18.3 Transfer functions and frequency response

Transfer functions and frequency response give you a mathematical way to describe how a system processes signals. Instead of tracking what happens sample by sample in the time domain, you can characterize an entire LTI system with a single function and then see exactly how it treats every frequency. This is the foundation for designing filters, control systems, and most signal processing tools you'll encounter.

Transfer Functions and Frequency Response

Defining Transfer Functions

A transfer function describes the input-output relationship of a linear time-invariant (LTI) system. It's defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming all initial conditions are zero:

$H(s) = \frac{Y(s)}{X(s)}$

Here, $X(s)$ is the input and $Y(s)$ is the output, both in the Laplace domain. The variable $s$ is a complex frequency variable ($s = \sigma + j\omega$).

Why is this useful? Because once you know $H(s)$, you can predict the output for any input, and you can read off stability, transient behavior, and steady-state response directly from its poles and zeros. The transfer function is typically a ratio of polynomials in $s$, and the roots of the numerator are the zeros while the roots of the denominator are the poles.

Frequency Response Characteristics

Frequency response tells you how a system handles sinusoidal inputs at different frequencies. You obtain it by substituting $s = j\omega$ into the transfer function, giving you $H(j\omega)$.

This is a complex-valued function, so it has two parts:

• Magnitude response, $|H(j\omega)|$: the ratio of output amplitude to input amplitude at frequency $\omega$. If $|H(j\omega)| = 2$ at some frequency, the system doubles the amplitude of a sinusoid at that frequency.
• Phase response, $\angle H(j\omega)$: the phase shift the system introduces between the input and output sinusoids at frequency $\omega$. A phase of $-90°$ means the output lags the input by a quarter cycle.

The key idea is that an LTI system can only change the amplitude and phase of a sinusoid. It can't create new frequencies on its own. So the frequency response completely describes what the system does to any sinusoidal input.

Bode Plots for Visualization

Bode plots are the standard way to graph frequency response. They consist of two separate plots:

• Magnitude plot: shows $|H(j\omega)|$ in decibels versus frequency on a logarithmic axis, where $|H(j\omega)|_{dB} = 20\log_{10}|H(j\omega)|$
• Phase plot: shows $\angle H(j\omega)$ in degrees versus frequency on a logarithmic axis

The log-log format is what makes Bode plots so practical. Multiplying transfer functions (cascading systems) turns into simple addition of their dB magnitudes, and decades of frequency get equal spacing on the axis.

You can also sketch Bode plots quickly by hand using straight-line (asymptotic) approximations. Each pole contributes a $-20$ dB/decade slope change, and each zero contributes a $+20$ dB/decade slope change. From these plots you can read off gain, bandwidth, and stability margins (gain margin and phase margin) at a glance.

System Characteristics

Gain and Bandwidth

Gain is how much the system amplifies or attenuates the input signal. DC gain is the gain at zero frequency, found by evaluating $H(0)$. For example, if $H(0) = 5$, the system multiplies any constant (DC) input by 5.

Bandwidth is the range of frequencies the system passes effectively. It's defined by the cutoff frequency $\omega_c$, where the magnitude response drops by 3 dB from its maximum. That 3 dB drop corresponds to the output amplitude falling to about 70.7% of the peak value (since $10^{-3/20} \approx 0.707$).

A wider bandwidth means the system can handle a broader range of frequencies. Audio amplifiers, for instance, need bandwidth covering roughly 20 Hz to 20 kHz to reproduce the full range of human hearing.

Resonance and Peaking

Resonance occurs when the magnitude response has a pronounced peak at some frequency $\omega_r$. This happens in systems with underdamped complex poles.

The quality factor $Q$ measures how sharp that peak is. A high $Q$ means a tall, narrow peak; a low $Q$ means a broad, gentle bump. Quantitatively, $Q$ is the ratio of the resonant frequency to the bandwidth of the peak.

Resonance is sometimes exactly what you want (tuned radio receivers use it to select one station's frequency), but in other contexts it can cause dangerous oscillations, especially in feedback control systems.

Stability and Transient Response

A system is stable if every bounded input produces a bounded output (this is called BIBO stability). You can determine stability directly from the transfer function:

• If all poles lie in the left-half plane (negative real parts), the system is stable.
• If any pole lies in the right-half plane (positive real part), the system is unstable.
• Poles on the imaginary axis represent the boundary case (marginally stable).

Transient response describes how the system behaves when disturbed, such as after a step input. Three quantities characterize it:

• Rise time: how quickly the output reaches the desired value
• Settling time: how long until the output stays within a small band (typically 2%) of the final value
• Overshoot: how far the output exceeds the final value before settling

These depend on pole locations. Poles far to the left in the $s$-plane give fast responses. Complex poles with small damping ratios produce oscillatory, underdamped responses with significant overshoot. Real, repeated poles give critically damped responses (fastest settling without overshoot), and widely separated real poles give overdamped responses.

Filter Types

Lowpass Filters

Lowpass filters pass low-frequency signals and attenuate high-frequency ones. The ideal lowpass filter would have a perfectly flat response up to the cutoff frequency and then drop to zero instantly, but real filters always have a gradual rolloff in the transition band.

Common lowpass filter designs involve different trade-offs:

• Butterworth: maximally flat magnitude in the passband, moderate rolloff
• Chebyshev: steeper rolloff than Butterworth, but allows ripple in the passband (Type I) or stopband (Type II)
• Bessel: most linear phase response (least distortion of signal shape), but the gentlest rolloff

Typical applications include anti-aliasing filters before analog-to-digital conversion, noise reduction, and smoothing sensor measurements by removing high-frequency noise.

Highpass Filters

Highpass filters do the opposite: they pass high-frequency signals and attenuate low-frequency ones. The ideal version would block everything below the cutoff and pass everything above, but again, practical designs have a gradual transition.

Highpass filters can be derived from lowpass designs through a frequency transformation (replacing $s$ with $1/s$ in the normalized transfer function). Applications include removing DC offset or slow drift from signals, isolating high-frequency components, and audio equalization where you want to boost treble relative to bass.

Bandpass Filters

Bandpass filters pass a specific range of frequencies (the passband) and attenuate everything outside it. They're characterized by:

• Center frequency, $\omega_0$: the middle of the passband
• Bandwidth, $BW$: the difference between the upper and lower cutoff frequencies ($BW = \omega_{c,upper} - \omega_{c,lower}$)

You can build a bandpass filter by cascading a lowpass and a highpass filter with appropriate cutoff frequencies, or by designing a single transfer function with complex pole pairs. The quality factor relates to bandwidth here too: $Q = \omega_0 / BW$.

Bandpass filters are everywhere in communications (selecting a specific radio channel from a crowded spectrum), instrumentation (isolating a vibration frequency of interest), and audio processing (extracting a particular frequency range from a recording).