Engineering & Technology · Electronics, Devices & Signal Processing

Filters, signals, spectra and frequency-domain reasoning

A sensor trace shows a slow oscillation with a faster ripple. A frequency description identifies the two rhythms; a filter description predicts which is changed more. Neither view replaces the other: time shows when values occur, while spectra expose repeated rates and their relative sizes.

Electrical Circuits supplies charge/current balance, stored energy, sinusoidal phase, rms and the unloaded RC response. Material Structure & Properties supplies the distinction between a material description and a complete product. This chapter explains bias and feedback before using gain. It introduces spectra through an explicit two-tone sum and sampling through points on known curves. All parameters, code intervals, timing values and noise examples are original teaching models; no physical construction or performance qualification is implied.

Back to the chapter

After reading, you can

Read time and frequency descriptions of one signal and predict how an ideal linear filter changes its components.

Build a spectrum from an explicit signal

A sine wave is specified by amplitude, frequency and phase. Adding two sinusoids produces a time waveform whose shape depends on all three for each component. A line spectrum lists their frequencies and amplitudes; phases must also be retained if the waveform is to be reconstructed.

Fourier methods generalise this representation to suitable periodic signals and to broader signal classes. A finite measured record needs choices about duration, sampling and processing; a displayed spectrum is not automatically the exact infinite-duration content. Here the signal is an analytic sum, so we know its two frequencies without estimating them from data.

Apply a linear response component by component

A linear, time-invariant model responds to a scaled sum by the same scaled sum of individual responses. In sinusoidal steady state, each input frequency keeps its frequency, while its amplitude and phase change according to the response at that frequency. A nonlinear circuit can create new components; the linear shortcut must not be applied after clipping without justification.

Low-pass, high-pass, band-pass and notch descriptions indicate broad frequency preferences. A useful specification also gives actual passband variation, attenuation, transition region, phase/timing and conditions. An ideal abrupt stop is a mathematical target, not an automatic property of a physical filter.

Retain phase and initial-state scope

Our original case reuses the electrical chapter’s ideal unloaded RC low-pass relation. At frequency ratio r=f/fc, magnitude is 1/√(1+r²) and phase is −arctan(r). Each sinusoid therefore has a different reduction and delay in phase. Do not multiply the entire mixed waveform by one gain number.

The figure shows input and output line amplitudes on the same axes. It intentionally omits a phase plot, so the caption reports both phases. A magnitude-only picture cannot show the full output waveform. A startup transient also requires initial-state information in addition to steady frequency response.

Filtering involves a decision about useful information. Removing high-frequency content can reduce unwanted ripple but can also round a wanted transition or suppress a meaningful event. The question is not whether the trace looks smoother; it is whether the response preserves what the application needs.

Two frequencies 20 Hz and 200 Hz are placed as labelled categories, not a linearly spaced frequency axis. Within each pair blue is input, orange output; vertical peak voltage 0–1 V at 280 units/V. Input amplitudes 1 and 0.30 V; ideal RC fc50 Hz gives output 0.9284767 and 0.0727607 V. Corresponding phases −21.8014° and −75.9638° relative to each input sine. Lines offset horizontally only to separate paired values; no other frequencies are represented.

Calculate the output of an original two-tone model

Let input x(t)=1.0 sin(2π·20t)+0.30 sin(2π·200t) volts, with t in seconds. Choose an ideal RC low-pass characteristic frequency fc=50 Hz, and consider only the steady periodic response. This is a constructed signal, not a sensor measurement.

At 20 Hz, r=0.4, gain≈0.9285 and phase≈−21.80°. At 200 Hz, r=4, gain≈0.2425 and phase≈−75.96°. The output component peak amplitudes are therefore 0.9285 V and 0.07276 V. Its expression retains each original frequency and adds the respective phase.

Over complete periods, the distinct zero-mean sinusoidal components are orthogonal, so the input rms is √[(1²+0.30²)/2]≈0.7382 V. Output rms is √[(0.9285²+0.07276²)/2]≈0.6585 V using unrounded components. This combined rms does not imply that both components experienced the same gain.

The fast/slow peak-amplitude ratio falls from 0.30 to about 0.07837. Whether that is sufficient depends on a stated ripple allowance and on the real filter’s tolerance, load and bandwidth.

Try the reasoning

Can two signals share amplitude spectra yet differ in time shape?
Show the answer and reasoning
Yes. Their phases can differ, so an amplitude spectrum alone is incomplete.
Would this linear model generate a new 40 Hz component from a 20 Hz sinusoid?
Show the answer and reasoning
No. A steady LTI response changes amplitude and phase at that frequency; harmonics require a different mechanism such as nonlinearity.
Does lower overall rms prove that all unwanted content has been removed?
Show the answer and reasoning
No. Compare each relevant component or band against the application’s requirements.

What to carry forward

Frequency response acts on components. Preserve phase, state assumptions and the meaning of useful bandwidth.

References

Teaching resources and research supporting this lesson’s explanations and examples.