Questions: Spectral Leakage and Windowing Trade-offs

Question 1

You are analyzing a signal that contains two components: a dominant sinusoid at full scale, and a weak sinusoid at −38 dB relative amplitude, separated by only 2 frequency bins. The rectangular window has side lobes at −13 dB. Which window is the best choice, and why?

• A. Rectangular — its narrow main lobe ensures the two components appear as distinct spectral peaks
• B. Hann — its −32 dB side lobes are nearly sufficient and its main-lobe broadening is moderate
• C. Blackman — its ~−58 dB side lobes suppress the strong component's leakage enough to reveal the −38 dB component
• D. Zero-pad the signal — this reduces side-lobe levels and improves dynamic range

Answer: (C)

The dominant challenge is dynamic range: the weak component at −38 dB will be buried in leakage from the strong component if side lobes are −13 dB (rectangular) or even −32 dB (Hann). Only a window with side lobes below −38 dB can prevent the leakage from masking the weak component. The Blackman window (~−58 dB) satisfies this. Zero-padding is wrong: it interpolates the spectrum more finely but does not change the side-lobe levels — the underlying leakage pattern is unchanged.

Question 2

A spectral analyst needs to distinguish two sinusoids of equal amplitude that are very close together in frequency. Which window property is the primary concern?

• A. Low side-lobe level, because high side-lobes spread energy that obscures the two separate peaks
• B. Narrow main-lobe width, because the two peaks must fit within the main lobe without merging
• C. High window amplitude, because a larger window amplitude improves signal-to-noise ratio
• D. A long window duration, because more samples always improve both resolution and dynamic range simultaneously

Answer: (B)

Frequency resolution — the ability to distinguish two closely spaced spectral peaks — is determined by the main-lobe width. If the main lobe is wider than the frequency separation between the two components, their peaks blur together into a single merged peak. For two equal-amplitude, closely spaced sinusoids, the rectangular window's narrow main lobe actually gives the best chance of resolving them. Side-lobe level matters when one component is much weaker than a nearby strong one; for equal-amplitude tones, it is the main lobe that is the binding constraint.

Question 3

Zero-padding a time-domain signal before computing the DFT reduces spectral leakage by evaluating the spectrum at more frequency points.

• T. True
• F. False

Answer: False

Zero-padding interpolates the DFT output — it evaluates the continuous DTFT at more closely spaced frequencies, making the spectral display appear smoother and peak locations more precise. But it does not change the underlying leakage. Leakage is determined entirely by the window applied to the data, not by the number of DFT output points. A zero-padded spectrum shows the same smearing as the non-zero-padded one, just sampled more densely. The only ways to reduce leakage are to choose a window with lower side lobes or to acquire more data.

Question 4

A window with a wider main lobe necessarily provides worse performance than a narrow-main-lobe window when the goal is detecting a weak sinusoid near a strong one.

• T. True
• F. False

Answer: False

For detecting a weak signal near a strong one, dynamic range (side-lobe suppression) is the critical property, not main-lobe width. A wide-main-lobe window like Blackman (~−58 dB side lobes) dramatically outperforms the narrow-main-lobe rectangular window (~−13 dB side lobes) for this task: the rectangular window's high side lobes bury the weak component entirely. 'Worse performance' depends on the task — wider main lobe means worse *frequency resolution*, but better *dynamic range*, which is what matters when a weak signal is near a strong one.

Question 5

You are analyzing an audio signal to find a harmonic at −40 dB relative to the fundamental, and the two are separated by 10 Hz. Explain why the rectangular window is a poor choice, and what window property you should prioritize instead.

Model answer: The rectangular window's side lobes reach only ~−13 dB below the main peak. A component at −40 dB will be completely buried in the side-lobe leakage from the fundamental — it will not be visible in the spectrum at all. The key property to prioritize is side-lobe suppression: you need a window whose side lobes are at least −40 dB (ideally lower, to provide margin). A Blackman window (~−58 dB) would work. The tradeoff is a wider main lobe, but since the two components are 10 Hz apart and frequency resolution is not the binding constraint, this is acceptable. Resolution and dynamic range are the two ends of the windowing trade-off; this scenario is firmly in the dynamic-range-limited regime.

The question requires students to correctly identify *which* axis of the trade-off is relevant to the task, and to recognize that the rectangular window's narrow main lobe — though often presented as its advantage — is irrelevant here, while its high side lobes are the fatal flaw. Good spectral analysis starts by asking: is this a resolution problem or a dynamic range problem?