Filter Types

Filters are the backbone of any signal processing system. They selectively pass or reject frequency components of a signal based on design specifications. Understanding the five fundamental filter types, their frequency responses, and their distinctions is foundational for DSP study and GATE preparation.

Core Concept: What a Filter Does

A digital filter is a mathematical operation that transforms an input sequence x[n] into an output sequence y[n] by selectively emphasizing or suppressing different frequency components. The filter is characterized by its frequency response H(e^jw), which determines how each frequency component of the input is scaled and phase-shifted at the output. The magnitude |H(e^jw)| defines the gain and the argument angle(H(e^jw)) defines the phase shift at each frequency w.

In practice, ideal brick-wall filters cannot be realized with a finite number of filter coefficients. The passband is the frequency range where |H(e^jw)| is close to 1 (signal passes through). The stopband is where |H(e^jw)| is close to 0 (signal is attenuated). The transition band lies between them.

Low Pass Filter

A low pass filter (LPF) passes all frequencies from 0 up to the cutoff frequency fc and attenuates all frequencies above fc. In discrete time, all frequencies are normalized to the range [0, pi] radians (or equivalently [0, fs/2] Hz). The LPF is the most fundamental filter type. An ideal LPF has H(e^jw) = 1 for |w| less than wc and 0 otherwise, with the ideal impulse response being a sinc function of infinite length.

High Pass Filter

A high pass filter (HPF) passes frequencies above the cutoff and attenuates those below. An HPF can be derived from an LPF by the spectral inversion: h_hp[n] = delta[n] - h_lp[n] for a symmetric FIR filter, or equivalently by multiplying the LPF impulse response by (-1)^n, which shifts the spectrum by pi radians.

Band Pass and Band Stop Filters

A band pass filter (BPF) passes a band of frequencies between a lower cutoff f1 and upper cutoff f2, rejecting all others. It is commonly used in communication receivers to select a single channel. A band stop filter (BSF), also called a notch filter, does the inverse: it rejects a specific frequency band while passing everything else. BSFs are used to eliminate interference at a known frequency, such as 50 Hz power line hum.

Allpass Filter

An allpass filter has a constant magnitude response of 1 at all frequencies, meaning it passes all frequency components without any attenuation. However, it has a non-constant phase response. Its purpose is to modify the phase of a signal without changing its spectral magnitude, typically to equalize the phase distortion introduced by other filters in the system.

Mathematical Expression

For a general linear time-invariant digital filter, the output is given by the difference equation: y[n] = sum(b_k * x[n-k]) - sum(a_k * y[n-k]). The transfer function in z-domain is H(z) = B(z)/A(z) where B(z) is the numerator polynomial of the feedforward coefficients and A(z) is the denominator polynomial of the feedback coefficients. For FIR filters, A(z) = 1, so there is no feedback.

Numerical Example

Given:
Sampling rate fs = 8000 Hz
Desired LPF cutoff = 2000 Hz
Normalized cutoff frequency wc needed

Why this formula applies:
Discrete-time frequency normalization: w = 2*pi*f / fs

Formula:
wc = 2 * pi * fc / fs (radians per sample)

Substitution:
wc = 2 * pi * 2000 / 8000
   = 2 * pi * 0.25
   = pi/2 radians per sample

Calculation:
wc = pi/2 ≈ 1.5708 radians/sample
This means the LPF cutoff is at w = pi/2 (quarter of Nyquist)

Final Answer:
Normalized cutoff wc = pi/2 radians/sample.
For HPF at same cutoff: wc_hp = pi - pi/2 = pi/2.
The HPF and LPF share the same cutoff but complementary responses.

Exam Tip: In GATE, digital filter cutoff frequencies are almost always expressed in normalized radians (0 to pi). Remember that pi corresponds to fs/2 (Nyquist), and 2*pi corresponds to fs. A common error is forgetting the factor of 2 in wc = 2*pi*fc/fs.

Quick Revision

• LPF: passes 0 to wc, stops wc to pi. Most fundamental filter type.
• HPF: passes wc to pi, stops 0 to wc. Derived from LPF by spectral inversion.
• BPF: passes w1 to w2 only. Used in channel selection and modulation.
• BSF (notch): stops w1 to w2, passes all others. Used to remove interference.
• Allpass: |H| = 1 for all w. Phase response is nonlinear. Used for phase equalization.
• Normalized frequency: wc = 2*pi*fc/fs. At Nyquist frequency fs/2, w = pi.
• Ideal filters have infinite-length impulse responses. Practical filters use windowing or optimization to approximate them.