Frequency response and filtering

Lesson 7 showed that an LTI system multiplies the spectrum of its input by its frequency response, H(jω). This lesson puts that fact to work. Amara's microphone channel needs three filters. A high-pass filter must remove the 60 Hz hum picked up from the cabinet wiring, a low-pass filter in front of the ADC must remove everything that the 8 kHz sampling rate cannot represent, and the noise detector needs a band-pass filter that keeps the frequencies where traffic is loud and people are sensitive. Designing them means reading and drawing frequency responses, expressing gains in decibels, and accepting that real filters only approximate the ideal shapes.

What the frequency response means

Because complex exponentials are eigenfunctions of LTI systems, a sinusoid x(t) = A cos(ωt + φ) passes through a stable LTI system and comes out as a sinusoid of the same frequency:

The magnitude |H(jω)| is the gain at that frequency and the angle ∠H(jω) is the phase shift. Measuring a filter on the bench means exactly this: drive it with a sine at one frequency, measure the ratio of output to input amplitude and the time shift between them, then step to the next frequency. For a system described by its impulse response, H(jω) is the Fourier transform of h(t); for a circuit, it comes from replacing each capacitor by 1/(jωC) and each inductor by jωL and solving the circuit as a voltage divider.

Decibels

Gains are usually quoted in decibels (dB), 20 log₁₀|H|, because filters span huge ranges and because the gains of cascaded stages, which multiply, add in decibels. A gain of 1 is 0 dB, 10 is +20 dB, 0.1 is −20 dB and 0.01 is −40 dB. The factor 1/√2 ≈ 0.707 is −3 dB, the point where the output power has fallen by half. A useful memory aid: every factor of 2 in amplitude is about 6 dB, and every factor of 10 is 20 dB. Frequencies are plotted on a logarithmic axis, measured in decades (factors of 10) or octaves (factors of 2).

The first-order low-pass filter and its Bode plot

The RC low-pass filter has frequency response

Its magnitude in decibels is −10 log₁₀(1 + (ω/ω_c)²) and its phase is −arctan(ω/ω_c). A Bode plot draws both against log frequency. Well below the corner frequency ω_c, the magnitude is close to 0 dB and the phase close to 0°. Well above it, the magnitude falls by 20 dB per decade, or 6 dB per octave, and the phase approaches −90°. At the corner itself, the magnitude is exactly −3 dB and the phase −45°. Drawing the two straight-line asymptotes, flat to the corner and then sloping at −20 dB per decade, gives a sketch that is never more than 3 dB wrong. For Amara's filter, RC = 0.16 ms, so ω_c = 6,250 rad/s and f_c = ω_c/2π ≈ 995 Hz.

High-pass, band-pass and notch filters

Swapping the resistor and capacitor turns the low-pass circuit into a high-pass filter, with H(jω) = (jω/ω_c)/(1 + jω/ω_c). Its magnitude rises at +20 dB per decade below the corner and is flat above it. Amara puts one in front of the microphone preamplifier with a corner at 200 Hz to suppress the 60 Hz hum. At 60 Hz, f/f_c = 0.3 and the gain is 0.3/√1.09 ≈ 0.287, or −10.8 dB. That is a useful reduction but not a cure, so she also re-routes the microphone cable away from the cabinet's power wiring. A first-order filter is gentle; to remove an interfering frequency much more strongly, engineers use steeper filters or a notch filter tuned to that one frequency.

A band-pass filter passes a range between two corners and attenuates both sides; cascading a high-pass and a low-pass filter makes a simple one. The noise detector uses a band-pass filter from about 300 Hz to 3 kHz, roughly the range where traffic noise and human hearing overlap most, while the anti-aliasing filter and the hum filter handle the edges. A notch, or band-stop, filter removes a narrow band and passes everything else.

Ideal filters and why they cannot be built

The ideal low-pass filter has gain 1 up to a cut-off frequency ω_c and gain 0 above it, with zero phase. It would remove all high-frequency noise and leave the wanted band untouched. Its impulse response, found from the inverse Fourier transform of a rectangle in frequency, is a sinc function in time, h(t) = (ω_c/π)·sin(ω_c t)/(ω_c t). That sinc extends forever in both directions, including negative time, so the ideal filter is non-causal: it would have to respond before its input arrived. Every real filter is therefore a compromise. Its specification lists a passband in which the gain may ripple by a small amount, a stopband in which the gain must stay below some level, and a transition band between them. Sharper transitions cost more filter stages, more components and more delay.

