Essential Signal Processing Filters

Why This Matters

Filters are the workhorses of signal processing—they're how you separate the signal you want from the noise you don't. Whether you're analyzing audio waveforms, designing communication systems, or preprocessing data for Fourier analysis, you'll need to understand not just what each filter does, but why its mathematical structure produces that behavior. The concepts here connect directly to frequency domain analysis, transfer functions, and the convolution theorem you've studied throughout this course.

You're being tested on your ability to match filter types to applications, compare their frequency responses, and explain tradeoffs like phase linearity vs. computational efficiency or sharp rolloff vs. passband ripple. Don't just memorize filter names—know what frequency characteristics each filter achieves, how it's implemented (FIR vs. IIR), and when you'd choose one over another.

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Filters by Frequency Selection

These filters define which frequencies survive and which get attenuated. The key concept is the cutoff frequency $f_c$, where the filter's gain drops to $\frac{1}{\sqrt{2}}$ (or $-3$ dB) of its maximum value.

Low-Pass Filter

• Passes frequencies below $f_c$—attenuates higher frequencies, effectively "smoothing" signals by removing rapid fluctuations
• Transfer function has poles arranged to create gain decay at high frequencies; the rolloff rate depends on filter order
• Applications include anti-aliasing before sampling, removing high-frequency noise from audio, and smoothing sensor data

High-Pass Filter

• Passes frequencies above $f_c$—blocks low-frequency components including DC offset (the constant term in your Fourier series)
• Complementary to low-pass in design; subtracting a low-pass output from the original signal yields a high-pass response
• Applications include removing baseline drift in biomedical signals and emphasizing edges in image processing

Band-Pass Filter

• Passes frequencies between $f_L$ and $f_H$—isolates a specific frequency band while attenuating everything outside
• Constructed mathematically by cascading a high-pass (cutoff $f_L$) with a low-pass (cutoff $f_H$), or designed directly
• Essential for communication systems where you need to extract one channel from a multiplexed signal

Band-Stop Filter (Notch Filter)

• Attenuates frequencies between $f_L$ and $f_H$—the inverse of a band-pass filter, removing a specific frequency band
• Narrow band-stop filters called notch filters target single frequencies like 60 Hz power line interference
• Constructed by summing low-pass and high-pass outputs, or using zeros placed at the rejection frequency

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Filters by Implementation Structure

The distinction between FIR and IIR filters is fundamental to digital signal processing. It determines stability, phase response, and computational cost.

Finite Impulse Response (FIR) Filter

• Impulse response has finite duration—output depends only on current and past inputs, with no feedback from previous outputs
• Always stable and achieves linear phase—meaning all frequencies experience the same time delay, preserving waveform shape
• Designed using windowing methods (Hamming, Hanning, Blackman) or the Parks-McClellan algorithm for optimal equiripple response

Infinite Impulse Response (IIR) Filter

• Impulse response extends infinitely—feedback structure means output depends on previous outputs, creating recursive computation
• More computationally efficient than FIR for achieving sharp frequency transitions; fewer coefficients needed for equivalent performance
• Can be unstable if poles lie outside the unit circle; typically introduces nonlinear phase, causing different delays at different frequencies

Moving Average Filter

• Simplest FIR filter—output is the arithmetic mean of the last $N$ input samples: $y[n] = \frac{1}{N}\sum_{k=0}^{N-1} x[n-k]$
• Frequency response is a sinc-like function with zeros at multiples of $\frac{f_s}{N}$, where $f_s$ is the sampling frequency
• Excellent for smoothing noisy time-series data but provides poor frequency selectivity compared to designed filters

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Filters by Frequency Response Shape

These classical analog filter designs (implemented digitally via transformations) represent different tradeoffs between passband flatness, rolloff steepness, and ripple.

Butterworth Filter

• Maximally flat passband—no ripple in the passband, providing the smoothest possible frequency response near DC
• Rolloff rate of $20n$ dB/decade, where $n$ is the filter order; higher orders give sharper transitions but require more computation
• Transfer function magnitude follows $|H(j\omega)|^2 = \frac{1}{1 + (\omega/\omega_c)^{2n}}$, derived from pole placement on the unit circle

Chebyshev Filter

• Steeper rolloff than Butterworth for the same order—achieves sharper frequency transitions at the cost of introducing ripple
• Type I has ripple in the passband; Type II has ripple in the stopband—choose based on which region tolerates imperfection
• Ripple parameter $\epsilon$ controls the tradeoff: more ripple allowed means steeper rolloff achieved

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Phase-Only Filtering

All-Pass Filter

• Passes all frequencies with unity gain—magnitude response is flat, but the phase response varies with frequency
• Used for phase equalization—corrects phase distortion introduced by other filters without affecting amplitude spectrum
• Transfer function has poles and zeros that are complex conjugate reciprocals: $H(z) = \frac{z^{-1} - a^*}{1 - az^{-1}}$

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Quick Reference Table

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Self-Check Questions

1. Which two filters could you combine to create a band-pass filter, and what parameters would you need to specify?

2. You're designing a filter for a real-time embedded system with limited processing power, but phase distortion is acceptable. Should you choose FIR or IIR, and why?

3. Compare and contrast Butterworth and Chebyshev filters: what does each optimize for, and when would you choose one over the other?

4. A biomedical signal has 60 Hz power line interference superimposed on it. Which filter type would you use to remove only that interference while preserving the rest of the signal?

5. Why does an FIR filter guarantee stability while an IIR filter might become unstable? Relate your answer to the filter's pole locations in the z-plane.