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## 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}

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

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.