Problem 1 (conversion techniques):

Consider a continuous-time linear and time-invariant system, defined by the differential equation,

y"(t) + 7 y'(t) + 10 y(t) = 6 x(t).

(a) Find the impulse response and transfer function of the system. Plot the impulse response and frequency response.

(b) Convert this system to a discrete-time system through impulse invariance, and find the impulse response and transfer function of the corresponding discrete system. Plot the impulse response and frequency response.

(c) Convert this system to a discrete-time system through bilinear transformation, and find the impulse response and transfer function of the corresponding discrete system. Plot the impulse response and frequency response.

Note that the frequency response of a discrete system is from -π to π.If time-domain sampling is involved in the process, you can use Δt to denote the sample spacing.

Problem 2 (Gradient and Laplacian operators):

Based on the same concept, design (a) the gradient and (b) Laplacian operators for hexagonal image data arrays, and compare them to the 3x3 gradient and Laplacian operators for standard rectangular data arrays. (Hint: The operators for hexagonal arrays are not in the standard 3x3 or 5x5 format.)

Problem 3 (Frequency-shift keying):

(a) Describe Fourier series expansion as a special case of AM modulation and demodulation.
(b) Describe the quadrature receivers as a special case of Fourier series expansion.

Problem 4 (2D Hilbert transform):

Utilize the 2D coin image to

(a) construct the Hilbert-transform image pair, based on the same phase-shift concept,
(b) evaluate the total energy of the image pair,and
(c) check the orthogonality of the image pair.

Attachment:- Coin-image.jpg