13.2 Quadrature Mirror Filters (QMF)

Quadrature Mirror Filters: Properties and Characteristics

Quadrature Mirror Filters (QMF) form the core of two-channel filter banks. They split a signal into two subbands (low-pass and high-pass), then reconstruct the original signal from those subbands. Understanding QMFs is essential because they're the foundation for multiresolution analysis and wavelet transforms.

The key idea: the high-pass filter is a "mirror image" of the low-pass filter, reflected around the quadrature frequency $\frac{\pi}{2}$. This symmetry, combined with orthogonality between the two filters, keeps the subband signals uncorrelated and enables clean reconstruction.

Definition and Structure

A QMF bank is a two-channel perfect reconstruction filter bank with two stages:

• Analysis stage: A pair of filters (low-pass $H_0(z)$ and high-pass $H_1(z)$) that split the input into two subband signals.
• Synthesis stage: A corresponding pair of filters ($G_0(z)$ and $G_1(z)$) that recombine the subbands to reconstruct the original signal.

The high-pass filter is derived from the low-pass filter by alternating the sign of its coefficients:

$H_1(z) = H_0(-z)$

This is the "quadrature mirror" relationship. The magnitude response of $H_1$ is a mirror image of $H_0$ around $\frac{\pi}{2}$. The impulse responses of the two filters are orthogonal, which ensures the subband signals carry non-redundant information.

Perfect Reconstruction and Applications

Perfect reconstruction (PR) means the output of the filter bank is an exact (possibly delayed) copy of the input, with no aliasing or amplitude distortion. Two conditions must hold for PR:

1. Alias cancellation: The aliasing introduced by downsampling in the analysis stage is exactly canceled by the synthesis filters.
2. No distortion: The combined transfer function of the analysis-synthesis path is a pure delay, i.e., $\hat{X}(z) = c \cdot z^{-d} X(z)$ for some constant $c$ and delay $d$.

QMF banks show up in several major applications:

• Audio/image compression: Splitting signals into subbands allows perceptually optimized coding.
• Subband coding: Each subband can be quantized and coded independently.
• Wavelet transforms: Iterated QMF banks on the low-pass subband produce the multiresolution decomposition used in discrete wavelet transforms.
• Wireless communications: Channelization divides available bandwidth into subbands for efficient spectrum use and interference reduction.

QMF Bank Design for Perfect Reconstruction

Filter Coefficient Selection

Designing a QMF bank starts with choosing the low-pass filter $H_0(z)$. The high-pass filter follows automatically from the mirror relationship. The filter coefficients must satisfy two main conditions:

• Power complementary property: The squared magnitude responses of the two filters sum to unity at all frequencies:

$|H_0(e^{j\omega})|^2 + |H_1(e^{j\omega})|^2 = 1$

This prevents amplitude distortion in the reconstructed signal.

• Alias cancellation condition: The synthesis filters are chosen so that aliasing terms (created by downsampling) cancel when the subbands are recombined.

In practice, true perfect reconstruction with FIR QMF filters of the original "classic" QMF formulation is only exactly achievable with trivial (two-tap) filters. For longer filters, designers either accept a small reconstruction error (near-perfect reconstruction) or use conjugate quadrature filters (CQF), which do achieve exact PR. You'll sometimes see both referred to loosely as "QMF" in the literature, so pay attention to context.

Design Techniques and Trade-offs

• Johnston's method is a classic approach that optimizes filter coefficients to minimize both reconstruction error and stopband energy. It produces near-PR designs with well-controlled frequency responses.
• Filter length (number of taps) directly affects performance. More taps give sharper frequency selectivity and less aliasing, but increase computational cost and processing delay.
• Stopband attenuation vs. transition bandwidth: A narrower transition band requires more taps. For real-time applications, you often accept a wider transition band to keep latency low.

The design process involves balancing three competing goals:

1. Reconstruction accuracy (how close to PR)
2. Computational efficiency (number of multiplications per sample)
3. Delay (total latency through the analysis-synthesis chain)

QMF Banks for Signal Decomposition and Reconstruction

Analysis and Synthesis Stages

Here's how a signal flows through a two-channel QMF bank, step by step:

Analysis (decomposition):

1. Filter the input $x[n]$ with the low-pass filter $H_0(z)$ to get the low-frequency subband.
2. Filter $x[n]$ with the high-pass filter $H_1(z)$ to get the high-frequency subband.
3. Downsample each subband by a factor of 2 (keep every other sample). This is the decimation step, denoted $\downarrow 2$.

Downsampling cuts the sample rate in half for each subband. Since each subband occupies roughly half the original bandwidth, this satisfies Nyquist and avoids wasting samples.

Synthesis (reconstruction):

1. Upsample each subband by a factor of 2 (insert a zero between every sample), denoted $\uparrow 2$.
2. Filter the upsampled low subband with synthesis filter $G_0(z)$ and the high subband with $G_1(z)$.
3. Sum the two filtered signals to produce the reconstructed output $\hat{x}[n]$.

If the PR conditions are met, $\hat{x}[n] = x[n-d]$ for some integer delay $d$.

Efficient Implementation Structures

Directly implementing the analysis and synthesis filters is straightforward but not optimal. Several techniques reduce computation:

• Polyphase representation: Splits each filter into its even- and odd-indexed coefficient subsequences. This lets you filter after downsampling (or before upsampling), cutting the number of multiply-accumulate operations roughly in half.
• Lattice structure: Exploits the orthogonality and symmetry of QMF coefficients to build a modular, numerically stable implementation. Each stage is parameterized by a single rotation angle, making it easy to scale to different filter lengths.
• Lifting scheme: Factors the polyphase matrix into a sequence of simple upper/lower triangular matrices and a scaling. This reduces multiplications further and allows in-place computation (no extra memory buffers needed). The lifting scheme is the basis for the integer-to-integer wavelet transforms used in JPEG 2000.

Applications in Audio and Image Processing

Audio (subband coding): The signal is split into frequency subbands using a QMF bank. Each subband is then quantized independently. Perceptual models guide bit allocation: low-frequency subbands (which carry more perceptual weight) get more bits, while high-frequency subbands get fewer. This is the principle behind codecs like MPEG audio.

Image processing: QMF banks extend naturally to 2D by applying them separately along rows and columns. One level of decomposition produces four subbands: LL (low-low), LH, HL, and HH. Iterating on the LL subband builds a multiresolution pyramid.

Wavelet-based image compression algorithms rely on this decomposition:

• EZW (Embedded Zerotree Wavelet): Exploits the fact that if a wavelet coefficient at a coarse scale is insignificant, its descendants at finer scales are likely insignificant too. This creates an embedded bitstream where you can truncate at any point for a valid (lower-quality) image.
• SPIHT (Set Partitioning in Hierarchical Trees): Refines the zerotree idea with a more efficient partitioning strategy, achieving strong compression performance. SPIHT is used in some medical and satellite imaging systems.

Both EZW and SPIHT depend on the QMF bank to produce the wavelet coefficient tree structure that makes their coding strategies possible.