FIR (Finite Impulse Response) and IIR (Infinite Impulse Response) notch filters emerge as two separate methods for shaping frequency responses in the field of digital signal processing. Whereas both filters aim to cut down on undesirable frequencies, they vary in terms of behavior and design.

This article explains the key distinctions between FIR and IIR notch filters, making it less difficult to choose the ideal filter for a specific application.

## Differences Between FIR and IIR Notch Filter

Here are the key differences between FIR and IIR notch filters:

### 1. Block Diagram and Corresponding Equations

FIR and IIR are fundamentally different in terms of cascading stages. Let’s have a deeper look.

**FIR Notch Filter**

A Finite Impulse Response (FIR) Notch Filter is a form of digital filter used in signal processing to attenuate or eliminate a particularly narrow range of frequencies while allowing other frequencies to pass through relatively unaltered.

**Image 1: FIR Notch Filter Block Diagram**

The frequency response equation of an FIR notch filter can be described as:

where:

– H(f) is the frequency response of the filter at frequency f,

– W_{n}(f) is the notch filter’s weight function, often a windowing function like Hamming or Blackman,

– is a complex exponential term that represents the phase shift introduced by the filter at frequency f,

– n represents the filter tap index.

**IIR Notch Filter**

An Infinite Impulse Response (IIR) Notch Filter, also known as a recursive score filter, is a sort of digital filter utilized in signal processing to suppress or eliminate a narrow range of frequencies while permitting other frequencies to pass through generally unaffected.

Unlike FIR filters, IIR filters utilize feedback in their filter structure, permitting them to attain similar filtering characteristics with fewer filter coefficients.

**Image 2: IIR Notch Filter Block Diagram**

The frequency response equation of an IIR notch filter can be represented as:

where:

– H(f) is the frequency response of the filter at frequency f,

– A is the filter’s attenuation factor at the notch frequency,

– f_{n} is the notch frequency,

– is a complex exponential term that introduces the phase shift due to the filter at frequency f.

### 2. Filter Response

An FIR notch filter has a finite impulse response, which implies that a limited number of input samples can affect its output. FIR filter uses only current and past digital input samples to get the current output sample value. It does not use any past output samples.

**Image 3: FIR Filter Response**

On the other hand, an IIR filter has infinite impulse response. The IIR filter uses the current input sample value, past input, and output samples to get the current output sample value.

**Image 4: IIR Filter Response**

### 3. Stability

As there are no feedback loops and poles in FIR filters, they are inherently stable.

On the other hand, IIR filters can be unstable if it is not designed carefully. They have both poles and zeros. They contain feedback loops, and the degree of instability is determined by the structure and filter coefficients.

### 4. Implementation Complexity

A high filter order could be essential to provide a sharp notch in the frequency response of the FIR filter, which would require more filter taps and be more computationally demanding.

In contrast to FIR filters, IIR filters can deliver sharp notches with lower filter orders. As a result, there may be fewer filter coefficients and less computational complexity.

### 5. Phase Response

FIR filters give a linear phase response, which is beneficial in applications (like audio processing) where maintaining the relationship between diverse frequency components in phase is fundamental.

Nonlinear phase responses of IIR filters may cause phase distortion in the filtered signal. This may be a drawback when phase coherence is vital.

### 6. Frequency Response Design

For designing FIR filters with exact frequency responses, strategies like windowing, least squares, or frequency-sampling strategies are typically utilized.

IIR filter design can be more challenging and might include techniques like pole-zero placement that require familiarity with the filter poles, zeros, and their locations inside the complex plane.

### What is the limitation of the IIR filter?

This filter can only be utilized when the analog filter is bandlimited. It is more challenging to apply these ideas, such as calculation noise and limit cycles, in fixed-point arithmetic.

## Conclusion

The trade-off between precision and efficiency determines whether FIR or IIR should be utilized in the notch filtering spectrum. IIR energizes computer effectiveness, but the linear phase of FIR provides accuracy. Making choices that are unique to the task at hand empowers effective signal processing.

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