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1 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING DIGITAL COMMUNICATION Spring 00 Yrd. Doç. Dr. Burak Kelleci
2 OUTLINE Quantization Pulse-Code Modulation
3 THE QUANTIZATION PROCESS A continuous signal has continuous range of amplitudes, in other words it has infinite number of amplitude levels. Any human sense can detect only finite amplitude differences. Therefore, the original continuous signal may be approximated by a signal constructed of discrete amplitudes. If the discrete amplitude levels have sufficiently close spacing, the discrete-level signal will be practically indistinguishable from the continouslevel signal.
4 THE QUANTIZATION PROCESS Amplitude quantization is defined as the process of transforming the sample amplitude m(nt s ) of a message signal m(t) at time t=nt s into a discrete amplitude v(nt s ) taken from a finite set of possible amplitudes. k
5 THE QUANTIZATION PROCESS In this class, we assume that the quantization process is memoryless and instantaneous, in other words the transformation at time t=nt s is not affected by earlier or later samples of message signal. The signal amplitude m is specified by the index k if it lies inside the interval m m m, k,, L k k k, where L is the total number of amplitude levels used in the quantizer. m k : decision levels or decision thresholds.
6 THE QUANTIZATION PROCESS The amplitudes vk are called representation levels or reconstruction levels, and the spacing between two adjacent representation levels is called a quantum or step-size. The mapping v=g(m) is the quantizer characteristic. Quantizers are uniform quantizer: representation levels are uniformly spaced. non-uniform quantizer: representation levels are not uniformly spaced. midtread: origin lies in the middle of tread of the staircaselike graph midrise: origin lies in the middle of a rising part of staircaselike graph.
7 THE QUANTIZATION PROCESS Midtread Uniform Quantizer Midtrise Uniform Quantizer
8 QUANTIZATION NOISE Quantization noise: Error between the input signal and quantized signal. Let s assume that the quantization noise is statistically independent from the signal and the message signal does not overload the quantizer, in other words the signal levels are within the quantizer maximum and minimum levels. Consider a continuous signal m in the range of (-m max, m max ). For midrise quantizer the step-size is m max L
9 QUANTIZATION NOISE
10 QUANTIZATION NOISE The quantization noise values are bounded by / q / If the step-size is sufficiently small, we can assume that quantization error is uniformly distributed random variable, and its effect is similar the thermal noise. Its probability density is f Q q 0 q otherwise
11 QUANTIZATION NOISE The mean of quantization error is zero, and its variance is Q q The quantization noise is proportional to the step-size. Reducing the step-size reduces the quantization noise. q f q Q dq dq
12 QUANTIZATION NOISE Let R denote the number of bits per sample. The number of levels area L R The step-size and the variance of quantization noise mmax R R Q mmax 3 R log L
13 QUANTIZATION NOISE The signal-to-noise ratio for a message signal of power P P 3P R SNR Q m max The SNR increases exponentially with increasing the number of bits per sample, R. Since R is proportional to required channel bandwidth, there is a tradeoff between the channel bandwidth and SNR like FM. Binary transmission of message provides a more efficient method than FM for the trade-off due to the exponential relationship.
14 QUANTIZATION NOISE Sinusoidal Modulating Signal Let s assume a sinusoidal signal with amplitude A m is applied to uniform quantizer. The average signal power on ohm is P A m The SNR of the quantized signal is SNR 0log P 0 Q 3Am A m / R 3 SNR.8 6R in db R L R SNR(dB)
15 PULSE-CODE MODULATION In pulse-code modulation (PCM) a message signal is represented by a sequence of coded pulses, which is accomplished by representing the signal in discrete form in both time and amplitude. The basic operations performed in a PCM transmitter is sampling, quantizing and encoding. The basic operations in the receiver are regeneration, decoding and reconstruction of the train of quantized samples.
16 PULSE-CODE MODULATION
17 SAMPLING & QUANTIZATION Prior sampling a low-pass filter is used to restrict the bandwidth half of the sampling frequency. This filter prevents aliasing of frequency components above half of the sampling rate. The sampled signals are then quantized to predetermined levels. The quantized signal is then encoded to binary bits for robust transmission.
18 QUANTIZATION Till now, we assumed uniform quantization. However, some signals, such as voice, spends different times at different levels. For example, voice signals are most of the time has low amplitude levels. The loud talks are not as common as the weak talks. Therefore, if step size of the quantizer changed to favor weak talks, the overall quality of the transmitted voice signal is improved. In practice, this non-uniform quantization is performed by passing the message signal through a compressor and then applying the compressed signal to a uniform quantizer.
