# Introduction and motivation

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In this module, we give a brief introduction to sub-band coding, its relation to transform coding, and its use in MPEG-style audio coding.
• Sub-band coding is a popular compression tool used in, for example, MPEG-style audio coding schemes (see [link] ).
• [link] illustrates a generic subband coder. In short, the input signal is passed through a parallel bank ofanalysis filters $\left\{{H}_{i}\left(z\right)\right\}$ and the outputs are “downsampled” by a factor of N . Downsampling-by- N is a process which passes every ${N}^{th}$ sample and ignores the rest, effectively decreasing the data rate by factor N . The downsampled outputs are quantized (using a potentially differentnumber of bits per branch—as in transform coding) for storage or transmission.Downsampling ensures that the number of data samples to store is not any larger than the number of data samples entering the coder;in [link] , N sub-band outputs are generated for every N system inputs.
• Relationship to Transform Coding:   Conceptually, sub-band coding (SC) is very similar to transform coding (TC).Like TC, SC analyzes a block of input data and produces a set of linearly transformed outputs, now called “subband outputs.”Like TC, these transformed outputs are independently quantized in a way that yields coding gain over straightforward PCM.And like TC, it is possible to derive an optimal bit allocation which minimizes reconstruction error variance for a specified average bit rate.In fact, an N -band SC system with length- N filters is equivalent to a TC system with $N×N$ transformation matrix T : the decimated convolution operation which defines the ${i}^{th}$ analysis branch of [link] is identical to an inner product between an N -length input block and ${\mathbsf{t}}_{i}^{t}$ , the ${i}^{th}$ row of T . (See [link] .) So what kind of frequency responses characterize the most-commonly used transformation matrices?Lets look at the DFT first. For the ${i}^{th}$ row, we have
$|{H}_{i}\left(\omega \right)|\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left|\sum _{n=0}^{N-1},{e}^{-j\frac{2\pi }{N}in},{e}^{-j\omega n}\right|\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left|\sum _{n=0}^{N-1},{e}^{-j\left(\omega +\frac{2\pi }{N}i\right)n}\right|\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left|\frac{sin\left(\frac{N}{2}\left(\omega +\frac{2\pi i}{N}\right)\right)}{sin\left(\frac{1}{2}\left(\omega +\frac{2\pi i}{N}\right)\right)}\right|.$
[link] plots these magnitude responses. Note that the ${i}^{th}$ DFT row acts as a bandpass filter with center frequency $2\pi i/N$ and stopband attenuation of $\approx 6$ dB. [link] plots the magnitude responses of DCT filters, where we see that they have even less stopband attenuation.
• Psycho-acoustic Motivations:   We have seen that N -band SC with length- N filters is equivalent to $N×N$ transform coding. But is transform coding the best technique to use in high qualityaudio coders? It turns out that the key to preserving sonic quality under high levels of compression is to shape the reconstruction error so that theear will not hear it . When we talk about psychoacoustics later in the course, we'll see thatthe properties of noise tolerated by the ear/brain are most easily described in the frequency domain.Hence, bitrate allocation based on psychoacoustic models is most conveniently performed when SC outputs represent signal componentsin isolated frequency bands . In other words, instead of allocating fewer bits to sub-band outputshaving a smaller effect on reconstruction error variance, we will allocate fewer bits to sub-band outputs having a smaller contributionto perceived reconstruction error. We have seen that length- N DFT and DCT filters give a $2\pi /N$ bandwidth with no better than 6 dB of stopband attenuation. The SC filters required for high-quality audio coding require muchbetter stopband performance, say $>90$ dB. It turns out that filters with passband width $2\pi /N$ , narrow transition bands, and descent stopband attenuation require impulse responselengths $\gg N$ . In N -band SC there is no constraint on filter length, unlike N -band TC. This is the advantage of SC over TC when it comes to audiocoding A similar conclusion resulted from our comparison of DPCM and TC of equal dimension N ; it was reasoned that the longer “effective” input length of DPCM with N -length prediction filtering gave performance improvement relative to TC. .
• To summarize, the key differences between transform and sub-band coding are the following.
1. SC outputs measure relative signal strength in different frequency bands, while TC outputs might not have a strictbandpass correspondence.
2. The TC input window length is equal to the number of TC outputs, while the SC input window lengthis usually much greater than number of SC outputs (16 $×$ greater in MPEG).
• At first glance SC implementation complexity is a valid concern. Recall that in TC, fast $N×N$ transforms such as the DCT and DFT could be performed using $\sim N{log}_{2}N$ multiply/adds! Must we give up this computational efficiency for better frequency resolution?Fortunately the answer is no ; clever SC implementations are built around fast DFT or DCT transforms and are very efficient as a result.Fast sub-band coding, in fact, lies at the heart of MPEG audio compression (see ISO/IEC 13818-3).

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