Higher-order filters

Cascading identical first-order stages with buffers between them multiplies their responses, which adds their decibel gains: two stages fall at −40 dB per decade, and n stages at −20n dB per decade. The corner becomes soft, though, because each stage is already 3 dB down at the nominal corner. Purpose-designed filters place their poles more cleverly. The Butterworth filter is the most common choice: its magnitude is as flat as possible in the passband, and for order n,

so it is exactly −3 dB at ω_c for every order and falls at −20n dB per decade beyond it. Other families trade flatness for sharpness: Chebyshev filters allow ripple in the passband to get a steeper transition, and Bessel filters give up sharpness to keep the phase close to linear. ELE-351 designs these in detail.

Phase, delay and distortion

Magnitude is only half of the frequency response. A filter whose phase is a straight line through the origin, ∠H = −ωt_d, delays every frequency by the same time t_d, so the waveform shape is preserved. If the phase is not linear, different frequencies are delayed by different amounts and a sharp waveform smears out, even when every frequency passes at full strength. The delay of a narrow group of frequencies around ω is the group delay, τ_g = −d∠H/dω. For the RC filter, τ_g = RC/(1 + (ω/ω_c)²): 0.16 ms at low frequencies, falling to half that at the corner. For an audio locator tone this is irrelevant, but in data links and in control loops phase and delay matter as much as gain.

import math

def butter_mag_db(f, fc, n):
    return -10 * math.log10(1 + (f / fc) ** (2 * n))

fc = 2000.0
print("   f (Hz)   n=1     n=2     n=3     n=4    (dB)")
for f in (300, 880, 2000, 4000, 6000, 8000):
    row = "  ".join(f"{butter_mag_db(f, fc, n):6.1f}" for n in (1, 2, 3, 4))
    print(f"{f:8d}  {row}")

# First-order RC low-pass: phase and group delay
RC = 0.16e-3
for f in (100, 995, 3000):
    w = 2 * math.pi * f
    phase = -math.degrees(math.atan(w * RC))
    tg = RC / (1 + (w * RC) ** 2)
    print(f"RC filter at {f:4d} Hz: phase {phase:6.1f} deg, group delay {tg*1e3:.3f} ms")

# High-pass hum filter, corner 200 Hz
for f in (60, 880):
    r = f / 200
    print(f"high-pass at {f} Hz: {20*math.log10(r/math.sqrt(1+r*r)):.1f} dB")

   f (Hz)   n=1     n=2     n=3     n=4    (dB)
     300    -0.1    -0.0    -0.0    -0.0
     880    -0.8    -0.2    -0.0    -0.0
    2000    -3.0    -3.0    -3.0    -3.0
    4000    -7.0   -12.3   -18.1   -24.1
    6000   -10.0   -19.1   -28.6   -38.2
    8000   -12.3   -24.1   -36.1   -48.2
RC filter at  100 Hz: phase   -5.7 deg, group delay 0.158 ms
RC filter at  995 Hz: phase  -45.0 deg, group delay 0.080 ms
RC filter at 3000 Hz: phase  -71.7 deg, group delay 0.016 ms
high-pass at 60 Hz: -10.8 dB
high-pass at 880 Hz: -0.2 dB

The table shows the order trade-off at a glance. Every order is −3.0 dB at the 2 kHz corner, and the attenuation at 6 kHz grows by about 9.5 dB per order, from 10 dB for a single stage to 38 dB for fourth order. The RC filter's group delay is a fraction of a millisecond, and the hum filter cuts 60 Hz by about 11 dB while costing the 880 Hz tone only a fifth of a decibel.

The 4 kHz row deserves a second look. At exactly half the sampling rate, even the fourth-order filter attenuates by only about 24 dB, and a first-order filter by 7 dB. Noise between 4 and 6 kHz is therefore only partly removed. Lesson 9 shows what happens to that leftover noise when the ADC samples it, and why Amara chose the corner, order and sampling rate together rather than one at a time. In practice, designers either raise the sampling rate to widen the gap between the band they want and half the sampling rate, or accept a steeper, more expensive filter. The same trade-off appears in every data-acquisition system, from audio interfaces to the vibration monitors on industrial pumps.