Jawaharlal Nehru Technological University Hyderabad B.Tech III Year II Semester DIGITAL SIGNAL PROCESSING Previous Question Paper (January/February - 2021 - R16)

Discrete-time signals are mathematical sequences represented by a function of an integer-valued independent variable n, denoted as x[n]. There are several standard ways to represent these signals, each serving specific analytical purposes.

1. Functional Representation: The signal is expressed as a mathematical formula or expression. For example, the unit step sequence is defined as x[n] = 1 for n ≥ 0 and x[n] = 0 for n < 0. This is useful for analytical derivations.

2. Graphical Representation: Signals are represented by plotting the values x[n] against the discrete time index n. This provides a visual intuition of the signal's behavior, such as its periodicity, symmetry, or decay properties.

3. Tabular Representation: The signal is presented as a table or a list of values corresponding to each index n. This is particularly useful for digital computer processing where data is stored in arrays.

4. Sequence Representation: A signal can be represented as a sequence of numbers enclosed in braces, with an arrow (↑) indicating the position of the sample at n = 0. For example, x[n] = {..., 0, 1, 2, 3, 0, ...} where the 2 is at n = 0. This is a concise notation for finite-duration signals.

5. Unit Impulse Decomposition: Any discrete-time signal can be represented as a weighted sum of shifted unit impulse sequences δ[n-k]. The mathematical representation is:

where the summation is over all integers k. This representation is fundamental to the study of Linear Time-Invariant (LTI) systems, as it allows the output of a system to be determined via convolution.

To realize the system in cascaded form, we first determine the system function H(z) by taking the Z-transform of the difference equation:

Rearranging the terms to find H(z) = Y(z)/X(z):

Next, we factor the denominator polynomial 1 - 3/4 z^-1 + 1/8 z^-2. The roots of the quadratic equation z² - 3/4 z + 1/8 = 0 are found using the quadratic formula, yielding roots at 1/2 and 1/4. Thus, the denominator factors as (1 - 1/2 z^-1)(1 - 1/4 z^-1).

The system function can be expressed as a product of two first-order sections:

The cascaded realization consists of two Direct Form I or II structures connected in series:

• Section 1 (H₁(z)): Implements the numerator (1 + 1/3 z^-1) and the denominator (1 - 1/2 z^-1). This involves an input x(n), a feedforward coefficient of 1/3, and a feedback coefficient of 1/2.
• Section 2 (H₂(z)): Implements the denominator (1 - 1/4 z^-1) with a feedback coefficient of 1/4. The output of the first section serves as the input to this second section, producing the final output y(n).