Copyright (c) 2017, Shujaat Khan All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 'AS IS' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Source Coding Represent Partitions Scalar quantization is a process that maps all inputs within a specified range to a common value. This process maps inputs in a different range of values to a different common value. Wavelab 6.
In effect, scalar quantization digitizes an analog signal. Two parameters determine a quantization: a and a. A quantization partition defines several contiguous, nonoverlapping ranges of values within the set of real numbers. To specify a partition in the MATLAB ® environment, list the distinct endpoints of the different ranges in a vector. For example, if the partition separates the real number line into the four sets. Codebook = -1, 0.5, 2, 3; is one possible codebook for the partition 0,1,3.
Determine Which Interval Each Input Is In The quantiz function also returns a vector that tells which interval each input is in. For example, the output below says that the input entries lie within the intervals labeled 0, 6, and 5, respectively. Here, the 0th interval consists of real numbers less than or equal to 3; the 6th interval consists of real numbers greater than 8 but less than or equal to 9; and the 5th interval consists of real numbers greater than 7 but less than or equal to 8. Partition = 3,4,5,6,7,8,9; codebook = 3,3,4,5,6,7,8,9; index,quants = quantiz(2 9 8,partition,codebook); Optimize Quantization Parameters. Section Overview Quantization distorts a signal. You can reduce distortion by choosing appropriate partition and codebook parameters.
However, testing and selecting parameters for large signal sets with a fine quantization scheme can be tedious. One way to produce partition and codebook parameters easily is to optimize them according to a set of so-called training data. Note The training data you use should be typical of the kinds of signals you will actually be quantizing. Example: Optimizing Quantization Parameters The lloyds function optimizes the partition and codebook according to the Lloyd algorithm. The code below optimizes the partition and codebook for one period of a sinusoidal signal, starting from a rough initial guess.
Then it uses these parameters to quantize the original signal using the initial guess parameters as well as the optimized parameters. The output shows that the mean square distortion after quantizing is much less for the optimized parameters. The quantiz function automatically computes the mean square distortion and returns it as the third output parameter. Ans = 0.0148 0.0024 Differential Pulse Code Modulation. Section Overview The quantization in the section requires no a priori knowledge about the transmitted signal. In practice, you can often make educated guesses about the present signal based on past signal transmissions. Using such educated guesses to help quantize a signal is known as predictive quantization.
The most common predictive quantization method is differential pulse code modulation (DPCM). The functions dpcmenco, and dpcmopt can help you implement a DPCM predictive quantizer with a linear predictor. DPCM Terminology To determine an encoder for such a quantizer, you must supply not only a partition and codebook as described in and, but also a predictor. The predictor is a function that the DPCM encoder uses to produce the educated guess at each step. A linear predictor has the form. Note The initial zero in the predictor vector makes sense if you view the vector as the polynomial transfer function of a finite impulse response (FIR) filter. Example: DPCM Encoding and Decoding A simple special case of DPCM quantizes the difference between the signal's current value and its value at the previous step.
Thus the predictor is just y(k) = x (k - 1). The code below implements this scheme. It encodes a sawtooth signal, decodes it, and plots both the original and decoded signals. The solid line is the original signal, while the dashed line is the recovered signals. The example also computes the mean square error between the original and decoded signals. Note The training data you use with dpcmopt should be typical of the kinds of signals you will actually be quantizing with dpcmenco. Example: Comparing Optimized and Nonoptimized DPCM Parameters This example is similar to the one in the last section.
However, where the last example created predictor, partition, and codebook in a straightforward but haphazard way, this example uses the same codebook (now called initcodebook) as an initial guess for a new optimized codebook parameter. This example also uses the predictive order, 1, as the desired order of the new optimized predictor. The dpcmopt function creates these optimized parameters, using the sawtooth signal x as training data. The example goes on to quantize the training data itself; in theory, the optimized parameters are suitable for quantizing other data that is similar to x. Notice that the mean square distortion here is much less than the distortion in the previous example.
Distor = 0.0063 Compand a Signal. Section Overview In certain applications, such as speech processing, it is common to use a logarithm computation, called a compressor, before quantizing. The inverse operation of a compressor is called an expander. The combination of a compressor and expander is called a compander.
The compand function supports two kinds of companders: µ-law and A-law companders. Its reference page lists both compressor laws. Example: µ-Law Compander The code below quantizes an exponential signal in two ways and compares the resulting mean square distortions. First, it uses the function with a partition consisting of length-one intervals. In the second trial, implements a µ-law compressor, quantiz quantizes the compressed data, and compand expands the quantized data.
