FIR vs. IIR

Attribute	FIR	IIR
Definition	Finite Impulse Response	Infinite Impulse Response
Filter Type	Non-recursive	Recursive
Impulse Response	Finite duration	Infinite duration
Stability	Always stable	May be unstable
Phase Response	Linear phase	Non-linear phase
Memory Requirement	Higher	Lower
Computational Complexity	Lower	Higher
Frequency Response	Can approximate ideal response	Exact response possible
Filter Design	Design methods available	Design methods available

Introduction

When it comes to digital signal processing, filters play a crucial role in manipulating and enhancing signals. Two commonly used types of filters are Finite Impulse Response (FIR) filters and Infinite Impulse Response (IIR) filters. While both FIR and IIR filters serve the purpose of filtering signals, they differ in various aspects, including their design, characteristics, and applications. In this article, we will explore and compare the attributes of FIR and IIR filters to gain a better understanding of their strengths and weaknesses.

Design

FIR filters are characterized by their impulse response, which is finite in length. This means that the output of an FIR filter is solely determined by a weighted sum of its input samples and the filter coefficients. The coefficients of an FIR filter are typically obtained through techniques such as windowing or frequency sampling. On the other hand, IIR filters have an impulse response that extends infinitely into the past and future. The coefficients of an IIR filter are obtained by solving a set of recursive difference equations, which allows for feedback within the filter structure.

Stability

One of the key differences between FIR and IIR filters lies in their stability characteristics. FIR filters are inherently stable due to their finite impulse response. Since there is no feedback within the filter structure, there is no possibility of instability. On the contrary, IIR filters can be unstable if not designed carefully. The feedback nature of IIR filters introduces the potential for poles in the transfer function to lie outside the unit circle, leading to instability. Therefore, stability analysis and design considerations are crucial when working with IIR filters.

Frequency Response

The frequency response of a filter describes how it affects different frequencies within a signal. FIR filters have a linear phase response, which means that all frequency components of the input signal experience the same delay. This property is desirable in applications where preserving the phase relationship between different frequency components is important, such as audio processing or image filtering. On the other hand, IIR filters may introduce nonlinear phase shifts, which can distort the phase relationship between different frequency components. However, IIR filters often offer sharper roll-off characteristics and can achieve a steeper transition between passbands and stopbands compared to FIR filters.

Computational Complexity

Another aspect to consider when comparing FIR and IIR filters is their computational complexity. FIR filters are generally more computationally intensive compared to IIR filters. This is because the output of an FIR filter is calculated by convolving the input signal with the filter coefficients, which requires a large number of multiplications and additions. In contrast, IIR filters can achieve similar filtering characteristics with fewer computations, as the recursive nature of their difference equations allows for more efficient implementation. Therefore, if computational resources are limited, IIR filters may be a more suitable choice.

Filter Length

The length of a filter refers to the number of taps or coefficients it possesses. FIR filters typically require a larger number of coefficients to achieve a desired frequency response compared to IIR filters. This is because FIR filters do not have the advantage of feedback, which limits their ability to achieve sharp roll-off characteristics with a small number of coefficients. On the other hand, IIR filters can achieve similar frequency responses with a smaller number of coefficients, thanks to their recursive nature. Therefore, if memory or computational resources are limited, IIR filters may be preferred due to their compactness.

Applications

Both FIR and IIR filters find applications in various domains of signal processing. FIR filters are commonly used in applications where linear phase response and stability are critical, such as audio equalization, echo cancellation, and image processing. Their ability to provide precise control over the frequency response makes them suitable for applications that require accurate filtering characteristics. On the other hand, IIR filters are often employed in applications where computational efficiency and compactness are prioritized, such as real-time audio processing, control systems, and wireless communication. The recursive nature of IIR filters allows for efficient implementation and reduced memory requirements.

Conclusion

In conclusion, FIR and IIR filters have distinct attributes that make them suitable for different applications. FIR filters offer stability, linear phase response, and precise control over the frequency response, but at the cost of higher computational complexity and larger filter lengths. On the other hand, IIR filters provide computational efficiency, compactness, and sharper roll-off characteristics, but require careful design considerations to ensure stability. Understanding the strengths and weaknesses of FIR and IIR filters is crucial in selecting the appropriate filter type for a given signal processing task, taking into account the specific requirements and constraints of the application.