Common LaTeX Equations for DSP Engineers #

The Discrete Time Fourier Transform #

X(k) = \sum_{n=0}^{N-1}x(n) \cdot e^{-j 2 \pi nk/N}, 0 \le k \le N-1 $$ X(k) = \sum_{n=0}^{N-1}x(n) \cdot e^{-j 2 \pi nk/N}, 0 \le k \le N-1 $$

FIR Filter / Convolution #

y(n) = \sum_{n=0}^{N-1}h(k) \cdot x(n-k) $$ y(n) = \sum_{n=0}^{N-1}h(k) \cdot x(n-k) $$

IIR Filter #

y(n) = \sum_{k=0}^{N}a_{k} \cdot y(n-k) + \sum_{r=0}^{M}b_{r} \cdot x(n-r)

$$ y(n) = \sum_{k=0}^{N}a_{k} \cdot y(n-k) + \sum_{r=0}^{M}b_{r}.x(n-r) $$

Discrete Correlation #

R_{xy} = \sum_{m=-\infty}^{\infty }x[m] y^*[m-k]

$$ R_{xy} = \sum_{m=-\infty}^{\infty }x[m] y^*[m-k]$$

Radar Return Power Equation #

P_{r}=\frac{P_t G A_e \sigma}{(4 \pi)^2 R^4}

$$ P_{r}=\frac{P_t G A_e \sigma}{(4 \pi)^2 R^4} $$

Radar Equation #

h(\tau, t) = \sum_{i}^{}a_i(t) e^{-j2 \pi f_c \tau_i(t)} \delta(\tau-\tau_i(t))

$$ h(\tau, t) = \sum_{i}^{}a_i(t) e^{-j2 \pi f_c \tau_i(t)} \delta(\tau-\tau_i(t)) $$

\tau_i(t) = \frac{2(R_i + v_i t)}{c}

$$ \tau_i(t) = \frac{2(R_i + v_i t)}{c} $$

where \( a_i e^{-j 2 \pi f_c \tau_i(t)} \) is the baseband time-varying gain of the reflected signal from target \( i \) and \( f_c \) is the center frequency.