Infinite Impulse Response (IIR) Filter

An Infinite Impulse Response (IIR) Filter is a impulse response filter that uses both current and past input values as well as past output values to compute the current output.

• Context:
  • It can (typically) be characterized by a Recursive Difference Equation, where each output is a weighted sum of the current and previous outputs.
  • It can (often) exhibit phase distortion because of its feedback mechanism, which causes the phase response to be nonlinear, unlike linear-phase FIR filters.
  • It can (typically) be designed using methods like the bilinear transform or impulse invariance, which convert analog filter designs to their digital counterparts.
  • ...
  • It can be used in real-time signal processing applications, where efficiency and quick response are crucial, such as in audio processing and communication systems.
  • It can have an impulse response that theoretically extends to infinity, although in practice, the response decays over time.
  • It can be described by its poles and zeros in the z-domain, where poles represent the system's natural frequencies and zeros represent the frequencies the filter eliminates.
  • It can be sensitive to quantization errors, especially in fixed-point arithmetic, which can potentially lead to numerical instability due to the recursive nature of its calculations.
  • ...
• Example(s):
  • A second-order Butterworth filter implemented as an IIR filter to smooth a noisy signal in an audio application.
  • A Chebyshev Type I filter used in a communication system to minimize the bandwidth of a signal while allowing for slight ripples in the passband.
  • An Elliptic Filter applied in radar signal processing to reduce interference while maintaining a steep cutoff.
  • A Notch Filter (implemented as an IIR filter) to remove a specific frequency from a signal, such as powerline noise at 60 Hz.
  • An IIR low-pass filter used in a music synthesizer to attenuate high frequencies and retain lower-frequency tones.
  • A Biquad Filter designed using IIR principles for real-time audio processing in digital equalizers.
  • An IIR high-pass filter applied to biomedical signals to remove low-frequency baseline drift in electrocardiogram (ECG) recordings.
  • A Savitzky-Golay IIR filter used in spectroscopy to smooth noisy signals while preserving the shape and features of the spectrum.
  • A State-Space IIR Filter used in control systems to model and filter real-time sensor data in robotics.
  • An Adaptive IIR Filter in wireless communication systems that adjusts its coefficients to dynamically filter noise in varying environments.
  • ...
• Counter-Example(s):
  • A Finite Impulse Response (FIR) filter, which does not use feedback and has a finite-duration impulse response.
  • A Moving Average Filter, which is a non-recursive filter with no feedback and finite impulse response.
• See: Digital Filter, Finite Impulse Response (FIR), Butterworth Filter, Chebyshev Filter, Impulse Response, Pole-Zero Plot, Signal Processing, Filter Design, Real-Time Systems, Discrete-Time Filter, Linear Time-Invariant System, Impulse Response, Finite Impulse Response, Electronic Filter, Digital Filter.

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/infinite_impulse_response Retrieved:2024-9-12.
  • Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response [math]\displaystyle{ h(t) }[/math] that does not become exactly zero past a certain point but continues indefinitely. This is in contrast to a finite impulse response (FIR) system, in which the impulse response does become exactly zero at times [math]\displaystyle{ t\gt T }[/math] for some finite [math]\displaystyle{ T }[/math] , thus being of finite duration. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as IIR systems or IIR filters.

    In practice, the impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction. For instance, analog electronic filters composed of resistors, capacitors, and/or inductors (and perhaps linear amplifiers) are generally IIR filters. On the other hand, discrete-time filters (usually digital filters) based on a tapped delay line employing no feedback are necessarily FIR filters. The capacitors (or inductors) in the analog filter have a "memory" and their internal state never completely relaxes following an impulse (assuming the classical model of capacitors and inductors where quantum effects are ignored). But in the latter case, after an impulse has reached the end of the tapped delay line, the system has no further memory of that impulse and has returned to its initial state; its impulse response beyond that point is exactly zero.

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