruple sums and delta functions, exploring the relationship with DFT rows and phase vectors. The goal might involve expanding terms over multiple indices and using properties like orthogonality.

Evaluating expansion of |r_k|^2

I'm working through the expression for ∣rk∣2|r_k|^2, breaking it down into sums of exponentials. By rearranging the sums, I find geometric series forming Kronecker deltas. Ultimately, the sum over nn produces a delta condition on kk and the mm sum imposes a condition on the difference between indices. This hints at certain structure and constraints on the terms.

Simplifying expression for |r_k|^2

I'm simplifying the expression for ∣rk∣2|r_k|^2 by focusing on the mod N constraints. After making a coordinate change by setting q=p−k(mod  N)q = p - k (\mod N) and q′=p′+k(mod  N)q' = p' + k (\mod N), I can reduce the expression to a sum over pp and p′p' terms involving fpf_p and fp−kf_{p-k}.

Considering simplifications for |r_0|^2

I’m looking at the case where k=0k = 0. The deltas simplify to p=qp = q and p′=q′p' = q', making aH=0a^H = 0, so everything becomes exp⁡(0)=1\exp(0) = 1. The sum of 1's over pp and p′p' ends up being N4N^4, but I’m trying to ensure this matches with ∣r0∣2|r_0|^2. The user wants me to continue simplifying, though I’m still not entirely sure about the normalization factor.

Step-by-step simplification with orthogonality identity

To simplify the expression, I’ll use the orthogonality identity for sums of complex exponentials. By applying this identity, the sums over nn and mm are eliminated through the use of modulo delta functions. With some careful substitution, the expression reduces to a simplified formula:

∣rk∣2=N2∑p∑p′exp⁡(j[fpH−fp−kH+fp′H−fp′+kH]v)|r_k|^2 = N^2 \sum_{p} \sum_{p'} \exp(j [f_p^H - f_{p-k}^H + f_{p'}^H - f_{p'+k}^H] v)

This step-by-step breakdown will help clarify the simplification.

Examining conjugacy and simplifying expression

I'm thinking the second sum could be a complex conjugate of the first, but I need to check if that holds for this expression. If I rename the index, it might work out as a conjugate under certain conditions. However, keeping the expression with delta functions feels cleaner. I’ll move forward with that approach for clarity in the final result.

Thought for 38s

回答

把你给的展开式直接代回去，然后先把关于 nn、mm 的和做掉（用离散傅里叶指数的正交性）就能大幅化简。

1) 先把 ∣fnx∣2∣fmx∣2\big|\mathbf f_n\mathbf x\big|^2\big|\mathbf f_m\mathbf x\big|^2 代回 ∣rk∣2|r_k|^2

∣rk∣2=∑n=1N∑m=1N∣fnx∣2∣fmx∣2ej2πk(n−m)N|r_k|^2=\sum_{n=1}^N\sum_{m=1}^N \big|\mathbf f_n\mathbf x\big|^2\big|\mathbf f_m\mathbf x\big|^2 e^{j\frac{2\pi k(n-m)}{N}}

代入你最后一行（把 ∑p,q,p′,q′\sum_{p,q,p',q'} 记为四重求和）：

∣rk∣2=∑n,m∑p,q,p′,q′exp⁡(jaHv) exp⁡ ⁣(−j2π(n−1)(p−q)N)exp⁡ ⁣(−j2π(m−1)(p′−q′)N)exp⁡ ⁣(j2πk(n−m)N).\begin{aligned} |r_k|^2 &=\sum_{n,m}\sum_{p,q,p',q'} \exp(j\mathbf a^H\mathbf v)\, \exp\!\left(-j\frac{2\pi(n-1)(p-q)}{N}\right) \exp\!\left(-j\frac{2\pi(m-1)(p'-q')}{N}\right) \exp\!\left(j\frac{2\pi k(n-m)}{N}\right). \end{aligned}

