Numerical Differentiation Methods

• Numerical differentiation approximates derivatives using finite difference formulas.
• Subtractive cancellation errors can arise when using the limit definition of a derivative for numerical approximation, especially with small $h$ values.

Finite Difference Approximations

• Numerical differentiation approximates derivatives using finite difference formulas.
• Forward, backward, and central difference approximations have different orders of accuracy.
• Central difference is generally more accurate than forward or backward difference.
• Error analysis using Taylor series helps determine the accuracy of these approximations.
• Smaller $h$ does not always mean more accuracy due to subtractive cancellation errors.

Approximations

• The forward difference is given by $f'(x_0)\approx \frac{f(x_0+h)-f(x_0)}{h}$
• The backward difference is given by $f'(x_0)\approx \frac{f(x_0)-f(x_0-h)}{h}$
• Central difference is given by $f'(x_0)\approx \frac{f(x_0+h)-f(x_0-h)}{2h}$
• The central difference approximation is generally more accurate than forward or backward difference approximations, which are first order accurate

Error Analysis

• Forward difference has an error term of $-\frac{h}{2}f''(x_0)+O(h^2)$, making it first-order accurate
• Backward difference has an error term of $\frac{h}{2}f''(x_0)+O(h^2)$, making it first-order accurate
• Central difference has an error term of $-\frac{h^2}{6}f'''(x_0)+O(h^4)$, making it second-order accurate

Higher-Order Approximations

• Higher-order approximations can be derived using Taylor series expansions to eliminate lower-order error terms
• A second-order accurate forward difference approximation can be derived as: $f'(x_0)\approx \frac{-f(x_0+2h)+4f(x_0+h)-3f(x_0)}{2h}$

Fourier Transform Properties

• $a \cdot f(t) + b \cdot g(t) \leftrightarrow a \cdot F(f) + b \cdot G(f)$ is the linearity property
• $f(at) \leftrightarrow \frac{1}{|a|} F(\frac{f}{a})$ is the time scaling property; Compression in time leads to expansion in frequency
• $f(t - t_0) \leftrightarrow e^{-j2\pi ft_0}F(f)$ is the time shifting property, where time shift leads to a phase shift in the frequency domain but the magnitude remains unchanged
• $e^{j2\pi f_0t}f(t) \leftrightarrow F(f - f_0)$ is the frequency shifting property, also known as modulation
• $f^(t) \leftrightarrow F^(-f)$ is the conjugation property
• $\frac{df(t)}{dt} \leftrightarrow j2\pi fF(f)$ is the time differentiation property
• $-jt f(t) \leftrightarrow \frac{dF(f)}{df}$ is the frequency differentiation property
• $\int_{-\infty}^{t} f(\tau) d\tau \leftrightarrow \frac{1}{j2\pi f} F(f) + \frac{1}{2}F(0)\delta(f)$ describes integration
• $f(t) * g(t) \leftrightarrow F(f)G(f)$ is the convolution property
• $f(t)g(t) \leftrightarrow F(f) * G(f)$ is the multiplication property
• $\int_{-\infty}^{\infty} |f(t)|^2 dt = \int_{-\infty}^{\infty} |F(f)|^2 df$ is Parseval's Theorem