19.3 Frequency-domain analysis and filtering

Frequency Response and Transfer Functions

Frequency-domain analysis lets you describe how a circuit behaves not in terms of time, but in terms of frequency. Instead of asking "what does the output look like over time?", you ask "how does this circuit treat each frequency that passes through it?" This perspective is what makes filter design and signal processing possible.

Frequency Response Characteristics

The frequency response of a system tells you two things at every frequency: how much the system amplifies or reduces the signal (magnitude), and how much it shifts the signal in time (phase).

• A system might pass low frequencies with no change but cut high frequencies in half.
• The frequency response captures both of these effects across the entire spectrum.
• Engineers use frequency response to evaluate stability, performance, and whether a circuit will do what they need it to do.

Transfer Function Representation

A transfer function is the math behind frequency response. For a linear time-invariant (LTI) system, it's the ratio of the output to the input in the frequency domain:

$H(\omega) = \frac{V_{out}(\omega)}{V_{in}(\omega)}$

You can derive transfer functions using the Fourier transform or the Laplace transform ($s = j\omega$ connects the two). The transfer function encodes the system's poles (frequencies where the response blows up) and zeros (frequencies where the response drops to zero). Together, poles and zeros determine the gain, bandwidth, and overall shape of the frequency response.

Bode Plot Visualization

A Bode plot is the standard way to visualize a transfer function. It consists of two graphs plotted against frequency on a logarithmic scale:

• Magnitude plot: Shows gain in decibels (dB) vs. frequency. Gain in dB is calculated as $20 \log_{10}|H(\omega)|$.
• Phase plot: Shows the phase shift (in degrees) between output and input vs. frequency.

The logarithmic frequency axis lets you see behavior across a huge range of frequencies at once. On a Bode plot, you can quickly spot:

• Cutoff frequencies (where gain drops by 3 dB)
• Resonant peaks (where gain spikes)
• Roll-off rate (how steeply the gain falls in the stopband)
• Gain margin and phase margin (indicators of system stability)

For a simple first-order low-pass filter, the magnitude plot is flat in the passband, then drops at 20 dB/decade past the cutoff frequency. A second-order system rolls off at 40 dB/decade.

Filter Types

Filters are circuits designed to pass certain frequencies and block others. The four basic types cover most practical needs.

Low-Pass and High-Pass Filters

A low-pass filter passes frequencies below its cutoff frequency and attenuates frequencies above it. Think of it as keeping the "slow" parts of a signal. A simple RC circuit with the output taken across the capacitor acts as a low-pass filter. Common uses include removing high-frequency noise from sensor readings or extracting bass frequencies in audio.

A high-pass filter does the opposite: it passes frequencies above the cutoff and attenuates those below it. Taking the output across the resistor in an RC circuit gives you a high-pass filter. This is useful for blocking DC offset, removing low-frequency hum, or isolating treble in audio signals.

Band-Pass and Band-Stop Filters

A band-pass filter passes only a specific range of frequencies (the passband) and attenuates everything outside that range. It has both a lower and an upper cutoff frequency. Radio receivers use band-pass filters to isolate one station's frequency from all the others on the spectrum.

A band-stop filter (also called a notch filter) does the reverse: it blocks a specific frequency range while passing everything else. A classic application is removing 60 Hz power line interference from a sensitive measurement. If you've ever seen a "60 Hz notch filter" on lab equipment, that's a band-stop filter tuned to reject that one frequency.

Filter Characteristics

Cutoff Frequency and Attenuation

The cutoff frequency ($f_c$) is the frequency where the filter's output power drops to half its passband value. In terms of voltage gain, this corresponds to a $-3 \text{ dB}$ drop, since:

$20 \log_{10}\left(\frac{1}{\sqrt{2}}\right) \approx -3 \text{ dB}$

This is why the cutoff frequency is often called the half-power point or the -3 dB point. It marks the boundary between the passband (where signals pass through mostly unchanged) and the stopband (where signals get reduced).

Attenuation describes how quickly the filter suppresses signals beyond the cutoff. This roll-off rate depends on the filter's order:

• A first-order filter rolls off at 20 dB/decade (equivalently, 6 dB/octave)
• A second-order filter rolls off at 40 dB/decade (12 dB/octave)
• An $n$th-order filter rolls off at $20n$ dB/decade

Higher-order filters give sharper transitions between passband and stopband, but they're more complex to build.

Bandwidth and Quality Factor

Bandwidth ($BW$) is the width of the frequency range that a filter passes. For a band-pass filter, it's the difference between the upper and lower $-3 \text{ dB}$ frequencies:

$BW = f_{upper} - f_{lower}$

The quality factor ($Q$) measures how selective or "sharp" a filter is. It's defined as the ratio of the center frequency to the bandwidth:

$Q = \frac{f_0}{BW}$

• A high Q (say, Q = 50) means a very narrow passband relative to the center frequency. The filter is highly selective and picks out a tight range of frequencies.
• A low Q (say, Q = 1) means a wide passband. The filter is less selective and lets a broad range through.

Q also relates to how a filter behaves in the time domain. High-Q filters ring longer when hit with a sudden input because they store energy efficiently and release it slowly. Low-Q filters settle quickly but don't discriminate between frequencies as well. This tradeoff between selectivity and transient response comes up constantly in filter design.