Subband Coding

Subband coding is a signal processing technique primarily used in digital audio, speech, and image compression. The main idea behind subband coding is to split a signal into different frequency components, called subbands, and then process each subband independently. By compressing these subbands separately, we can achieve more efficient data representation and compression.

This method is widely used in applications like audio codecs (MP3, AAC), image compression (JPEG 2000), and video compression (MPEG standards).

Key Concepts in Subband Coding

To fully understand subband coding, let’s dive into the mathematical and conceptual components involved.

1. Decomposition of the Signal

The first step in subband coding is decomposing the signal into different frequency subbands. This is typically done using filters—specifically, bandpass filters. These filters separate the signal into components that represent different parts of the frequency spectrum.

Let’s assume the input signal is a discrete-time signal $x[n]$. The goal is to decompose this signal into multiple subbands. Each subband represents a different frequency range, and we process each one separately.

Mathematically, we apply a filter bank to the signal. A filter bank consists of several filters that divide the signal into different frequency bands. Suppose we want to decompose the signal into two subbands: a low-frequency subband $x_L[n]$ and a high-frequency subband $x_H[n]$.

This can be written as:

Where:

• $h_L[n]$ is a low-pass filter.
• $h_H[n]$ is a high-pass filter.
• $*$ represents convolution.

2. Downsampling

Once the signal is filtered into subbands, we can reduce the sampling rate of each subband without losing important information. This is because each subband now contains a limited range of frequencies, so we can represent it using fewer samples.

The process of reducing the sampling rate is called downsampling. For instance, if the original signal $x[n]$ was sampled at a rate $f_s$, we can downsample each subband by a factor of 2 (or higher). This reduces the data size significantly, which is crucial for compression.

Mathematically, downsampling by a factor of 2 can be written as:

This step effectively halves the number of samples in each subband, reducing the overall data size.

3. Quantization

After downsampling, each subband is quantized to reduce the precision of the signal values. Quantization is a crucial step in data compression as it allows the signal to be represented using fewer bits. The trade-off is that quantization introduces some distortion into the signal, but careful design can minimize the impact.

Quantization can be uniform or non-uniform. In uniform quantization, the signal values are mapped to fixed intervals, while in non-uniform quantization, the intervals may vary based on the signal characteristics.

For example, let’s assume that the downsampled subband $x_L^\downarrow[n]$ takes continuous values between 0 and 1. In a 4-bit uniform quantization scheme, each value is mapped to one of 16 possible levels. The quantized version of the signal, $\hat{x}_L^\downarrow[n]$, would have values restricted to these levels.

Mathematically, this can be expressed as:

Where $Q(\cdot)$ represents the quantization function.

4. Encoding

Once quantization is completed, the subbands are encoded using an entropy coding technique like Huffman coding or arithmetic coding. This step is crucial for achieving high compression ratios. The encoding process takes advantage of the fact that some signal values or patterns occur more frequently than others and assigns shorter codes to frequent patterns and longer codes to infrequent ones.

5. Reconstruction (Decoding)

To reconstruct the original signal, the inverse operations of the encoding process are applied. The process of reconstruction involves:

1. Decoding the quantized and encoded data.
2. Upsampling each subband to the original sampling rate.
3. Filtering each subband using the corresponding reconstruction filters.
4. Summing the filtered subbands to reconstruct the original signal.

Mathematically, the reconstruction of the original signal $\hat{x}[n]$ from the low-pass and high-pass subbands $\hat{x}_L^\uparrow[n]$ and $\hat{x}_H^\uparrow[n]$ (upsampled versions) is given by:

Where the subbands are first upsampled and then passed through corresponding reconstruction filters, $g_L[n]$ and $g_H[n]$, which are complementary to the analysis filters $h_L[n]$ and $h_H[n]$.

Example: Simple Subband Coding of a Signal

Let’s illustrate the subband coding process with a simple example.

Step 1: Signal Decomposition

Consider a discrete-time signal $x[n]$ that consists of both low and high-frequency components. We apply a two-channel filter bank to decompose the signal into a low-frequency subband and a high-frequency subband.

Step 2: Downsampling

We downsample both subbands by a factor of 2, reducing the number of samples in each subband by half.

Step 3: Quantization

The downsampled subbands are then quantized using a simple uniform quantizer. Let’s say we use 8 bits for the low-frequency subband and 4 bits for the high-frequency subband. This reduces the bit rate of the signal.

Step 4: Encoding

We encode the quantized subbands using Huffman coding, which compresses the data based on the frequency of occurrence of each quantized value.

Step 5: Reconstruction

At the receiver, we reverse the process by decoding the Huffman codes, upsampling the subbands, filtering them, and summing them to reconstruct the original signal $\hat{x}[n]$.

Mathematical Summary

1. Analysis filters:

   $$x_L[n] = x[n] * h_L[n], \quad x_H[n] = x[n] * h_H[n]$$

2. Downsampling:

   $$x_L^\downarrow[n] = x_L[2n], \quad x_H^\downarrow[n] = x_H[2n]$$

3. Quantization:

   $$\hat{x}_L^\downarrow[n] = Q(x_L^\downarrow[n]), \quad \hat{x}_H^\downarrow[n] = Q(x_H^\downarrow[n])$$

4. Upsampling:

   $$\hat{x}_L^\uparrow[n] = \hat{x}_L^\downarrow[n/2], \quad \hat{x}_H^\uparrow[n] = \hat{x}_H^\downarrow[n/2]$$

5. Reconstruction:

   $$\hat{x}[n] = \hat{x}_L^\uparrow[n] + \hat{x}_H^\uparrow[n]$$

In digital signal processing (DSP), wavelets and multiresolution analysis are fundamental tools used for signal decomposition and reconstruction, particularly useful in applications like image processing, compression, and data analysis.

1. Filter Banks and Their Importance

A filter bank is a set of filters that decompose a signal into multiple components, each representing a different frequency band. The primary goal of using filter banks is to separate the input signal into subbands—low and high frequency components—that can be processed independently and then recombined.

1.1 Two-Band Filter Bank

Consider a two-band filter bank consisting of two filters:

• $h_0(n)$ (low-pass filter)
• $h_1(n)$ (high-pass filter)

These filters split the input signal $f(n)$ into two frequency bands: low and high frequency. The analysis filter bank decomposes the signal, and the synthesis filter bank is used to reconstruct the signal.

The input signal is divided as follows:

• The low-pass filter $h_0(n)$ captures the low-frequency components of the signal.
• The high-pass filter $h_1(n)$ captures the high-frequency components.

After decomposition, the signal can be processed and later combined in the synthesis filter bank to reconstruct the original signal, represented as:

where $g_0(n)$ and $g_1(n)$ represent the filtered subbands.

1.2 Modulated Filters

The filters can also be modulated versions of each other, meaning they have alternating signs in their coefficients. For example, if $h_1(n)$ is a reflection or order-reversed version of $h_0(n)$, it can be modulated as:

This property is important in perfect reconstruction, where the analysis and synthesis filters are carefully designed to guarantee no loss of information during signal processing.