Filter Fundamentals

Filters are fundamental building blocks in signal processing and communication systems. A filter is a two-port network designed to pass signals within a specified frequency range while attenuating signals outside that range. Understanding the four basic filter types, their passive prototype structures, and the key frequency-domain parameters is essential for both GATE and practical circuit design.

Why Filters Exist

In any real communication or signal processing system, the signal of interest occupies a specific band of frequencies while unwanted signals, noise, or interference occupy other frequency bands. A filter selectively allows the desired frequency components to pass through while rejecting others. The word prototype refers to a normalised, idealised filter design (with cutoff frequency of 1 rad/s and source/load impedance of 1 ohm) from which practical filters are derived by frequency and impedance scaling.

Low Pass Filter (LPF)

A low pass filter passes all frequencies from DC (0 Hz) up to the cutoff frequency fc while attenuating frequencies above fc. The ideal LPF has a rectangular magnitude response: unity gain in the passband and zero gain in the stopband. The passive prototype LPF is a ladder network of series inductors and shunt capacitors. Inductors in series present high impedance to high-frequency signals, and capacitors in shunt provide a low-impedance path to ground for high-frequency signals, creating the low-pass characteristic.

The transfer function of a first-order passive RC LPF is H(jw) = 1 / (1 + jwRC). The -3dB cutoff frequency is wc = 1/RC radians/second, or fc = 1/(2*pi*RC) Hz. At this frequency, the magnitude of H equals 1/sqrt(2) and the phase shift is -45 degrees.

High Pass Filter (HPF)

A high pass filter is the dual of the LPF. It passes frequencies above fc and attenuates frequencies below fc. The passive prototype HPF is obtained by swapping inductors and capacitors in the LPF prototype: series capacitors and shunt inductors. A series capacitor blocks DC and low frequencies while passing high frequencies. The first-order HPF transfer function is H(jw) = jwRC / (1 + jwRC). At f = fc, the magnitude is again 1/sqrt(2) and the phase is +45 degrees.

Band Pass Filter (BPF)

A band pass filter passes a band of frequencies between a lower cutoff f1 and an upper cutoff f2 and attenuates frequencies outside this range. The bandwidth BW = f2 - f1, and the centre frequency f0 = sqrt(f1 * f2). For a passive BPF, the prototype is typically realised with a series LC resonator in the series arm (low impedance at resonance, passing signals near f0) and a parallel LC resonator in the shunt arm (high impedance at resonance, preventing signals near f0 from going to ground). The quality factor Q = f0/BW describes selectivity: a higher Q gives a narrower, sharper passband.

Band Stop Filter (BSF)

A band stop filter (also called a band reject filter or notch filter) is the complement of the BPF. It attenuates a specific band of frequencies while passing all others. The passive BSF prototype uses a parallel LC resonator in the series arm (high impedance at resonance, blocking signals near f0) and a series LC resonator in the shunt arm (low impedance at resonance, diverting signals near f0 to ground). A narrow-band BSF centred at a single frequency is often called a notch filter, used to eliminate a specific interference frequency such as 50 Hz power line hum.

Key Parameters

Several parameters characterise filter performance. The passband ripple specifies the allowed variation in gain within the passband. The stopband attenuation specifies the minimum attenuation (in dB) outside the passband. The transition band is the frequency range between the passband edge and the stopband edge where the filter response transitions. An ideal filter has zero transition band width, but practical filters always have a finite transition region whose width depends on filter order.

Solved Example

Given:
Passive RC Low Pass Filter: R = 10 kohm, C = 1 nF
Input: 1 V peak sinusoidal signal

Why this formula applies:
For a first-order RC LPF, the -3dB cutoff and output voltage magnitude are standard results.

Formula:
f_c = 1 / (2 * pi * R * C)
|H(jf)| = 1 / sqrt(1 + (f/f_c)^2)

Substitution for f_c:
f_c = 1 / (2 * 3.14159 * 10000 * 1e-9)
f_c = 1 / (62.832e-6)

Calculation:
f_c = 15,915 Hz ≈ 15.9 kHz

At f = 2*f_c = 31.83 kHz:
|H| = 1 / sqrt(1 + (2)^2) = 1 / sqrt(5) = 0.447 = -6.99 dB ≈ -7 dB

Final Answer:
Cutoff frequency = 15.9 kHz.
At twice the cutoff frequency, the output is attenuated to 44.7% of input, or approximately -7 dB.
For each order of filter, attenuation beyond cutoff is -20 dB/decade for first-order filters.

Exam Tip: A first-order passive filter rolls off at -20 dB/decade beyond cutoff. An n-th order filter rolls off at -20n dB/decade. GATE often asks the attenuation at a specific frequency multiple of fc using |H| = 1/sqrt(1+(f/fc)^2n).

• LPF passes 0 to fc; HPF passes above fc; BPF passes f1 to f2; BSF stops f1 to f2.
• Passive LPF prototype: series L and shunt C ladder. HPF: swap L and C. BPF and BSF use LC resonators.
• Cutoff frequency of first-order RC: fc = 1/(2*pi*RC). At fc, |H| = 0.707 = -3 dB.
• First-order filter rolls off at -20 dB/decade; n-th order rolls off at -20n dB/decade.
• BPF centre frequency f0 = sqrt(f1*f2); bandwidth BW = f2 - f1; Q = f0/BW.

Quick Revision

• LPF: series inductors + shunt capacitors. HPF: series capacitors + shunt inductors (dual of LPF).
• fc (RC LPF/HPF) = 1/(2*pi*RC). At fc, magnitude = 1/sqrt(2), phase = +/-45 deg.
• BPF: f0 = sqrt(f1*f2), BW = f2-f1, Q = f0/BW. Higher Q = narrower band.
• Rolloff: -20n dB/decade for n-th order filter beyond cutoff.
• Trap: The -3dB frequency is defined as the frequency where |H| = 1/sqrt(2), not where |H| = 0.5.
• BSF (notch filter) rejects a narrow band; used to suppress 50 Hz hum in audio systems.
• Prototype filter: normalised to wc = 1 rad/s and 1 ohm impedance; scaled for actual use.