Among these methods, the probability density function pdf optimized vector quantization vq scheme has been shown to be superior to those based on training data 6, 7. In scalar quantization, each sample is quantized independently. Uniform quantizer with 128 levels, n 2 7, 7 bits per sample speech typically limited to 4khz sample at 8khz ts 18000 125 s 8000 samples per second at 7 bits per sample 56 kbps differential pcm speech samples are typically correlated instead of coding samples independently, code the difference between samples. Quantization based on a novel sampleadaptive product. The gain codebook is optimized by the lloydmax algo rithm. Sigmadelta adc with reduced sample rate multibit quantizer. Furthermore, since f, is a logconcave function, the lloydmax algorithm converges to the globally optimum gain codebook. Optimized compressive image sensing system over mobile. The optimized codebook has the same vector size as. Due to the symmetry of the pdf and the quantizer design, the mse distortion can.
The essence of pq is to decompose the highdimensional vector space into the. No training vectors are needed for this since the pdf is known exactly l. For uniform pdfx max x f x x 12x max want to uniformly quantize an rv x ux max,x max assume that desire m rls for r. The output sample is reconstructed with either hard reconstruction rule or soft reconstruction rule. Hence the simple inputindependent additive noise model is only an approximation to the noise in the pdf optimized quantizer. The optimal partition for the sample is given by 5 the overall distortion for the entire system is given by 6 we examine two approaches to reconstruct the sample. Indx, quant, distor quantizsig, partition, codebook outputs.
All of these are examples of quantization, but the signals. Design of sample adaptive product quantizers for noisy. Codebook is a length n vector that contains the output set. The lloyd algorithm functions as follows source distribution. Adaptive quantization an overview sciencedirect topics. Due to the symmetry of the pdf and the quantizer design, the mse distortion can be written as. Optimized gft gft is eigen matrix of the graph laplacian. Pdf quantization based on a novel sampleadaptive product.
The lloyd algorithm functions as follows source distribution f xx is assumed to be known. Uniform quantizer all intervals of the same size i. Simulation results show that mcoqd provides more robust. Nearest neighbor and centroid conditions still apply for a channel optimized quantizer, but they now depend on both the channel properties and source distribution. Lloydmax quantizers are optimal as scalar quantizers. Shroff, member, ieee abstract in this paper, we propose a novel feedforward quantized by a codebook is given by adaptive quantization scheme called the sample adaptive prod uct quantizer sapq. Solution depends on input pdf and can be done numerically for commonly used. A channel optimized quantizer is designed according to the channel and source distribution in order to minimize the distortion between the source value and estimate at the receiver. Pdf pdf optimized parametric vector quantization of. Design and analysis of binary scalar quantizer of laplacian. Quantizers can be designed to operate on scalar or vector signals and a comprehensive overview of the topic is available in 2.
For the mae optimized quantizer, the pmf for the reconstruction symbols is given by p 1 1 m mzln2 0 e x mdx e x. Also it is assumed that the pdf used to optimize a quantizer qi. Rather than allocate a uniform step size, as would be optimum for a uniform pdf, the lloyd max approach identifies decision boundaries according to the mean values of equal area partitions of the pdf curve. A product quantizer can generate an exponentially large codebook at very low memorytime cost.
Pdf twosource channel optimized quantization jamie. For a fourdimensional vector quantizer having a rate of 1 bit sample transmitted through a noisy channel, the softdecoding channel optimized quantizer can achieve about 33. Recall that a pdf fx satisfies the following properties. Pdf channel optimized sample adaptive product quantization. The quantizer and encoder together is called as analog to digital converter. Rao, fellow, ieee abstracta computationally efficient, high quality, vector msvq 3 and splitvq 4 which employ suboptimal quantization scheme based on a parametric probability. Uniform scalar quantization the simplest example of sq is uniform scalar quantization usq.
Softdecoding vector quantizer using reliability information. For a binary source alphabet an optimal context quantizer can be computed exactly by a fast. Outer intervals may be an exception reconstruction usually the midpoint is selected midrise quantizer zero is not an output level midtread quantizer zero is an output level 12. Must send side info backward adaptive uses past values to choose. Under what circumstances is a nonuniform quantizer. In this adc scheme, the input process is oversampled sampled above its nyquist rate and quantized using a lowresolution quantizer usually 1bit. Optimized product quantization 1 optimized product quantization. Finding an optimal solution to the above problem results in a quantizer sometimes called a mmsqe minimum meansquare quantization error solution, and the resulting pdfoptimized nonuniform quantizer is referred to as a lloydmax quantizer, named after two people who independently developed iterative methods to solve the two sets of simultaneous equations resulting from. A nonuniform pdf optimized quantizer produces a smaller msqe mean squared quantization error compared to a uniform one at the same number of distinct output levels. Pdf optimized parametric vector quantization of speech. Shroff, member, ieee abstract in this paper, we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq.
Assuming the estimated pdf to be a good approximation to the original pdf, the mixture model parameters, which define the estimated pdf, have all. Pdf in image compression contextbased entropy coding is commonly used. We present a scalar quantization scheme named lossybit entropyconstrained scalar quantization lbecsq that is able to reduce the d. Optimized product quantization tiezheng ge, kaiming he, qifa ke, and jian sun abstractproduct quantization pq is an effective vector quantization method. Principle component analysis pca, estimate rx from samples p. Pdf optimized parametric vector quantization of speech line. Observe the number of output values that falls in outer levels and inner levels if they match the assumed pdf. Let us repeat the earlier experiment with changing input variance and distributions and see the performance of the jayant quantizer compared to the pdf optimized quantizer. Sampling provides a discretetime representation of a continuous waveform. Non uniform quantization functions optimal output alphabets. Method i for finding optimal quantizers for a squarederror distortion measure. Nonlinear tonetransfer curve of quantizer, showing a linear region.