The output shows that the distortion is smaller for the second scheme. This is because equal-length intervals are well suited to the logarithm of sig, but not well suited to sig. The figure shows how the compander changes sig. Mu = 255;% Parameter for mu-law compander sig = -4.1:4; sig = exp(sig);% Exponential signal to quantize V = max(sig);% 1. Quantize using equal-length intervals and no compander. Install windows 98 bochs. index,quants,distor = quantiz(sig,0:floor(V),0:ceil(V));% 2.
Use same partition and codebook, but compress% before quantizing and expand afterwards. Compsig = compand(sig,Mu,V, 'mu/compressor'); index,quants = quantiz(compsig,0:floor(V),0:ceil(V)); newsig = compand(quants,Mu,max(quants), 'mu/expander'); distor2 = sum((newsig-sig).^2)/length(sig); distor, distor2% Display both mean square distortions.
Plot(sig);% Plot original signal. Hold on; plot(compsig,'r-');% Plot companded signal. Legend('Original','Companded','Location','NorthWest') The output and figure are below. Note For long sequences from sources having skewed distributions and small alphabets, arithmetic coding compresses better than Huffman coding. To learn how to use arithmetic coding, see. Create a Huffman Code Dictionary in MATLAB Huffman coding requires statistical information about the source of the data being encoded.
In particular, the p input argument in the huffmandict function lists the probability with which the source produces each symbol in its alphabet. For example, consider a data source that produces 1s with probability 0.1, 2s with probability 0.1, and 3s with probability 0.8. The main computational step in encoding data from this source using a Huffman code is to create a dictionary that associates each data symbol with a codeword. The commands below create such a dictionary and then show the codeword vector associated with a particular value from the data source. Sig = repmat(3 3 1 3 3 3 3 3 2 3,1,50);% Data to encode symbols = 1 2 3;% Distinct data symbols appearing in sig p = 0.1 0.1 0.8;% Probability of each data symbol dict = huffmandict(symbols,p);% Create the dictionary.
Hcode = huffmanenco(sig,dict);% Encode the data. Dhsig = huffmandeco(hcode,dict);% Decode the code.
Arithmetic Coding. Section Overview Arithmetic coding offers a way to compress data and can be useful for data sources having a small alphabet.
The length of an arithmetic code, instead of being fixed relative to the number of symbols being encoded, depends on the statistical frequency with which the source produces each symbol from its alphabet. For long sequences from sources having skewed distributions and small alphabets, arithmetic coding compresses better than Huffman coding. The arithenco and arithdeco functions support arithmetic coding and decoding. Represent Arithmetic Coding Parameters Arithmetic coding requires statistical information about the source of the data being encoded. In particular, the counts input argument in the arithenco and arithdeco functions lists the frequency with which the source produces each symbol in its alphabet. You can determine the frequencies by studying a set of test data from the source.
The set of test data can have any size you choose, as long as each symbol in the alphabet has a nonzero frequency. For example, before encoding data from a source that produces 10 x's, 10 y's, and 80 z's in a typical 100-symbol set of test data, define. Quantized = Columns 1 through 6 -1.0000 -1.0000 -1.0000 -1.0000 0.5000 0.5000 Columns 7 through 12 2.0000 2.0000 2.0000 2.0000 2.0000 3.0000 Column 13 3.0000 Scalar Quantization Example 2 This example illustrates the nature of scalar quantization more clearly.
After quantizing a sampled sine wave, it plots the original and quantized signals. The plot contrasts the x's that make up the sine curve with the dots that make up the quantized signal.
The vertical coordinate of each dot is a value in the vector codebook.
Although humans are well equipped for analog communications, analog transmission is not particularly efficient. When analog signals become weak because of transmission loss, it is hard to separate the complex analog structure from the structure of random transmission noise. If you amplify analog signals, it also amplifies noise, and eventually analog connections become too noisy to use. Digital signals, having only 'one-bit' and 'zero-bit' states, are more easily separated from noise. They can be amplified without corruption. Digital coding is more immune to noise corruption on long-distance connections. Also, the world's communication systems have converted to a digital transmission format called pulse code modulation (PCM).
PCM is a type of coding that is called 'waveform' coding because it creates a coded form of the original voice waveform. This document describes at a high level the conversion process of analog voice signals to digital signals. There are no specific requirements for this document. This document is not restricted to specific software and hardware versions. For more information on document conventions, refer to the. PCM is a waveform coding method defined in the ITU-T G.711 specification.
The first step to convert the signal from analog to digital is to filter out the higher frequency component of the signal. This make things easier downstream to convert this signal. Most of the energy of spoken language is somewhere between 200 or 300 hertz and about 2700 or 2800 hertz. Roughly 3000-hertz bandwidth for standard speech and standard voice communication is established. Therefore, they do not have to have precise filters (it is very expensive). A bandwidth of 4000 hertz is made from an equipment point if view.
