Questions: Orthogonal Signal Decomposition and Basis Functions

Question 1

A signal needs to be decomposed using an orthogonal basis and you want to find c₃ — how much of basis function φ₃ is present. Which procedure correctly exploits orthogonality?

• A. Compute the inner product of the signal with all other basis functions, then subtract their contributions to isolate c₃
• B. Solve a system of N equations simultaneously, because changing one coefficient always affects the fit of the others
• C. Compute ⟨x, φ₃⟩/‖φ₃‖² — orthogonality ensures this inner product isolates c₃ without reference to any other coefficient
• D. Take the Fourier transform of the signal and read off c₃ from the spectrum at the corresponding frequency

Answer: (C)

Orthogonality means each basis function is independent of all others — zero inner product means no overlap. This independence is precisely what allows c₃ to be computed as a single inner product without involving any other coefficient. Option B describes what you'd need with a non-orthogonal basis, where coupling forces you to solve a system. Option A is a more laborious version of the wrong approach.

Question 2

Why does using a non-orthogonal basis create problems for signal decomposition?

• A. Non-orthogonal bases cannot span the full signal space, so exact reconstruction is impossible
• B. Basis functions with nonzero inner products overlap, so their coefficients are coupled — extracting them requires solving a linear system, and noise spreads across all coefficients
• C. Non-orthogonal decompositions always produce complex-valued coefficients even for real signals
• D. Non-orthogonal bases require more basis functions to represent the same signal

Answer: (B)

Non-orthogonality means the basis functions have nonzero mutual overlap. Because they share 'content,' knowing how much of one is in the signal doesn't tell you how much of another is — the coefficients are coupled. Extracting them requires solving a linear system, and noise or errors in any coefficient propagate to all others. This is why orthogonal structures (Fourier, DCT, wavelets) dominate practical signal processing.

Question 3

In an orthogonal basis, knowing how much of one basis function is present in a signal tells you nothing about how much of any other basis function is present.

• T. True
• F. False

Answer: True

This is the direct consequence of zero inner product. Orthogonality means basis functions measure completely independent aspects of the signal. The coefficient c₃ captures only what φ₃ contributes; c₄ captures only what φ₄ contributes. There is no cross-contamination — which is exactly what allows each coefficient to be computed as a single isolated inner product.

Question 4

Any basis that spans the full signal space (a complete basis) is also orthogonal.

• T. True
• F. False

Answer: False

Completeness and orthogonality are independent properties. A basis is complete if it can represent any signal in the space; it is orthogonal if its basis functions have zero pairwise inner products. Non-orthogonal complete bases exist and can represent any signal — they just require solving coupled systems to extract coefficients. Orthogonality is an additional property that makes coefficient extraction efficient and noise-stable.

Question 5

Explain why orthogonality is so valuable for signal decomposition. What goes wrong if you use a non-orthogonal basis instead?

Model answer: Orthogonality means the basis functions have zero inner product — they measure independent aspects of the signal. This independence lets each coefficient be computed as a single inner product without coupling to any other coefficient. With a non-orthogonal basis, the functions overlap, so their coefficients are coupled: you must solve a system of equations to extract them, and errors or noise in one coefficient contaminate all others. Orthogonal bases eliminate this coupling, making decomposition stable, efficient, and amenable to independent coefficient editing — the foundation of filtering, compression, and spectral analysis.

The vector analogy is useful: orthogonal basis vectors in 2D let you project onto x̂ and ŷ independently. Non-orthogonal basis vectors mean projecting onto one axis shifts what you see on the other. The same logic applies to signals, just with inner products replacing dot products.