Root-raised-cosine filter

Key takeaways

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

Reference excerpt

In signal processing, a root-raised-cosine filter (RRC), sometimes known as square-root-raised-cosine filter (SRRC), is frequently used as the transmit and receive pulse shaping filter in a digital communication system to perform matched filtering. This helps in constraining the occupied bandwidth of the waveform without introducing intersymbol interference (ISI). The combined response of two such filters is that of the raised-cosine filter. It obtains its name from the fact that its frequency response, H r r c ( f ) {\displaystyle H_{rrc}(f)} , is the square root of the frequency response of the raised-cosine filter, H r c ( f ) {\displaystyle H_{rc}(f)} :

H r c ( f ) = H r r c ( f ) ⋅ H r r c ( f ) {\displaystyle H_{rc}(f)=H_{rrc}(f)\cdot H_{rrc}(f)}

or:

| H r r c ( f ) | = | H r c ( f ) | {\displaystyle |H_{rrc}(f)|={\sqrt {|H_{rc}(f)|}}}

or:

h r c ( t ) = ( h r r c ∗ h r r c ) ( t ) {\displaystyle h_{rc}(t)=\left(h_{rrc}*h_{rrc}\right)(t)}

where ∗ {\displaystyle *} is being used to refer to convolution

Why it is required To have minimum ISI (Intersymbol interference), the overall response of transmit filter, channel response and receive filter has to satisfy Nyquist ISI criterion. The raised-cosine filter is the most popular filter response satisfying this criterion. To get the Signal-to-noise ratio benefits of matched filtering, half of this filtering is done on the transmit side and half is done on the receive side. On the receive side, the channel response, if it can be accurately estimated, can also be taken into account so that the overall response is that of a raised-cosine filter.

Mathematical description

The RRC filter is characterised by two values; β, the roll-off factor, and Ts the reciprocal of the symbol-rate. The impulse response of such a filter can be given as:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Root-raised-cosine filter

Start with the simplest possible case. Write down what Root-raised-cosine filter 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 Root-raised-cosine filter 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 Root-raised-cosine filter 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 Root-raised-cosine filter

In research: Root-raised-cosine filter 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 Root-raised-cosine filter 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: Root-raised-cosine filter is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear filters, Telecommunication theory, so understanding it makes those chapters shorter.
In everyday life: Look for Root-raised-cosine filter 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.

How to study Root-raised-cosine filter in 20 minutes

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

Frequently asked questions

What is Root-raised-cosine filter in simple terms?

In signal processing, a root-raised-cosine filter (RRC), sometimes known as square-root-raised-cosine filter (SRRC), is frequently used as the transmit and receive pulse shaping filter in a digital communication system to perform matched filtering. This helps in constraining the occupied bandwidth…

Why does Root-raised-cosine filter 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 Root-raised-cosine filter?

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 Root-raised-cosine filter.