# How does one interpret an element of the “transfer matrix” used to calculate frequency domain granger causality (via VAR models)?

I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize autoregressive models convert the time-domain parameters to a frequency-based representation (i.e. take the Fourier transform of the original parameters), and then take the inverse of that representation to get the so-called "transfer matrix". This transfer matrix is referenced in many causality-related papers, and is the backbone of all freq. domain measures. Here is the derivation for the transfer matrix:

$$a_{i,j}(k)$$ - Autoregressive coefficient describing influence of node $$j$$ on node $$i$$ at lag=$$k$$

$$p$$ - Model-order (i.e. how many past time-points to use to predict the present)

$$\textbf{A}(k)$$ - $$Q$$-by-$$Q$$ autoregressive parameter matrix at lag=$$k$$. This matrix is estimated via model fitting.

$$U(t)$$ - Innovation noise (i.e. the difference between the model's predictions and observed data at time $$t$$.)

\begin{align} {Y(t)} = \sum_{k=1}^{p}\textbf{A}(k)Y(t-k) + U(t) \end{align}

$$\bar{\textbf{A}}(f)$$ - Frequency representation of $$\textbf{A}(k)$$

$$\bar{{a}}_{i,j}(f)$$ - Frequency representation of autoregression describing influence of node $$j$$ on node $$i$$ at freqeuncy $$f$$

$$\bar{a}_{i,j}(f) = \begin{cases} 1-\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k} & \text{if i=j} \\ -\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k)} & \text{otherwise} \end{cases}$$

\begin{align} \text{where } \textbf{j}=\sqrt{-1} \end{align}

Note that above piecewise equation stems from taking the FT of model coefs. At least as far as I understand it. I'm not actually positive as to why this equation is piecewise and not identical for all $$i$$/$$j$$ combinations. I haven't worked with FTs very much.

$$\textbf{H}(f)$$ - Transfer matrix

\begin{align} \textbf{H}(f)=\bar{\textbf{A}}^{-1}(f) \end{align}

Now for my question--does anyone have an intuitive definition for what the transfer matrix, or a given element of the transfer matrix ($$h_{i,j}(f)$$), actually represents? I can easily and intuitively understand what $$a_{i,j}(k)$$ represents, but I've yet to fully grasp what $$h_{i,j}(f)$$ represents. How does one interpret such a value in layman's terms?