Digital Signal Processing Formulas
1. Signals and Systems
Discrete-time signal
\[ x[n] \quad \text{where } n \in \mathbb{Z} \]
Linear Time-Invariant (LTI) System
\[ y[n] = \sum_{k=-\infty}^{\infty} h[k]x[n-k] \]
2. Fourier Analysis
Discrete Fourier Transform (DFT)
\[ X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N} \]
Inverse Discrete Fourier Transform (IDFT)
\[ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j2\pi kn/N} \]
3. Z-Transform
Z-Transform
\[ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n} \]
Inverse Z-Transform
\[ x[n] = \frac{1}{2\pi j} \oint X(z)z^{n-1}dz \]
4. Sampling and Reconstruction
Nyquist-Shannon Sampling Theorem
\[ f_s > 2f_{max} \]
where \(f_s\) is the sampling frequency and \(f_{max}\) is the highest frequency component in the signal.
5. Digital Filters
FIR Filter
\[ y[n] = \sum_{k=0}^{M-1} b_k x[n-k] \]
IIR Filter
\[ y[n] = \sum_{k=0}^{M} b_k x[n-k] - \sum_{k=1}^{N} a_k y[n-k] \]