Questions: Filter Banks and Multiband Signal Decomposition

Question 1

A student designs a 2-band filter bank by pairing a Butterworth lowpass filter with a Butterworth highpass filter at the same cutoff, downsampling each output by 2, then upsampling and summing to reconstruct the signal. Why will this not achieve perfect reconstruction?

• A. Butterworth filters have non-ideal stopbands that always pass some signal energy, causing distortion
• B. Downsampling introduces aliasing within each band, and the synthesis stage must be specifically designed to cancel this aliasing — standard Butterworth filters are not designed to do this
• C. Perfect reconstruction requires at least 4 bands; a 2-band bank is inherently too coarse
• D. The cutoff frequencies must differ between the highpass and lowpass filters to avoid spectral overlap at the boundary

Answer: (B)

Downsampling each sub-band by 2 folds residual out-of-band energy (from the filter's imperfect stopband) back into the passband — this is aliasing. The synthesis filters cannot undo this aliasing unless they are specifically designed together with the analysis filters so that aliasing terms cancel when the bands are summed. Off-the-shelf Butterworth filters carry no such guarantee. Perfect reconstruction imposes joint constraints on the entire analysis-synthesis system.

Question 2

In a critically-sampled K-band filter bank, each analysis output is downsampled by K. What is the total data rate across all K outputs relative to the original input?

• A. K times higher — each band produces a separate stream at the original sample rate
• B. The same as the input — K outputs at 1/K the sample rate each multiplies back to the original rate
• C. K times lower — downsampling discards most of the signal
• D. It depends on the passband width of each individual filter

Answer: (B)

Critical sampling is designed to preserve the total data rate. Each of K bands is downsampled by K (keeping every K-th sample), so its sample rate drops to f_s/K. Across K bands: K × (f_s/K) = f_s. The total rate equals the input rate, making critical sampling maximally efficient — no redundancy. The tradeoff is that the aliasing introduced by downsampling must be actively canceled by the synthesis stage.

Question 3

A filter bank that achieves perfect reconstruction guarantees that the reconstructed output is exactly equal to the original input signal, with no distortion or aliasing error.

• T. True
• F. False

Answer: True

Perfect reconstruction (PR) is precisely defined: ŷ[n] = x[n − d] for some fixed delay d. The output is identical to the input up to a constant time delay, with no amplitude distortion, phase distortion, or aliasing. Achieving PR requires the analysis and synthesis filter pairs to jointly satisfy specific mathematical constraints that cancel all aliasing introduced by downsampling.

Question 4

Once the analysis filter bank has been designed, the synthesis filters can be chosen freely and independently to best match the reconstruction application.

• T. True
• F. False

Answer: False

In a PR filter bank, the analysis and synthesis filter pairs are linked by tight mathematical constraints — you cannot design one without the other. The synthesis filters must be chosen so that aliasing components introduced during downsampling in the analysis stage cancel exactly when the sub-band outputs are recombined. Choosing synthesis filters independently will generally violate these constraints and produce aliasing artifacts in the reconstruction.

Question 5

Why does critical sampling in a filter bank introduce aliasing, and how does perfect reconstruction address this?

Model answer: Critical sampling downsamples each analysis output by K. Any energy outside the passband (from imperfect filter stopbands) folds back into the passband during downsampling — this is aliasing. Perfect reconstruction addresses this not by eliminating aliasing in each band separately, but by designing the analysis and synthesis filter pairs so that the aliasing terms from all K bands cancel exactly when the synthesis stage sums them together. The synthesis bank orchestrates a system-wide cancellation of aliasing, which requires the filter pairs to be designed jointly rather than independently.

This is the central design challenge of filter banks. Aliasing is unavoidable in critically-sampled systems, but it can be made to cancel. The QMF (quadrature mirror filter) solution for 2-band banks and the paraunitary framework for K-band banks both exploit this cancellation principle. The key insight is that aliasing cancellation is a property of the entire analysis-synthesis system, not of individual filters.