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This band-limiting filter is used to prevent aliasing (antialiasing). This happens when the input analog voice signal is undersampled, defined by the Nyquist criterion as Fs 2(BW) Fs = Sampling frequency BW = Bandwidth of original analog voice signal Figure 1: Analog Sampling After you filter and sample (using PAM) an input analog voice signal, the next step is to digitize these samples in preparation for transmission over a Telephony network.
The process of digitizing analog voice signals is called PCM. The only difference between PAM and PCM is that PCM takes the process one step further. PCM decodes each analog sample using binary code words.
PCM has an analog-to-digital converter on the source side and a digital-to-analog converter on the destination side. PCM uses a technique called quantization to encode these samples. Figure 2: Pulse Code Modulation - Nyquist Theorem Quantization is the process of converting each analog sample value into a discrete value that can be assigned a unique digital code word. As the input signal samples enter the quantization phase, they are assigned to a quantization interval.
All quantization intervals are equally spaced (uniform quantization) throughout the dynamic range of the input analog signal. Each quantization interval is assigned a discrete value in the form of a binary code word. The standard word size used is eight bits. If an input analog signal is sampled 8000 times per second and each sample is given a code word that is eight bits long, then the maximum transmission bit rate for Telephony systems using PCM is 64,000 bits per second. Figure 2 illustrates how bit rate is derived for a PCM system. Each input sample is assigned a quantization interval that is closest to its amplitude height.
If an input sample is not assigned a quantization interval that matches its actual height, then an error is introduced into the PCM process. This error is called quantization noise. Quantization noise is equivalent to the random noise that impacts the signal-to-noise ratio (SNR) of a voice signal. SNR is a measure of signal strength relative to background noise. The ratio is usually measured in decibels (dB). If the incoming signal strength in microvolts is Vs, and the noise level, also in microvolts, is Vn, then the signal-to-noise ratio, S/N, in decibels is given by the formula S/N = 20 log10(Vs/Vn).
SNR is measured in decibels (dB). The higher the SNR, the better the voice quality. Quantization noise reduces the SNR of a signal.
Therefore, an increase in quantization noise degrades the quality of a voice signal. Figure 3 shows how quantization noise is generated. For coding purpose, an N bit word yields 2N quantization labels. Figure 3: Analog to Digital Conversion One way to reduce quantization noise is to increase the amount of quantization intervals. The difference between the input signal amplitude height and the quantization interval decreases as the quantization intervals are increased (increases in the intervals decrease the quantization noise). However, the amount of code words also need to be increased in proportion to the increase in quantization intervals. This process introduces additional problems that deal with the capacity of a PCM system to handle more code words.
Uniform Quantization Of Images
SNR (including quantization noise) is the single most important factor that affects voice quality in uniform quantization. Uniform quantization uses equal quantization levels throughout the entire dynamic range of an input analog signal.
Therefore, low signals have a small SNR (low-signal-level voice quality) and high signals have a large SNR (high-signal-level voice quality). Since most voice signals generated are of the low kind, having better voice quality at higher signal levels is a very inefficient way of digitizing voice signals. To improve voice quality at lower signal levels, uniform quantization (uniform PCM) is replaced by a nonuniform quantization process called companding. Companding refers to the process of first compressing an analog signal at the source, and then expanding this signal back to its original size when it reaches its destination. The term companding is created by combining the two terms, compressing and expanding, into one word. At the time of the companding process, input analog signal samples are compressed into logarithmic segments.
Each segment is then quantized and coded using uniform quantization. The compression process is logarithmic.
The compression increases as the sample signals increase. In other words, the larger sample signals are compressed more than the smaller sample signals. This causes the quantization noise to increase as the sample signal increases. A logarithmic increase in quantization noise throughout the dynamic range of an input sample signal keeps the SNR constant throughout this dynamic range. The ITU-T standards for companding are called A-law and u-law. A-law and u-law are audio compression schemes (codecs) defined by Consultative Committee for International Telephony And Telegraphy (CCITT) G.711 which compress 16-bit linear PCM data down to eight bits of logarithmic data.
A-law Compander Limiting the linear sample values to twelve magnitude bits, the A-law compression is defined by this equation, where A is the compression parameter (A=87.7 in Europe), and x is the normalized integer to be compressed. U-law Compander Limiting the linear sample values to thirteen magnitude bits, the u-law (u-law and Mu- law are used interchangeably in this document) compression is defined by this equation, where m is the compression parameter (m =255 in the U.S. And Japan) and x is the normalized integer to be compressed. A-law standard is primarily used by Europe and the rest of the world.