注意

ej2πk(n−m)N=ej2πk(n−1)N  e−j2πk(m−1)Ne^{j\frac{2\pi k(n-m)}{N}}=e^{j\frac{2\pi k(n-1)}{N}}\;e^{-j\frac{2\pi k(m-1)}{N}}

于是把 nn 和 mm 的项分开：

∣rk∣2=∑p,q,p′,q′exp⁡(jaHv)[∑n=1Nej2πN(k−(p−q))(n−1)][∑m=1Ne−j2πN(k+(p′−q′))(m−1)].\begin{aligned} |r_k|^2 &=\sum_{p,q,p',q'}\exp(j\mathbf a^H\mathbf v) \left[\sum_{n=1}^N e^{j\frac{2\pi}{N}(k-(p-q))(n-1)}\right] \left[\sum_{m=1}^N e^{-j\frac{2\pi}{N}(k+(p'-q'))(m-1)}\right]. \end{aligned}

2) 用指数正交性把 ∑n\sum_n、∑m\sum_m 化成 Kronecker delta

用恒等式（离散傅里叶基正交）：

∑n=1Nej2πN(n−1)ℓ={N,ℓ≡0 (mod N)0,otherwise.\sum_{n=1}^N e^{j\frac{2\pi}{N}(n-1)\ell} = \begin{cases} N,& \ell\equiv 0\ (\mathrm{mod}\ N)\\ 0,& \text{otherwise}. \end{cases}

定义模 NN 的“delta”：

δN(ℓ)={1,ℓ≡0 (mod N)0,otherwise.\delta_N(\ell)= \begin{cases} 1,& \ell\equiv 0\ (\mathrm{mod}\ N)\\ 0,& \text{otherwise}. \end{cases}

则

∑n=1Nej2πN(k−(p−q))(n−1)=N δN ⁣(k−(p−q)),\sum_{n=1}^N e^{j\frac{2\pi}{N}(k-(p-q))(n-1)} = N\,\delta_N\!\big(k-(p-q)\big), ∑m=1Ne−j2πN(k+(p′−q′))(m−1)=N δN ⁣(k+(p′−q′)).\sum_{m=1}^N e^{-j\frac{2\pi}{N}(k+(p'-q'))(m-1)} = N\,\delta_N\!\big(k+(p'-q')\big).

所以立刻得到非常紧的形式：

∣rk∣2=N2∑p,q,p′,q′exp⁡(jaHv)  δN ⁣(k−(p−q))  δN ⁣(k+(p′−q′))\boxed{ |r_k|^2 = N^2\sum_{p,q,p',q'} \exp(j\mathbf a^H\mathbf v)\; \delta_N\!\big(k-(p-q)\big)\; \delta_N\!\big(k+(p'-q')\big) }

其中

aH=fpH−fqH+fp′H−fq′H.\mathbf a^H=\mathbf f_p^H-\mathbf f_q^H+\mathbf f_{p'}^H-\mathbf f_{q'}^H.

3) 用 delta 直接消掉 q,q′q,q'（变成双重和）

δN(k−(p−q))\delta_N(k-(p-q)) 强制

p−q≡k (mod N)⇒q≡p−k (mod N),p-q\equiv k\ (\mathrm{mod}\ N)\quad\Rightarrow\quad q\equiv p-k\ (\mathrm{mod}\ N),

δN(k+(p′−q′))\delta_N(k+(p'-q')) 强制

p′−q′≡−k (mod N)⇒q′≡p′+k (mod N).p'-q'\equiv -k\ (\mathrm{mod}\ N)\quad\Rightarrow\quad q'\equiv p'+k\ (\mathrm{mod}\ N).

令“取模回到 {1,…,N}\{1,\dots,N\}”的索引算子

⟨t⟩N:=((t−1) mod N)+1,\langle t\rangle_N := ((t-1)\bmod N)+1,

则

q=⟨p−k⟩N,q′=⟨p′+k⟩N.q=\langle p-k\rangle_N,\qquad q'=\langle p'+k\rangle_N.

