Table 1 Forward/backward difference with periodic boundary condition in convolution/matrix form and their Fourier transform
Forward difference | Backward difference | |
|---|---|---|
Convolution | \(\partial ^{+}_{x} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}+1] - f \left [k_{1}, k_{2}\right ]\) | \(\partial ^{-}_{x} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}] - f [\!k_{1}, k_{2}-1]\) |
form | \(\partial ^{+}_{y} f[\!{\mathbf {k}}] = f[\!k_{1}+1, k_{2}] - f [\!k_{1}, k_{2}]\) | \(\partial ^{-}_{y} f[\!{\mathbf {k}}] = f[\!k_{1}, k_{2}] - f [\!k_{1}-1, k_{2}]\) |
Matrix | \(\left [ \partial ^{+}_{x} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = {\mathbf {f}} \, {\mathbf {D}_{\mathbf {n}}^{\text {T}}}\) | \(\left [ \partial ^{-}_{x} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = - {\mathbf {f}} \, {\mathbf {D}_{\mathbf {n}}}\) |
form | \(\left [ \partial ^{+}_{y} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = {\mathbf {D}_{\mathbf {m}}} {\mathbf {f}}\) | \(\left [ \partial ^{-}_{y} f[\!{\mathbf {k}}] \right ]_{{\mathbf {k}} \in \Omega } = - {\mathbf {D}_{\mathbf {m}}^{\text {T}}} {\mathbf {f}}\) |
Fourier | \(\left (e^{j \omega _{2}} - 1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) | \(- \left (e^{-j\omega _{2}} - 1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) |
transform | \(\left (e^{j\omega _{1}} - 1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) | \(- \left (e^{-j \omega _{1}} - 1 \right) F \left (e^{j {\boldsymbol {\omega }}} \right)\) |