science

Impulse invariance

Impulse invariance 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 Impulse invariance rather than just read about it. In short: Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuou…

Key takeaways

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

Reference excerpt

Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the Nyquist frequency of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.

Discussion The continuous-time system's impulse response, h c ( t ) {\displaystyle h_{c}(t)} , is sampled with sampling period T {\displaystyle T} to produce the discrete-time system's impulse response, h [ n ] {\displaystyle h[n]} .

h [ n ] = T h c ( n T ) {\displaystyle h[n]=Th_{c}(nT)\,}

Thus, the frequency responses of the two systems are related by

H ( e j ω ) = 1 T ∑ k = − ∞ ∞ T H c ( j ω T + j 2 π T k ) {\displaystyle H(e^{j\omega })={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{TH_{c}\left(j{\frac {\omega }{T}}+j{\frac {2{\pi }}{T}}k\right)}\,}

If the continuous time filter is approximately band-limited (i.e. H c ( j Ω ) < δ {\displaystyle H_{c}(j\Omega )<\delta } when | Ω | ≥ π / T {\displaystyle |\Omega |\geq \pi /T} ), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2T) Hz):

H ( e j ω ) = H c ( j ω / T ) {\displaystyle H(e^{j\omega })=H_{c}(j\omega /T)\,} for | ω | ≤ π {\displaystyle |\omega |\leq \pi \,}

Comparison to the bilinear transform Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The bilinear transform is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does.

Effect on poles in system function If the continuous poles at s = s k {\displaystyle s=s_{k}} , the system function can be written in partial fraction expansion as

H c ( s ) = ∑ k = 1 N A k s − s k {\displaystyle H_{c}(s)=\sum _{k=1}^{N}{\frac {A_{k}}{s-s_{k}}}\,}

Thus, using the inverse Laplace transform, the impulse response is

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Impulse invariance

Start with the simplest possible case. Write down what Impulse invariance 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 Impulse invariance 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 Impulse invariance 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 Impulse invariance

In research: Impulse invariance 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 Impulse invariance 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: Impulse invariance is common in secondary-school and first-year university syllabi. It links to neighbouring topics Digital signal processing, Filter theory, so understanding it makes those chapters shorter.
In everyday life: Look for Impulse invariance 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 Impulse invariance in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Impulse invariance 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 Impulse invariance out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Impulse invariance in simple terms?

Why does Impulse invariance 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 Impulse invariance?

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 Impulse invariance.