代回 aH\mathbf a^H 得到

∣rk∣2=N2∑p=1N∑p′=1Nexp⁡ ⁣(j(fpH−f⟨p−k⟩NH+fp′H−f⟨p′+k⟩NH)v)\boxed{ |r_k|^2 = N^2\sum_{p=1}^N\sum_{p'=1}^N \exp\!\Big( j\big(\mathbf f_p^H-\mathbf f_{\langle p-k\rangle_N}^H +\mathbf f_{p'}^H-\mathbf f_{\langle p'+k\rangle_N}^H\big)\mathbf v \Big) }

而且因为指数里是“和”，还能写成乘积（纯代数分解）：

∣rk∣2=N2(∑p=1Ne j(fpH−f⟨p−k⟩NH)v)(∑p′=1Ne j(fp′H−f⟨p′+k⟩NH)v)\boxed{ |r_k|^2 = N^2\left(\sum_{p=1}^N e^{\,j(\mathbf f_p^H-\mathbf f_{\langle p-k\rangle_N}^H)\mathbf v}\right) \left(\sum_{p'=1}^N e^{\,j(\mathbf f_{p'}^H-\mathbf f_{\langle p'+k\rangle_N}^H)\mathbf v}\right) }

如果你能补充一句：fn\mathbf f_n 是否就是 DFT 矩阵的第 nn 行/列、以及 v\mathbf v 是否为实向量（或保证 (fpH−fqH)v(\mathbf f_p^H-\mathbf f_q^H)\mathbf v 为实数），那我还可以把上面的第二个括号进一步变成第一个括号的共轭，从而写成一个漂亮的 ∣⋅∣2|\cdot|^2 形式。

問題

我们现在要求\exp(j\mathbf{a}^H\mathbf{v})这一项一点也不要动！不要往里面放k！我们看看能不能用什么性质，把m、n用p、q、p',q'表示

思考

Explaining sums with Kronecker delta constraints

The user wants to avoid inserting k into the exponent. We can express sums over m,n using p,q,p',q' by applying Kronecker deltas, which constraints the indices. However, we need to demonstrate the sums yielding the link without introducing k into a. We’ll stop at the delta stage and consider reparameterization.

Clarifying orthogonality and Kronecker delta relations

The sums over n give a delta enforcing k = p - q mod N. The n’s themselves aren't directly expressed. When summing over n, we get a delta between k and p - q, without involving n. If they want to represent sums through p, we can align phases, but sum averages out over n. So, the result is a delta expression, using DFT orthogonality.

Thought for 11s

回答

可以的，而且完全不需要动 exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v)（更不会把 kk “塞进” a\mathbf a 里）。关键是把 n,mn,m 的求和用 DFT 指数正交性直接“做掉”。

1) 先把所有与 n,mn,m 有关的指数合并，但保持 exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v) 原样

你已经有

∣rk∣2=∑n=1N∑m=1N∑p,q,p′,q′exp⁡(jaHv)  e−j2π(n−1)(p−q)N  e−j2π(m−1)(p′−q′)N  ej2πk(n−m)N.|r_k|^2=\sum_{n=1}^N\sum_{m=1}^N \sum_{p,q,p',q'} \exp(j\mathbf a^H\mathbf v)\; e^{-j\frac{2\pi(n-1)(p-q)}{N}}\; e^{-j\frac{2\pi(m-1)(p'-q')}{N}}\; e^{j\frac{2\pi k(n-m)}{N}} .

