Filter Banks

A filter bank is a collection of bandpass filters that together decompose a wideband signal into multiple frequency subbands, each of which can be processed independently. Filter banks are the core signal processing structure behind audio compression, image processing, OFDM communications, and subband coding. In multirate DSP, filter banks always combine filtering with decimation at the analysis stage and interpolation at the synthesis stage, making them intimately tied to the concepts of interpolation, decimation, and polyphase decomposition.

Core Concept: Analysis and Synthesis Banks

A filter bank consists of two parts working together. The analysis bank at the input applies M bandpass filters Hk(z) to the incoming signal, each centered at a different frequency band, followed by decimation by M. This produces M subband signals each running at the reduced rate Fs/M. The synthesis bank at the output performs the reverse: each subband is upsampled by M, filtered by a synthesis filter Fk(z), and all M outputs are summed to reconstruct the original signal.

The critical design question is how to choose the analysis filters Hk(z) and synthesis filters Fk(z) such that the reconstructed output y[n] is an exact (or near-exact) replica of the input x[n], possibly with a constant delay. This property is called perfect reconstruction (PR). Achieving PR requires two simultaneous conditions: all aliasing introduced by decimation must be cancelled by the synthesis bank, and the overall magnitude and phase response must be distortion-free.

In a critically sampled (or maximally decimated) M-band filter bank, the total number of samples per unit time is conserved: M branches each at rate Fs/M gives the same total sample rate as the input at Fs. This property is exploited in subband audio coding systems such as MP3, where each subband is independently quantized and encoded based on perceptual importance.

Mathematical Expression

The z-domain analysis of a two-channel (M=2) filter bank gives the following expression for the reconstructed output Y(z) in terms of the input X(z) and its alias X(-z):

Y(z) = (1/2) . [H0(z)F0(z) + H1(z)F1(z)] . X(z) + (1/2) . [H0(-z)F0(z) + H1(-z)F1(z)] . X(-z)

For perfect reconstruction, the alias term must vanish:

H0(-z)F0(z) + H1(-z)F1(z) = 0 (alias cancellation)

And the signal term must equal a pure delay:

H0(z)F0(z) + H1(z)F1(z) = 2z^(-d) (distortion-free reconstruction)

The quadrature mirror filter (QMF) bank achieves alias cancellation by choosing H1(z) = H0(-z) and F0(z) = H0(z), F1(z) = -H1(z). However, exact PR with QMF requires additional conditions on H0(z), and in practice near-PR designs are used with power-complementary filter pairs.

Practical Understanding: Subband Coding

In subband coding, the signal is split into subbands by the analysis bank, and each subband is individually encoded and transmitted. At the receiver, the subbands are decoded and passed to the synthesis bank to reconstruct the original. The advantage of this approach is that different subbands can be encoded with different bit depths and compression ratios based on their perceptual importance or energy content.

In audio coding (MP3, AAC), a 32-band or 64-band filter bank (derived from a prototype lowpass filter modulated to M center frequencies using a DFT or cosine modulation) is used. The output of each band is quantized based on psychoacoustic masking thresholds. Bands where the signal is masked by louder adjacent bands are encoded with fewer bits or discarded, achieving high compression ratios without perceptible quality loss.

DFT filter banks use a single prototype lowpass filter h0[n] and derive all M bandpass filters by modulating: Hk(z) = H0(z . e^(-j2pik/M)). This means the entire bank can be implemented as a single polyphase filter structure followed by a DFT computation, making it highly efficient. This structure is fundamental to OFDM systems in wireless communications.

Given:
Two-channel (M=2) QMF filter bank
Analysis filter H0(z): lowpass, cutoff pi/2
H1(z) = H0(-z): highpass (QMF condition)
Synthesis: F0(z) = H0(z), F1(z) = -H1(z) = H0(z)

Why this formula applies:
QMF alias cancellation: H0(-z)F0(z) + H1(-z)F1(z) = 0
Verify with the QMF synthesis choice.

Formula:
Alias term = H0(-z).F0(z) + H1(-z).F1(z)

Substitution:
H1(z) = H0(-z), F0(z) = H0(z), F1(z) = -H0(-z)
Alias = H0(-z).H0(z) + H0(z).(-H0(-z))

Calculation:
Alias = H0(-z).H0(z) - H0(z).H0(-z) = 0
Alias cancellation is achieved exactly.

Signal term = H0(z).H0(z) + H0(-z).H0(-z)
            = |H0(z)|^2 + |H0(-z)|^2
For PR: |H0(z)|^2 + |H0(-z)|^2 = 2 (power complementary condition)

Final Answer:
Alias is cancelled exactly for any H0(z) with QMF synthesis choices.
Perfect reconstruction requires additionally H0 to be power-complementary.
Example: If H0 is a half-band filter with |H0(e^jw)|^2 + |H0(e^j(w-pi))|^2 = 1, then PR holds (up to scale).

Exam Tip: In GATE, the key QMF condition is H1(z) = H0(-z). Alias cancellation with synthesis F0(z) = H0(z), F1(z) = -H1(z) eliminates the aliased term exactly. Perfect reconstruction additionally requires the power-complementary condition on H0.

Mechanism Summary

• Analysis bank: M filters Hk(z) followed by decimation by M. Each subband output runs at rate Fs/M.
• Synthesis bank: M upsamplers by M followed by synthesis filters Fk(z). Outputs are summed.
• Perfect reconstruction requires alias cancellation (aliased term in Y(z) = 0) and distortion-free signal term (constant gain, linear phase delay only).
• QMF bank achieves alias cancellation with H1(z) = H0(-z), F0(z) = H0(z), F1(z) = -H1(z). PR additionally requires H0 to be power-complementary.
• DFT filter bank derives M bandpass filters from a single prototype via frequency modulation, implemented efficiently with polyphase + DFT.

Quick Revision

• Analysis bank: Hk(z) then downsample M. Synthesis bank: upsample M then Fk(z) then sum.
• Critically sampled (maximally decimated): M subbands, each at Fs/M, total rate = Fs preserved.
• QMF condition: H1(z) = H0(-z). Ensures alias cancellation with F0(z) = H0(z), F1(z) = -H1(z).
• Perfect reconstruction: alias cancellation + power complementary condition |H0|^2 + |H0(-z)|^2 = constant.
• DFT filter bank: Hk = H0(z.e^(-j2pik/M)), implemented as polyphase structure + IDFT. Efficient, used in OFDM and audio coding.
• Exam trap: confusing alias cancellation with distortion-free reconstruction. Both must hold simultaneously for perfect reconstruction.
• Real applications: MP3 uses 32-band modified DCT filter bank; OFDM uses DFT filter bank with hundreds of subbands.