Discrete Time Systems Implementation MCQ Quiz – Objective Question with Answer for Discrete Time Systems Implementation

We know that, the general linear constant coefficient difference equation characterizing an LTI discrete time system is given by the expression

y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)

The difference equation of the LTI system is given as

y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)

By applying the z-transform on both sides of the above equation and by rearranging the obtained equation, we get the rational system function as

H(z)=\(\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}\)

The computations in the given equation can be arranged into equivalent sets of difference equations. Each set of equations defines a computational procedure or an algorithm for implementing the system.

From each set of equations, we can construct a block diagram consisting of an interconnection of delay elements, multipliers, and adders is used in the realization of a system.

Computational complexity is one of the factors which is used in the implementation of the system. It refers to the numbers of Arithmetic operations (Additions, multiplications, and divisions).

According to the recent developments in the design and fabrication of rather sophisticated programmable DSPs, other factors, such as the number of times a fetch from memory is performed or the number of times a comparison between two numbers is performed per output sample, have become important in assessing the computational complexity of a given realization of a system.

Memory requirements refer to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values.

The parameters of the system must necessarily be represented with finite precision. The computations that are performed in the process of computing output from the system must be rounded off or truncated to fit within the limited precision constraints of the computer or hardware used in the implementation. Thus, Finite word length effects refer to the quantization effects that are inherent in any digital implementation of the system, either in hardware or software.

The  finite word length effects are

• Parameters of the system must be represented with finite precision
• Computations are truncated to fit in the limited precision
• Whether the computations are performed in fixed-point or floating-point arithmetic

Apart from the three factors given in the question, other factors such as, whether the structure or the realization lends itself to parallel processing or whether the computations can be pipelined are also the factors that influence our choice of the realization of the system.