๐ Topic Summary
An orthonormal basis is a set of vectors that are orthogonal (perpendicular) to each other and have a length (or norm) of 1. These bases simplify many calculations in linear algebra and signal processing. Parseval's Identity provides a crucial link between the 'energy' of a function or signal and the coefficients of its expansion in terms of an orthonormal basis. It states that the sum of the squares of the coefficients equals the integral of the square of the function's magnitude. It's incredibly useful in signal analysis and data compression.๐
This quiz tests your knowledge of these key principles. Let's begin! ๐
๐ง Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Orthonormal Basis |
A. A sequence of coefficients representing a signal in a given basis. |
| 2. Parseval's Identity |
B. A basis where all vectors are orthogonal and have a norm of 1. |
| 3. Orthogonal |
C. A vector with a magnitude (or length) of 1. |
| 4. Unit Vector |
D. Describes vectors which are perpendicular to each other. |
| 5. Coefficients |
E. Relates the energy of a function to the sum of the squared magnitudes of its coefficients in an orthonormal basis. |
โ๏ธ Part B: Fill in the Blanks
Complete the following paragraph using the words provided (not all words will be used):
Parseval's Identity is a cornerstone result when working with ________ bases. It essentially says that the total ________ of a function is equal to the sum of the squares of its ________ when expressed in terms of that basis. If we have an orthonormal basis {$e_n$}, then for any function $f$, Parseval's Identity gives us: $$\sum ||^2 = \int |f(x)|^2 dx$$. Here, $$ are the _______.
( orthonormal, orthogonal, energy, coefficients, vectors )
๐ค Part C: Critical Thinking
Explain in your own words how Parseval's Identity could be used to approximate a complex signal using a finite number of basis functions. What are the trade-offs involved?