U-law is used by North America and Japan. Both are linear approximations of logarithmic input/output relationship. Both are implemented using eight-bit code words (256 levels, one for each quantization interval). Eight-bit code words allow for a bit rate of 64 kilobits per second (kbps). This is calculated by multiplying the sampling rate (twice the input frequency) by the size of the code word (2 x 4 kHz x 8 bits = 64 kbps). Both break a dynamic range into a total of 16 segments:. Eight positive and eight negative segments.
Each segment is twice the length of the preceding one. Uniform quantization is used within each segment.
Both use a similar approach to coding the eight-bit word:. First (MSB) identifies polarity.
Bits two, three, and four identify segment. Final four bits quantize the segment are the lower signal levels than A-law.
Different linear approximations lead to different lengths and slopes. The numerical assignment of the bit positions in the eight-bit code word to segments and the quantization levels within segments are different. A-law provides a greater dynamic range than u-law.
u-law provides better signal/distortion performance for low level signals than A-law. A-law requires 13-bits for a uniform PCM equivalent. U-law requires 14-bits for a uniform PCM equivalent. An international connection needs to use A-law, u to A conversion is the responsibility of the u-law country. At the time of the PCM process, the differences between input sample signals are minimal.
Differential PCM (DPCM) is designed to calculate this difference and then transmit this small difference signal instead of the entire input sample signal. Since the difference between input samples is less than an entire input sample, the number of bits required for transmission is reduced. This allows for a reduction in the throughput required to transmit voice signals. Using DPCM can reduce the bit rate of voice transmission down to 48 kbps. How does DPCM calculate the difference between the current sample signal and a previous sample?
The first part of DPCM works exactly like PCM (that is why it is called differential PCM). The input signal is sampled at a constant sampling frequency (twice the input frequency). Then these samples are modulated using the PAM process. At this point, the DPCM process takes over. The sampled input signal is stored in what is called a predictor.
The predictor takes the stored sample signal and sends it through a differentiator. The differentiator compares the previous sample signal with the current sample signal and sends this difference to the quantizing and coding phase of PCM (this phase can be uniform quantizing or companding with A-law or u-law). After quantizing and coding, the difference signal is transmitted to its final destination. At the receiving end of the network, everything is reversed. First the difference signal is dequantized. Then this difference signal is added to a sample signal stored in a predictor and sent to a low-pass filter that reconstructs the original input signal. DPCM is a good way to reduce the bit rate for voice transmission.
However, it causes some other problems that deal with voice quality. DPCM quantizes and encodes the difference between a previous sample input signal and a current sample input signal.
DPCM quantizes the difference signal using uniform quantization. Uniform quantization generates an SNR that is small for small input sample signals and large for large input sample signals. Therefore, the voice quality is better at higher signals. This scenario is very inefficient, since most of the signals generated by the human voice are small.
Uniform Quantization In Matlab
Voice quality needs to focus on small signals. To solve this problem, adaptive DPCM is developed. Adaptive DPCM (ADPCM) is a waveform coding method defined in the ITU-T G.726 specification. ADPCM adapts the quantization levels of the difference signal that generated at the time of the DPCM process. How does ADPCM adapt these quantization levels? If the difference signal is low, ADPCM increases the size of the quantization levels.
If the difference signal is high, ADPCM decreases the size of the quantization levels. So, ADPCM adapts the quantization level to the size of the input difference signal. This generates an SNR that is uniform throughout the dynamic range of the difference signal. Using ADPCM reduces the bit rate of voice transmission down to 32 kbps, half the bit rate of A-law or u-law PCM. ADPCM produces 'toll quality' voice just like A-law or u-law PCM. Coder must have feedback loop, using encoder output bits to recalibrate the quantizer. Applicable as ITU Standards G.726.
Uniform Quantization Code
Turn A-law or Mu-law PCM samples into a linear PCM sample. Calculate the predicted value of the next sample. Measure the difference between actual sample and predicted value. Code difference as four bits, send those bits. Feed back four bits to predictor. Feed back four bits to quantizer.
Copyright (c) 2011, Nikesh Bajaj All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. Neither the name of the nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 'AS IS' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Hello, plz it is very Urgent in my thesis, I need to use convoultional encoder with code rate =1/2 or 3/4 with puncturing and the decode the data using vitdec (soft-decision viterbi decoder) with NO Quantization. I am writing the code but I have mistake in vitdec stage. Can someone help me in correcting this code plz.% msg=rand(1,10)0.5;% Random data% Convoultional Encoder trel = poly2trellis(6,65,57);% Define trellis code = convenc(msg,trel);% Encode.%decoding the msg using vitdec tblen = 35;% Traceback length decode = vitdec(code,trel,tblen,'cont','soft',3);% bit error must be zeroo ber= decode -code; thanks [email protected].