science

Time-variant system

Time-variant system is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Time-variant system rather than just read about it. In short: A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application. In other words, a time delay or time advance of input not only shifts the output signal in time but also changes other parameters and behavior.

Key takeaways

Time-variant system belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Time-variant system to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Time-variant system from memory before moving on to harder problems.

Reference excerpt

A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application. In other words, a time delay or time advance of input not only shifts the output signal in time but also changes other parameters and behavior. Time variant systems respond differently to the same input at different times. The opposite is true for time invariant systems (TIV).

Overview There are many well developed techniques for dealing with the response of linear time invariant systems, such as Laplace and Fourier transforms. However, these techniques are not strictly valid for time-varying systems. A system undergoing slow time variation in comparison to its time constants can usually be considered to be time invariant: they are close to time invariant on a small scale. An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change. Other linear time variant systems may behave more like nonlinear systems, if the system changes quickly – significantly differing between measurements. The following things can be said about a time-variant system:

It has explicit dependence on time. It does not have an impulse response in the normal sense. The system can be characterized by an impulse response except the impulse response must be known at each and every time instant. It is not stationary in the sense of constancy of the signal's distributional frequency. This means that the parameters which govern the signal's process exhibit variation with the passage of time. See Stationarity (statistics) for in-depth theoretics regarding this property.

Linear time-variant systems Linear-time variant (LTV) systems are the ones whose parameters vary with time according to previously specified laws. Mathematically, there is a well defined dependence of the system over time and over the input parameters that change over time.

y ( t ) = f ( x ( t ) , t ) {\displaystyle y(t)=f(x(t),t)}

In order to solve time-variant systems, the algebraic methods consider initial conditions of the system i.e. whether the system is zero-input or non-zero input system.

Examples of time-variant systems The following time varying systems cannot be modelled by assuming that they are time invariant:

The Earth's thermodynamic response to incoming Solar irradiance varies with time due to changes in the Earth's albedo and the presence of greenhouse gases in the atmosphere. Discrete wavelet transform, often used in modern signal processing, is time variant because it makes use of the decimation operation.

See also Control system Control theory System analysis Time-invariant system

References

Worked examples

Example 1 — a first encounter with Time-variant system

Start with the simplest possible case. Write down what Time-variant system claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Time-variant system before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Time-variant system ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Time-variant system

In research: Time-variant system appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Time-variant system in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Time-variant system is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, so understanding it makes those chapters shorter.
In everyday life: Look for Time-variant system outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Time-variant system in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Time-variant system means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Time-variant system out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Time-variant system in simple terms?

Why does Time-variant system matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Time-variant system?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Time-variant system.