In cascade form realisation of FIR system, _____ adders and _____ multipliers are required for (M - 1)^th order FIR transfer function. 

Explanation:

Cascade Form Realisation of FIR System

Definition: A Finite Impulse Response (FIR) filter is a type of digital filter that responds to a finite number of input samples before settling to zero. In the cascade form realization of an FIR system, the overall transfer function is implemented as a series of second-order sections (biquads) and possibly one first-order section if the order of the filter is odd. This method is preferred for its numerical stability and ease of implementation.

Working Principle: In cascade form realization, the FIR filter of order \( M-1 \) is decomposed into a series of smaller sections, each of which can be implemented with fewer computational resources. Each section is either a first-order or a second-order filter. These sections are then connected in series (cascaded) to achieve the desired overall filter response.

Advantages:

• Improved numerical stability compared to direct form implementations.
• Easier to design and implement using second-order sections.
• Modular structure allows for flexible design and implementation.

Disadvantages:

• Potentially higher computational complexity compared to direct form for certain applications.
• Requires careful scaling to avoid overflow in fixed-point implementations.

Correct Option Analysis:

The correct option is:

Option 2: \((M - 1)\) adders and \( M \) multipliers are required for \((M - 1)\)th order FIR transfer function.

To understand why this is the correct option, let's analyze the requirements for implementing an \((M - 1)\)th order FIR filter. An FIR filter of order \((M - 1)\) has \( M \) coefficients, denoted as \( \{b_0, b_1, \ldots, b_{M-1}\} \). The general form of the FIR filter equation is:

\( y[n] = b_0 x[n] + b_1 x[n-1] + \ldots + b_{M-1} x[n-(M-1)] \)

In cascade form realization, each section typically involves two main operations:

• Multiplication of the input or intermediate signal by the filter coefficients.
• Addition of the resulting products to form the output or intermediate signal.

For an \((M - 1)\)th order FIR filter, we need \( M \) multipliers because each coefficient \( b_i \) is used to multiply the corresponding input or intermediate signal. Additionally, we need \((M - 1)\) adders because we have to sum \( M \) products to get the final output. This is why option 2 is correct.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \((M + 1)\) adders and \((M - 1)\) multipliers.

This option is incorrect because it overestimates the number of adders and underestimates the number of multipliers. For an \((M - 1)\)th order FIR filter, we only need \((M - 1)\) adders, not \((M + 1)\). The number of multipliers should be \( M \), not \((M - 1)\).

Option 3: \((M - 1)\) adders and \((M + 1)\) multipliers.

This option is incorrect because it overestimates the number of multipliers needed. For an \((M - 1)\)th order FIR filter, we need exactly \( M \) multipliers, not \((M + 1)\). The number of adders \((M - 1)\) is correct, though.

Option 4: \( M \) adders and \((M - 1)\) multipliers.

This option is incorrect because it overestimates the number of adders needed. For an \((M - 1)\)th order FIR filter, we only need \((M - 1)\) adders. The number of multipliers should be \( M \), not \((M - 1)\).

Conclusion:

Understanding the cascade form realization of FIR filters is essential for correctly identifying the computational requirements for their implementation. An FIR filter of order \((M - 1)\) requires \((M - 1)\) adders and \( M \) multipliers. This modular and stable approach makes it suitable for various applications, despite its potential complexity compared to direct form implementations.