I. Definition of Adjoint Operator

  • For any vectors (or functions) $x$ and $y$, the adjoint operator $F^H$ satisfies:
$$\langle F x,\;y\rangle \;=\;\langle x,\;F^H y\rangle$$
  • where the discrete inner product is defined as
$$\langle u,v\rangle = \sum_n u_n^*\,v_n.$$

II. General Steps for Deriving the Adjoint

  1. Combine Unknowns Merge all unknowns (or functions) into a single large vector $x$.

  2. Write Forward Operator Express the forward operator as a matrix (or operator) $F$ multiplied by $x$:

$$y = F x.$$

  1. Calculate Inner Product $\langle F x, y\rangle$ Expand $F x$ using summation or integral notation.

  2. Rearrange and Extract In the summation/integral, “extract” $x$ and place it in the first component of the inner product:

$$\langle F x, y\rangle = \langle x,\,\dots\rangle.$$
  1. Identify Adjoint Operator The “$\dots$” represents $F^H y$.

III. Matrix Example

  1. For matrix $A$, where the inner product is a discrete sum:
$$\langle A x,\,y\rangle = \sum_i (A x)_i^*\,y_i = \sum_{j} x_j^* \Bigl(\sum_i A_{ij}^*\,y_i\Bigr) = \langle x,\,A^H y\rangle.$$
  1. Therefore, $A^H$ is the complex conjugate transpose.

IV. Composite Operator Example

Forward:

$$y = A_1\,I_w + A_2\,I_f$$

Combine unknowns:

$$x = \begin{bmatrix}I_w\\I_f\end{bmatrix}, \quad F = [\,A_1\;\;A_2\,], \quad y = F x$$

Derive Adjoint:

$$\begin{aligned} \langle F x, y\rangle &= \langle A_1 I_w + A_2 I_f,\;y\rangle \\ &= \langle I_w,\;A_1^H y\rangle + \langle I_f,\;A_2^H y\rangle \\ &= \bigl\langle x,\;\begin{bmatrix}A_1^H y\\ A_2^H y\end{bmatrix}\bigr\rangle. \end{aligned}$$

Therefore

$$F^H y = \begin{bmatrix}A_1^H y\\ A_2^H y\end{bmatrix}.$$

V. Practical Tips

  • Remember the Definition: The adjoint operator moves $F$ from the “left” to the “right” side of the inner product, with transpose/conjugate operations.
  • Process Operators Term by Term: Handle matrices, convolutions, derivatives, etc., by deriving from the perspective of inner products.
  • Check Equation Count: Adjoint operators are often simpler than inverse operators and don’t require invertibility, only satisfying the inner product relationship.

Tip: > – First write out the specific expression for $\langle F x,y\rangle$ on paper, gradually extracting $x$;
– Adjoint operators often involve transpose, conjugate, convolution kernel reversal, differential sign flipping, and similar operations.