What is the difference between FIR and IIR filters?

FIR filters implement direct convolution with a finite-length impulse response — always stable, can have linear phase, but need many taps for sharp cutoffs. IIR filters use recursive feedback — more efficient but can be unstable and have nonlinear phase.

How do you filter in the frequency domain?

FFT the signal, multiply by the desired frequency response H(ω), and inverse FFT. This is equivalent to convolving with the impulse response h(t) but is computationally faster for long filters. Remember to use zero-padding to avoid circular convolution artifacts.

What is the impulse response of a filter?

The impulse response h(t) or h[n] is the output when the input is a unit impulse (Dirac delta or Kronecker delta). It completely characterizes any LTI filter — once you know h, you can predict the output for any input via convolution. The LAPLACE Calculator can compute impulse responses from transfer functions.

Convolution and Filtering

Quick Answer

Filtering IS convolution: applying a filter to a signal means convolving the signal with the filter's impulse response. A low-pass filter smooths by convolving with a wide kernel (like a Gaussian). A high-pass filter sharpens by convolving with a kernel that computes differences. In the frequency domain (via the convolution theorem), filtering becomes multiplication: Y(ω) = X(ω)·H(ω), where H(ω) is the filter's frequency response. FIR filters directly implement discrete convolution; IIR filters implement it recursively.

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Filtering as Convolution: The Fundamental Connection

When a signal x(t) passes through a linear time-invariant (LTI) system with impulse response h(t), the output is y(t) = x(t) * h(t). This is the superposition integral — the mathematical statement that any input can be decomposed into scaled, shifted impulses, each producing a scaled, shifted copy of the impulse response, which sum to give the total output. The impulse response completely characterizes the filter's behavior. A filter that averages over a 1-second window has an impulse response that is a rectangle of width 1 second. A filter that computes the derivative has an impulse response consisting of two opposing impulses.

Key Formulas

FIR Filters: Direct Implementation of Convolution

A Finite Impulse Response filter implements y[n] = Σ h[k]·x[n−k] for k = 0 to N−1. This is exactly discrete convolution with the filter coefficients h[k]. The 'finite' means the impulse response has only N nonzero values. FIR filters are inherently stable (no feedback, no poles) and can achieve exactly linear phase (symmetric coefficients), which preserves signal shape. The tradeoff is that achieving a sharp frequency cutoff requires many coefficients (long impulse response), which means more computation. A 100-tap FIR filter requires 100 multiply-accumulate operations per output sample.

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IIR Filters: Recursive Convolution

An Infinite Impulse Response filter uses feedback: y[n] = Σ b[k]·x[n−k] − Σ a[k]·y[n−k]. The impulse response is theoretically infinite (exponentially decaying), so computing the convolution directly would be impossible. Instead, the recursive structure computes each output sample using only a few previous inputs and outputs. IIR filters achieve sharper frequency cutoffs with far fewer coefficients than FIR filters — a 2nd-order IIR can match a 50-tap FIR for basic low-pass filtering. The tradeoff: IIR filters can be unstable (if poles are outside the unit circle) and generally have nonlinear phase, which can distort signal shape.

Frequency Domain Filtering: Multiplication Instead of Convolution

The convolution theorem lets us filter in the frequency domain: Y(ω) = X(ω)·H(ω). Take the FFT of the input, multiply by the desired frequency response, and inverse FFT. This approach is conceptually simple — just zero out the frequencies you don't want — and computationally efficient for long filter kernels. An ideal low-pass filter sets H(ω) = 1 below the cutoff and H(ω) = 0 above. However, ideal brick-wall filters create ringing (Gibbs phenomenon) in the time domain because their impulse response (a sinc function) extends infinitely. Practical filters use windowed or transition-band designs to manage this tradeoff.

Choosing the Right Filter Implementation

For real-time processing with low latency, IIR filters are preferred because they need fewer computations per sample. For applications requiring linear phase (audio, seismic, biomedical), FIR filters are essential. For offline processing of long data blocks, FFT-based frequency-domain filtering is fastest. For adaptive filtering (noise cancellation, echo cancellation), FIR structures with LMS or RLS adaptation algorithms are standard because their stability is guaranteed. The LAPLACE Calculator can help analyze filter transfer functions in both continuous (analog) and discrete (digital) domains using Laplace and Z transforms respectively.

Frequently Asked Questions

Filtering IS convolution. Passing a signal through any linear time-invariant filter produces an output equal to the convolution of the input with the filter's impulse response. In the frequency domain, this convolution becomes multiplication of spectra.

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