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A Simple Lossy Audio Coding Scheme Based on the DWT A Simple Lossy Audio Coding Scheme Based on the DWT Vikas Kumar Gupta*, Prashant Kumar Khevariya* and José Salvado** * Department of Electrical Engineering Indian Institute of Technology – Roorkee Haridwar, Uttarakhand Roorkee, India Telf: +91 1332 285239; Fax: +91 1332 277587; e-mail: {vik4uuee , khareuee }@iit.ernet.in ** Department of Electrical Engineering EST – IPCB Av. do Empresário, s/n – Castelo Branco Telf: +351 272 339 300; Fax: +351 272 339 399; e-mail: [email protected] Abstract — This paper proposes a simple scheme for audio ratios (e.g. up to 25:1 and more) but are more susceptible coding that does not use perceptual models. The audio coder to high rate artifacts. is based on the discrete wavelet transform to decorrelate With the improvement of circuit technology and signals, computed through the lifting scheme, and followed by telecommunication bandwidth, there have been a great Huffman coding. The evaluation of the coding scheme is amount of interest in the high-quality, in which a large presented by using some .wav audio test files, coded for amount of data need to be processed, compressed and different conditions, and also includes subjective evaluation. transmitted in real time. But there exist a trade-off between Experimental results show the compression ratios achieved, the compression ratio and reconstruction quality. Higher the degradation of the signals expressed as values of signal to compression ratios imply more degradation on the sound noise ratio and the changes in spectrum information. quality, when reconstructed. Any compression scheme should consider these facts and should also provide a good control mechanism to use the limited bandwidth and 1. Introduction maintain acceptable quality, keeping in mind the computational complexity. The computational complexity Audio coding or audio compression algorithms are used to directly relates to the processing speed and it should be as obtain compact digital representations of high-fidelity less as possible so as to reduce the propagation delay (wideband) audio signals for the purpose of efficient during real time processing. transmission over larger distances and storage. This Typical perceptual audio compression schemes, e.g. becomes a necessity because of the bandwidth constraint MPEG audio, represented generically in figure 1, use and to reduce the transmission time between the wirelessly signal analysis, psychoacoustic models and bit allocation connected systems. The main objective in audio coding is and coding blocks. At the ISO/MPEG layer III (MP3) to represent the signal with a minimum number of bits coding scheme, the time to frequency mapping block while achieving transparent signal reproduction i.e. includes a polyphase analysis filter banks followed by generating output audio that cannot be distinguished from decimation of a factor of 32, feeding a modified discrete the original input signal. cosine transform (MDCT) and adaptive segmentation Based on the signal model or in the analysis/synthesis block, also connected with the psychoacoustic model. The techniques to encode audio signals one can classify audio bit allocation block includes block companding, coders into one of the following four types [1]: linear quantization and Huffman coding [3]. predictive, transform based, sub-band and sinusoidal. This paper proposes a simple audio coding and decoding Algorithms can also be classified as lossy or lossless, (CoDec) scheme based on sub-band analysis with according to the nature of the coding scheme. In lossy polyphase filter banks, using the discrete wavelet audio schemes compression is achieved by exploring transform (DWT). In the proposed CoDec we do not use information that is perceptually irrelevant. On the other any perceptual model in order to reduce computational hand lossless audio coding schemes the information is complexity. To evaluated the audio CoDec we used some “merely” packed in order to obtain a different and more .WAV test files, and two families of wavelets: the compact (though efficient) representation of data. Lossless Daubechies 2 (DB2) orthogonal wavelet and the audio coding techniques yield in general high quality audio biorthogonal (4,4) wavelet. Evaluation tests included the without artifacts at high frequency or at high rates. Lossy measurement of the compression ratio, the signal to noise audio coding techniques allow for better compression ratio, spectrum modification and perceptual quality. prototype function ψ (t ) known as mother wavelet . Scaling Time to Bit/noise Encoded PCM frequency allocation, Bitstream bitstream is achieved by multiplying t by a scale factor, normally a audio mapping filter quantizer, and formatting input a bank coding power of two, ψ (2t ), a ∈ℤ . Since the prototype function has finite support, it can be scaled and translated to cover Acillary data a − ∈ Psychoacoustic (optional) the all signal i.e. ψ (2t b ), b ℤ . The wavelet model decomposition of a signal can be represented