Spectral graph signal processing is traditionally built on self-adjoint Laplacians, where orthogonal eigenbases yield an energy-preserving Fourier transform and a variational frequency ordering via a real Dirichlet form. Directed networks break self-adjointness: the combinatorial directed Laplacian $L=D_{\mathrm{out}}-A$ is generally non-normal, so eigenvectors are non-orthogonal and classical Parseval identities and Rayleigh-quotient orderings do not apply. This paper develops a Laplacian-centric harmonic analysis for directed graphs that remains exact at the algebraic level while explicitly quantifying the geometric distortion induced by non-normality. We (i) define a Biorthogonal Graph Fourier Transform (BGFT) for $L$ using dual left/right eigenbases and show that vertex energy equals a Gram-metric quadratic form in BGFT coordinates, (ii) introduce a directed variational semi-norm $TV_{\mathcal{G}}(x)=\|Lx\|_2^2$ and prove sharp two-sided BGFT-domain bounds controlled by singular values of the eigenvector matrix, and (iii) derive sampling and reconstruction guarantees with explicit stability constants that separate sampling-set informativeness from eigenvector geometry. Finally, we provide reproducible simulations comparing a normal directed cycle to perturbed non-normal digraphs and show that filtering and reconstruction robustness track $κ(V)$ and the Henrici departure-from-normality $Δ(L)$, validating the theoretical predictions.

翻译：传统谱图信号处理建立在自伴拉普拉斯算子的基础上，其正交特征基可产生能量守恒的傅里叶变换，并通过实狄利克雷形式实现变分频率排序。定向网络破坏了自伴性：组合有向拉普拉斯算子$L=D_{\mathrm{out}}-A$通常是非正规的，因此特征向量非正交，经典帕塞瓦尔恒等式和瑞利商排序不再适用。本文为有向图发展了一套以拉普拉斯算子为核心的调和分析框架，该框架在代数层面保持精确性，同时显式量化由非正规性引起的几何畸变。我们（i）利用左右对偶特征基为$L$定义双正交图傅里叶变换（BGFT），并证明顶点能量等于BGFT坐标下的格拉姆度量二次型；（ii）引入有向变分半范数$TV_{\mathcal{G}}(x)=\|Lx\|_2^2$，并证明由特征向量矩阵奇异值控制的严格双向BGFT域边界；（iii）推导具有显式稳定性常数的采样与重构保证，将采样集信息量与特征向量几何结构分离。最后，我们通过可重复仿真比较正规有向环与扰动非正规有向图，证明滤波与重构鲁棒性跟踪$κ(V)$和亨里西非正规性度量$Δ(L)$，验证了理论预测。