把 ej2πk(n−m)Ne^{j\frac{2\pi k(n-m)}{N}} 拆成

ej2πk(n−m)N=ej2πk(n−1)N  e−j2πk(m−1)Ne^{j\frac{2\pi k(n-m)}{N}} = e^{j\frac{2\pi k(n-1)}{N}}\;e^{-j\frac{2\pi k(m-1)}{N}}

于是 nn 与 mm 的项完全分离：

∣rk∣2=∑p,q,p′,q′exp⁡(jaHv)  ∑n=1Nej2πN(k−(p−q))(n−1)⏟只剩 n 的和∑m=1Ne−j2πN(k+(p′−q′))(m−1)⏟只剩 m 的和.\begin{aligned} |r_k|^2 &=\sum_{p,q,p',q'} \exp(j\mathbf a^H\mathbf v)\; \underbrace{\sum_{n=1}^N e^{j\frac{2\pi}{N}(k-(p-q))(n-1)}}_{\text{只剩 }n\text{ 的和}} \underbrace{\sum_{m=1}^N e^{-j\frac{2\pi}{N}(k+(p'-q'))(m-1)}}_{\text{只剩 }m\text{ 的和}}. \end{aligned}

到这里为止，exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v) 完全没动。

2) 用正交性把 n,mn,m 求和变成“选择条件”（Kronecker delta）

用离散傅里叶正交性：

∑n=1Nej2πN(n−1)ℓ=N δN(ℓ),δN(ℓ)={1,ℓ≡0 (mod N)0,otherwise.\sum_{n=1}^N e^{j\frac{2\pi}{N}(n-1)\ell} = N\,\delta_N(\ell), \quad \delta_N(\ell)= \begin{cases} 1,& \ell\equiv 0\ (\mathrm{mod}\ N)\\ 0,& \text{otherwise}. \end{cases}

所以

∑n=1Nej2πN(k−(p−q))(n−1)=N δN ⁣(k−(p−q)),\sum_{n=1}^N e^{j\frac{2\pi}{N}(k-(p-q))(n-1)} = N\,\delta_N\!\big(k-(p-q)\big), ∑m=1Ne−j2πN(k+(p′−q′))(m−1)=N δN ⁣(k+(p′−q′)).\sum_{m=1}^N e^{-j\frac{2\pi}{N}(k+(p'-q'))(m-1)} = N\,\delta_N\!\big(k+(p'-q')\big).

代回去得到一个把 n,mn,m 完全消掉的式子：

∣rk∣2=N2∑p,q,p′,q′exp⁡(jaHv)  δN ⁣(k−(p−q))  δN ⁣(k+(p′−q′))\boxed{ |r_k|^2 = N^2\sum_{p,q,p',q'} \exp(j\mathbf a^H\mathbf v)\; \delta_N\!\big(k-(p-q)\big)\; \delta_N\!\big(k+(p'-q')\big) }

注意：这里 kk 只出现在两个 δN(⋅)\delta_N(\cdot) 里，没有进入 exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v)，因此满足你“exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v) 一点也不要动”的要求。

3) 你说的“把 m,nm,n 用 p,q,p′,q′p,q,p',q' 表示”——更准确地说是：用 p,q,p′,q′p,q,p',q' 约束掉 m,nm,n

严格来说，做完求和后 m,nm,n 已经不存在了，所以不是“把 m,nm,n 表示出来”，而是：

  • nn-求和把贡献限制为： k≡(p−q)(modN)k\equiv (p-q)\pmod N
  • mm-求和把贡献限制为： k≡−(p′−q′)(modN)k\equiv -(p'-q')\pmod N

你也可以把上式写成“受限求和”的形式（仍然不动指数项）：

∣rk∣2=N2 ⁣ ⁣ ⁣∑p,q,p′,q′p−q≡k (N)p′−q′≡−k (N)exp⁡(jaHv)\boxed{ |r_k|^2 = N^2\!\!\!\sum_{\substack{p,q,p',q'\\ p-q\equiv k\ (N)\\ p'-q'\equiv -k\ (N)}} \exp(j\mathbf a^H\mathbf v) }

如果你愿意，我还能在不把 kk 放进 exp⁡(jaHv)\exp(j\mathbf a^H\mathbf v)的前提下，把四重求和换成“先对差分 d=p−qd=p-q、d′=p′−q′d'=p'-q' 求和”的形式（让结构更像卷积/相关），但核心化简就已经是上面这一步： 用正交性消掉 m,nm,n，留下对 (p,q,p′,q′)(p,q,p',q') 的模 NN 约束条件。

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