Finally, the discussion of the jayant quantizer was motivated by the need for robustness in the face of changing input statistics. Channel optimized vector quantization covq, as a joint sourcechannel coding scheme, has proven to perform well in compressing a source and making the resulting quantizer robust to channel noise. More recent examples of applications include clustering analysis 150. Pdf optimization of blockadaptive quantization for sar. The lloyd max algorithm 10 is a wellknown approach to designing nonuniform quantizers optimized according to the prevailing pdf of the input signal.
The structure of the quantizer is captured in the quantities and given in 4. A lloyd quantizer is the optimal quantizer minimum distortion for a given number of. Scalar and vector quantization national chiao tung. Sample amplitude x is uniformly distributed between a and a. Ece 5578 multimedia communication lec 08 transform. Optimized product quantization 1 optimized product. Based on the scenes studied it was concluded that a reduction from 8 to 4bits sample was the maximum data reduction ratio appropriate for.
Lloydmax quantizer lmq 1 that is optimized to match the probability density function of a specific signal. Entropyconstrained scalar quantization with a lossy. Pdf pdf optimized parametric vector quantization of speech. Design of sample adaptive product quantizers for noisy channels. Figure 1 describes the basic case of memoryless scalar quantization using b bits sample, or n2b levels. Design of sample adaptive product quantizers for noisy channels zahir raza, fady alajaji, senior member, ieee, and tamas linder, senior member, ieee abstractchannel optimized vector quantization covq has proven to be an effective joint sourcechannel coding technique that makes the underlying quantizer robust to channel noise. Bit uniform quantizer an overview sciencedirect topics.
In this adc scheme, the input process is oversampled sampled above its nyquist rate and quantized using a lowresolution quantizer usually 1. Rao, fellow, ieee abstracta computationally efficient, high quality, vector msvq 3 and splitvq 4 which employ suboptimal quantization scheme based on a parametric probability density. Note that for the optimum quantizer, the output signal variance is less than that of the input. Solution depends on input pdf and can be done numerically for commonly used pdfs e. Multiplechannel optimized quantizers for rayleigh fading. Pdf optimal context quantization in lossless compression of. Abstractsthis paper extends the channel optimized quantization scheme of farvardin and vaishampayan 1 to two parallel channels. The decision thresholds of the softdecision detectors are optimized to achieve minimum endtoend distortion. Uniform quantizer with 128 levels, n 27, 7 bits per sample speech typically limited to 4khz sample at 8khz. Assuming that you know the pdf of the samples to be quantized design the quantizers step so that it is optimal for that pdf. Sample adaptive product quantizer sapq dong sik kim, member, ieee, and ness b. Empirical quantizer design empirical quantizer designs, based on unquantized train ing samples that are typical of sensor data, are useful when a design using closed form models becomes complex or im practical e. Download citation pdf optimized parametric vector quantization of. Also known as pdf optimized quantizer given m, the optimal bi and yi that minimize mse satisfy.
Optimal quantization values not necessarily the region midpoints. Define quantization error and optimum scalar quantizer design criteria. Assuming that you know the pdf of the samples to be quantized design the quantizer s step so that it is optimal for that pdf. The quantizer output is fed to an encoder which generates the binary code for every sample. The resultant mcoqd subsumes the multiple description scalar quantizer mdsq. If this quantizer is pdfoptimized, the quantization error x. Multiple description quantizer design using a channel. The extended multiplechannel optimized quantizer design mcoqd framework is applied to discrete memoryless channels with erasures.
We assume that the random variable input x has a known probability density function pdf with zero mean. This is a structurally constrained vector quantizer that uses unions of product codebooks. Optimal tradeoff between sampling rate and quantization. Sampler unit samples the input signal and these samples are then fed to the quantizer which outputs the quantized values for each of the samples.
Due to the symmetry of the pdf and the quantizer design, the mse distortion can be. Pdf softdecoding vector quantizer using reliability. Germany, june 29 july 4 structured spherical codes. In this work we are interested in the tradeoff between these two quantities in ad conversion using sigmadelta modulation sdm. In order to design the optimized quantizer, the statistical properties of the encoded data are considered. Optimized product quantization 1 optimized product quantization tiezheng ge, kaiming hey, qifa ke, and jian sun abstractproduct quantization pq is an effective vector quantization method. Quantizer designirrespective of the distortion measure used, the source pdf has, in principle, the necessary and the sufficient information to build an optimal finite rate quantizer for the source. The essence of pq is to decompose the highdimensional vector. On noisy source vector quantization via a subspace constrained. The lloyd quantizer that is constructed using an iterative procedure provides the minimum distortion for a given number of reconstruction levels, in other words, it generates the pdf optimized scalar quantizer.
Pdf optimization of blockadaptive quantization for sar raw. Express the resulting distortion as function of aand the variance. Index terms combined sourcechannel coding, turbocodes. For uniform pdf x max x f x x 12x max want to uniformly quantize an rv x ux max,x max assume that desire m rls for r.
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