19 QUANTIZATION There are two different standard for the compressing function. -law: used in North America and Japan A-law: used in Europe -law encoding is defined as v sgn m the decoding is defined as m sgn ln m ln v m v m
20 QUANTIZATION is a positive constant and =0 corresponds to uniform quantization. The slope of the compression curve or in other words the derivative of m with respect to v is d m d v m ln -law is not strictly logarithmic. For low input levels ( m <<) it is linear and logarithmic for high input levels ( m >>)
21 Normalized Output v Normalized Output v QUANTIZATION -law Compression and Decompression law Compression Law = =5 =00 -law Decompression Law = 0. =5 = Normalized Input m Normalized Input m
22 QUANTIZATION A-law compression law is defined as A m m ln A A v sgn m ln A m m lna A For A=, A-law corresponds to uniform quantizer. A-law decompressing law is defined as A v ln A m sgn v v lna e A v lna v ln A
23 QUANTIZATION A is a positive constant and A= corresponds to uniform quantization. The slope of the compression curve or in other words the derivative of m with respect to v is d m d v A lna ln A A-law shows also similar performance as -law. It behaves linear for small signal levels and shows logarithmic behavior for high signal levels. m A m m A
24 Normalized Output v Normalized Output v QUANTIZATION -law Compression and Decompression A-law Compression Law A= A=0 A=00 A-law Decompression Law A= 0. A=0 A= Normalized Input m Normalized Input m
25 Signal Level Signal Level Signal Level Signal Level QUANTIZATION Original Time (s) -0.5 Original Quantized Time (s) Original Quantized Time (s) -0.5 Original Quantized Time (s)
26 SNR (db) QUANTIZATION Comparasion of SNRs of -law (=00) and A-law (A-00) Uniform Quantization -Law Quantization A-Law Quantization Signal Level (db) 6-bit quantization is used Below 55dB, all methods show the same performance, since the signal is below the minimum step-size -Law and A-Law improves the SNR between -55dB and - 5dB. For signal levels above -5dB, uniform quantization shows better performance.
27 ENCODING After sampling and quantization the continuous message signal is limited to discrete set of values, but nor suited to transmission over a line or radio path. To make this signal more robust to noise, interference and other channel degradations, it needs to be encoded to a more appropriate form of signal. Representing each of this discrete set of values is called a code. One of these discrete events in a code is called code element or symbol.
28 ENCODING In a binary code, each symbol is assigned to one of two distinct values. The main reason for using the binary code is that the maximum advantage over the effects of noise in a transmission medium is obtained by using a binary code, because binary symbol withstands a relatively high level of noise and is easy to regenerate. Suppose that, in a binary code each words consists of R bits (binary digit). Therefore, R is the number of bits per sample. This code represents R distinct levels.
29 QUANTIZATION AND ENCODING IN MATLAB & SIMULINK In Matlab round function can be used to model the quantization. For example, the following code quantizes the input signal between - and where corresponds in binary. It also encodes the signal for step-size of 0.5. Therefore, minimum signal level that does not overload the quantizer is -x0.5=-0.5v Maximum signal level that does not overload the quantizer is x0.5=0.5v. The quantizer output will assign -0.5V to 00, -0.5 to 0, 0 to 0 and 0.5 to
30 Output Code QUANTIZATION AND ENCODING IN MATLAB & SIMULINK % Quantize between -0.5V and 0.5V % with a step size of 0.5V 3 t=0:e-3:; m=-:/(length(t)-):; Delta=0.5; % saturate m ms=m; ms(m<(-*delta))=-*delta; ms(m>(*delta))=*delta; % Quantize the signal mq=round((ms+*delta)/delta); figure(); plot(m,mq); xlabel('input Signal Level'); ylabel('output Code'); Input Signal Level Note that this code generates midtread response
31 Output Code QUANTIZATION AND ENCODING IN MATLAB & SIMULINK % Quantize between -0.5V and 0.5V % with a step size of 0.5V 3 t=0:e-3:; m=-:/(length(t)-):; Delta=0.5; % saturate m ms=m; ms(m<(-.5*delta))=-.5*delta; ms(m>(.5*delta))=.5*delta; % Quantize the signal mq=round((ms+.5*delta)/delta); figure(); plot(m,mq); xlabel('input Signal Level'); ylabel('output Code'); Input Signal Level Note that this code generates midtrise response
32 QUANTIZATION AND ENCODING IN MATLAB & SIMULINK In Simulink, there is Quantizer component, but this component has no limits and the user can only control the step size. Another method is using Lookup Table to model the quantizer.
33 EXAMPLE In the following PCM signal binary 0 is represented by -V and binary is represented by V. The code word consists of 3 bits and step-size is V. Find the sampled version of analog signal from which this binary signal is derived.
34 EXAMPLE - SOLUTION Symbol Code Word Amplit ude 00 V 00 V 3 0 3V V 5 0 5V 6 0 6V
35 EXAMPLE A PCM system that uses a uniform quantizer is followed by a 7-bit binary encoder. The bit rate of the system is equal to 50Mbit/s. What is the maximum message bandwidth for this system? Determine the output signal-to-quantization noise ratio when full-load sinusoidal modulating wave of frequency MHz is applied to the input.
36 EXAMPLE - SOLUTION Sampling the message signal with bandwidth W at Nyquist rate, and using R-bit code to represent each sample of the message signal gives the following bit duration. T b W max T S R WR 50Mbit / s 7 The output signal-to-quantization ratio is T b WR 3.57MHz SNR.8 6R dB 0log 0
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