as Fig .1. Diagram of a generic perceptual audio coding scheme[2]. = ft()∑∑ cabψ ab () t (3) a b The rest of the paper is organized as follows. The discrete Where ()2t=a2 (2 a t − b ) and c are coefficients wavelet transforms and its implementation through the ψab ψ ab lifting scheme is explained in Section II. In Section III the that can be computed trough the wavelet transform. structure of the proposed audio CoDec is presented and The wavelet transform of a signal x( t ) is defined as explained and compared with the most common coding schemes for audio signals. Evaluation tests and − = 1 * t b experimental results are shown and discussed in section IV Wabx (,)∫ xt () ψ dt (4) a ℝ a and finally, in Section VI, we present the main + conclusions. where a ∈ ℝ is the scale or dilation parameter, b ∈ ℝ is The work that supports this paper was carried out by the the translation parameter, ψ *(t ) denotes the complex first two authors during a two-month internship at EST- conjugate of (t ) and 1 a normalizes energy along IPCB, on summer 2007 [4]. ψ scales. By taking the discrete values of the scale and translation parameters (a=2−j , b = k 2 − j ) one can obtain 2. The DWT and the Lifting Scheme the discrete wavelet transform (DWT) given by: In signal analysis, a signal f( t ) is represented as a − j − − 1t− k 2 weighted sum of building blocks of basis functions wWk=(2,2)j j = xt () ψ * dt . k, j x − ∫ − j (5) = 2 j ℝ 2 ft()∑ ckψ k () t (1) k where ψk are basis functions and ck are weight The basic idea of the wavelet transform is to exploit the coefficients. Since the basis functions are fixed, the high correlation structure in most real signals, and build a information about the signal is carried by the coefficients. sparse approximation. The correlation structure is typically Choosing sinusoids as the basis functions in (1) yields the local in space (or time) and frequency; the samples that are Fourier representation. To detect spikes or well localized close each other are more correlated than ones that are far high frequency components in the signal, one needs a apart. Wavelet analysis has good scale-frequency and representation which contains information about both, the localization properties, allows the decomposition of signals time and frequency behaviour of the signal. However, with different levels of detail. For a multi-resolution [6] decomposition scheme of a signal with p levels, one has resolution in time (∆t) and resolution in frequency (∆ω ) to successively apply the analysis filters to the resulting cannot be arbitrarily small at the same time as their product approximation sub-sequence. is lower bounded by the Heisenberg’s uncertainty principle B. The Lifting Scheme ∆t ⋅ ∆ω ≥ 1 2 (2) This means that one must trade off time resolution for Daubechies and Sweldens proposed and described how any frequency resolution, or vice versa. However, low- discrete wavelet transform (or a two sub-band filtering frequency events are usually spread in time (non local) and scheme) with finite filters can be decomposed into a finite high-frequency events are usually concentrated (localized) sequence of simple filtering steps and this ladder structure in time. Therefore, it is possible to obtain good time- is called Lifting Steps [7]. Mathematically, this frequency information of a signal by choosing the basis decomposition corresponds to a polyphase matrix functions to act as cascaded octave band-pass filters which representation of the wavelet filters or DWT, as shown in repeatedly split the bandwidth of the signal in half. On the figure 2. Consider that all filters are of FIR type (for more other hand sinusoids cannot provide information about the extensive sub-band transform). At the analysis side one has time behaviour of a signal as they have infinite support. a pair of analysis filters hɶ (low-pass) and gɶ (high-pass) The solution is to use basis functions that have finite (compact) support and different widths. followed by sub-sampling by a factor of 2. A. The Discrete Wavelet Transform For a wavelet representation [5], the set of basis functions { } ψk are scaled and translated versions of the same Fig .2. Polyphase representation of the DWT At the synthesis side coefficients are first up-sampled and The DWT also has strong relations with sub-band analysis then passed trough a pair of synthesis filters h and g (low and filter banks and can be computed by the lifting and high pass, respectively). scheme, which is based on polyphase matrix Then one has to factor the polyphase matrices into decomposition. On the other hand the wavelet filters are elementary matrices, using the Euclidean algorithm. FIR filters that can have linear phase characteristics (in the In [7] Daubechies and Sweldens showed how to obtain the case of biorthogonal wavelets). algorithms by factoring the wavelet transform into lifting The DWT is applied to the audio input signal with three steps for the orthogonal and bi-orthogonal wavelet levels of decomposition, thus concentrating energy in families.
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