Jawaharlal Nehru Technological University Hyderabad B.Tech III Year DIGITAL SIGNAL PROCESSING Previous Question Paper (November/December - 2020 - R16)

A Linear Time-Invariant (LTI) system is characterized by two fundamental properties: linearity and time-invariance.

1. Linearity: A system is linear if it satisfies the principle of superposition. This means the response to a weighted sum of inputs is equal to the weighted sum of the individual responses. If y1(n) is the response to x1(n) and y2(n) is the response to x2(n), then the response to a(x1(n)) + b(x2(n)) must be a(y1(n)) + b(y2(n)).
2. Time-Invariance: A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. If y(n) is the response to x(n), then the response to x(n-k) must be y(n-k) for any integer k.

These properties allow LTI systems to be completely characterized by their impulse response, h(n), where the output is the convolution of the input x(n) and the impulse response h(n).

To find the impulse response h(n), we take the Z-transform of the given difference equation y(n) - 2y(n-1) + 4y(n-2) = x(n) + x(n-1) assuming zero initial conditions.

The transfer function H(z) is defined as Y(z)/X(z):

To find h(n), we express the denominator in terms of complex poles. The roots of 1 - 2z^(-1) + 4z^(-2) = 0 are z^(-1) = (2 ± sqrt(4 - 16))/8 = (1 ± i*sqrt(3))/4 = 0.5 * e^(±i*pi/3). Using partial fraction expansion or inverse Z-transform properties for complex poles, the impulse response takes the form of a damped sinusoid:

Given the magnitude r = sqrt(4) = 2 and theta = pi/3, the impulse response is: