$$X[k] = \begin{bmatrix} 10 & -2+j2 & -2 & -2-j2 \end{bmatrix}$$
3.1 The DFT of the sequence $x[n] = 1, 2, 3, 4$ is:
$$x[n_1, n_2] = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$
$$h[n] = 0.5^n u[n]$$
is:
1.1 (a) The range of values that can be represented by 12-bit signed binary numbers is -2048 to 2047.
$$y[n] = x[2n]$$
2.2 The impulse response of the system is $h[n] = \delta[n] + 2\delta[n-1] + 3\delta[n-2]$.
$$X[k_1, k_2] = \begin{bmatrix} 10 & -2 \ -2 & -2 \end{bmatrix}$$
5.1 The FIR filter with a length of 3 and coefficients $b_0 = 1, b_1 = 2, b_2 = 3$ has a transfer function: $$X[k] = \begin{bmatrix} 10 & -2+j2 & -2
2.1 (a) The even part of the signal $x[n] = \cos(0.5\pi n)$ is $x_e[n] = \cos(0.5\pi n)$.
This solution manual provides a comprehensive set of solutions to the problems and exercises in the 3rd edition of Sanjit K. Mitra's "Digital Signal Processing". The solutions are intended to help students understand the concepts and principles of digital signal processing.