Sliding DFT

Key takeaways

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

Reference excerpt

In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single sample apart (hopsize − 1). The calculation for the sliding DFT is closely related to Goertzel algorithm.

Definition Assuming that the hopsize between two consecutive DFTs is 1 sample, then

F t + 1 ( n ) = ∑ k = 0 N − 1 f k + t + 1 e − j 2 π k n / N = ∑ m = 1 N f m + t e − j 2 π ( m − 1 ) n / N = e j 2 π n / N [ ∑ m = 0 N − 1 f m + t e − j 2 π m n / N − f t + f t + N ] = e j 2 π n / N [ F t ( n ) − f t + f t + N ] . {\displaystyle {\begin{aligned}F_{t+1}(n)&=\sum _{k=0}^{N-1}f_{k+t+1}e^{-j2\pi kn/N}\\&=\sum _{m=1}^{N}f_{m+t}e^{-j2\pi (m-1)n/N}\\&=e^{j2\pi n/N}\left[\sum _{m=0}^{N-1}f_{m+t}e^{-j2\pi mn/N}-f_{t}+f_{t+N}\right]\\&=e^{j2\pi n/N}\left[F_{t}(n)-f_{t}+f_{t+N}\right].\end{aligned}}}

From this definition above, the DFT can be computed recursively thereafter. However, implementing the window function on a sliding DFT is difficult due to its recursive nature, therefore it is done exclusively in a frequency domain.

Sliding windowed infinite Fourier transform It is not possible to implement asymmetric window functions into sliding DFT. However, the IIR version called sliding windowed infinite Fourier transform (SWIFT) provides an exponential window and the αSWIFT calculates two sDFTs in parallel where slow-decaying one is subtracted by fast-decaying one, therefore a window function of w ( x ) = e − x α − e − x β {\displaystyle w(x)=e^{-x\alpha }-e^{-x\beta }} .

References

Worked examples

Example 1 — a first encounter with Sliding DFT

Start with the simplest possible case. Write down what Sliding DFT 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 Sliding DFT 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 Sliding DFT 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 Sliding DFT

In research: Sliding DFT 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 Sliding DFT 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: Sliding DFT is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fast Fourier transforms, Signal processing stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Sliding DFT 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 Sliding DFT in 20 minutes

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

Frequently asked questions

What is Sliding DFT in simple terms?

In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single sample apart (hopsize − 1). The calculation for the sliding DFT is closely related to Goertzel algorithm.

Why does Sliding DFT 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 Sliding DFT?

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 Sliding DFT.