Chapter Sampling and Quantization.1 Analog and Digital Signal In order to invetigate ampling and quantization, the difference between analog and digital ignal mut be undertood. Analog ignal conit of continuou value for both ae. Conider an electrical ignal whoe horizontal ai repreent time in econd and whoe vertical ai repreent amplitude in volt. The horizontal ai ha a range of value from zero to infinity with every poible value in between. Thi make the horizontal ai continuou. The vertical ai i alo continuou allowing the ignal amplitude to aume any value from zero to infinity. For every poible value in time there i a correponding value in amplitude for the analog ignal. Digital ignal on the other hand have dicrete value for both the horizontal and vertical ae. The ae are no longer continuou a they were with the analog ignal. In thi chapter, time will be ued a the quantity for the horizontal ai and volt will be ued for the vertical ai.. Introduction to Sampling The motivation for ampling and quantizing i the need to tore a ignal in a digital format. In order to convert an analog ignal to a digital ignal, the analog ignal mut be ampled and quantized. Sampling take the analog ignal and dicretize the time ai. After ampling, the time ai conit of dicrete point in time rather than continuou value in time. The reulting ignal after ampling i called a dicrete ignal, 3
4 ampled ignal, or a dicrete-time ignal. The reulting ignal after ampling i not a digital ignal. Even though the horizontal ai ha dicrete value the vertical ai i not dicretized. Thi mean that for any dicrete point in time, there are an infinite number of allowed value for the ignal to aume in amplitude. In order for the ignal to be a digital ignal, both ae mut be dicrete..3 Introduction to Quantization Since a dicrete ignal ha dicrete point in time but till ha continuou value in amplitude, the amplitude of the ignal mut be dicretized in order to tore it in digital format. The value of the amplitude mut be rounded off to dicrete value. If the vertical ai i divided into mall window of amplitude, then every value that lie within that window will be rounded off (or quantized) to the ame value. For eample, conider a waveform with window ize of 0.5 volt tarting at 4 volt and ending at +4 volt. At a dicrete point in time, any amplitude between 4.0 volt and 3.5 volt will be recorded a 3.75 volt. In thi eample the center of each 0.5-volt window (or quantization region) wa choen to be the quantization voltage for that region. Reaon for chooing the center a the quantization voltage will be dicued in ection.7. In thi eample the dynamic range of the ignal i 8 volt. Since each quantization region i 0.5 volt there are 16 quantization region included in the dynamic range. It i important that there are 16 quantization region in the dynamic range. Since a binary number will repreent the value of the amplitude, it i important that the number of quantization region i a power of two. In thi eample, 4 bit will be required to repreent each of the 16 poible value in the ignal amplitude.
5.4 Sampling Continuou-Time Signal Sampling a continuou-time ignal generate a dicrete-time ignal. Thi i accomplihed by multiplying the continuou-time ignal (the analog ignal) by a erie of unit impule. The reult i the original ignal information only at point in time where an impule occur. The proce dicard information about the original ignal at all other value in time..4.1 Dirac Delta Function The erie of unit pule ued to ample a ignal i a erie of Dirac delta function. Conider the Dirac delta function: δ (t) = Dirac Delta Function = 0,, t 0 t = 0 (.1) delta function: The following two relation define the continuou-time unit impule, or Dirac δ (t) = 0 for t 0 (.) δ( t ) dt = 1 for t = 0 (.3) The Dirac delta function i a unit pule in which the duration approache zero but the area remain unity. Thi mean a the width of the pule approache zero, the amplitude of the pule mut approach infinity to maintain a unit area [1]. Let g(t) equal a rectangular pule of width T that i an even function of time: g( t) = g( t) (.4) then δ ( t) = lim g( t) (.5) T 0
6 The dicrete-time verion of the unit impule function i defined by [1] 0 n 0 δ[ n ] = (.6) 1 n = 0 The Dirac delta function i the derivative of the unit tep function with repect to time, and therefore the unit tep function i the integral of the Dirac delta function with repect to time..4. Time-Domain Impule Sampling Conider a train of equally paced unit impule. Thi i called the Dirac comb and i defined a follow: δ T ( ) = δ( t kt k = 0 t ) (.7) where T i the ampling interval or ampling period. Impule ampling i performed by multiplying the continuou-time ignal (t) by the impule train. The dicrete-time ignal can be repreented mathematically by the following equation [8]: k = 0 ( t) = ( t) δ ( t) = ( t) δ( t kt ) (.8) T where (t) = continuou-time ignal (t) = dicrete-time ignal f = ampling frequency or ample rate = ( T ) From equation (.8) it i gathered that the dicrete-time ignal i the original ignal multiplied by a train of unit impule equally paced by the ampling interval [8]. 1
7 To eamine (t) in the frequency domain, the Fourier tranform i performed on (t): I{ ( t)} = X ( f ) = X ( f ) * I k = 0 δ( t kt ) = X ( f ) * k = δ( f kf ) (.9) where * denote convolution [3]. Multiplication in the time domain correpond to convolution in the frequency domain. A a reult, the pectral content of the ampled ignal ha the ame pectral content a the original ignal plu copie of the original ignal pectral content centered at integral multiple of the ampling frequency. Thi mean that in the frequency domain (t) look like a periodic repreentation of the original ignal frequency pectrum. Thee copie of the original ignal pectral content are called aliae. If the ampling rate i high enough, the aliae can be filtered out by low-pa or bandpa filter with cut-off frequencie outide the pectral content of the original ignal. Thi filtering of the aliae will recover the original ignal [3]..5 Nyquit Rate A dicued in ection (.4), the aliae caued by ampling can be removed if the ampling rate i high enough. Let the highet frequency component in the original ignal magnitude pectrum be called f and the lowet frequency component be called f min. The original ignal frequency content etend from f min to f. After ampling, the ampled ignal frequency content ha aliae (copie of the original ignal frequency content) centered at f = alia _ center kf (.10) where k = ± 0, 1,, 3, etc.
8 The lowet frequency component in the alia i given by the epreion: f = kf (.11) alia _ low f and the highet frequency component in the alia i given by the epreion: f alia _ high kf + f = (.1) A the ampling frequency i lowered, the firt alia (k = ± 1) i the alia that will merge into the original ignal pectral content firt. In thi cae f = ± f (.13) alia _ low f and f alia _ high ± f + f = (.14) Thi mean that the original ignal can be recovered without ditortion a long a the following relationhip hold: or f f > f (.15) Nyquit rate = f > f (.16) Therefore, the ampling rate mut be at leat twice a high a the highet frequency component in the original ignal magnitude pectrum. Thi rate, however, i merely a lower limit for the ideal cae. In practical ytem it i important to provide guardband between the original ignal pectral content and the aliae [3]. Increaing the ampling rate increae the ditance between the original ignal pectral content and the aliae. In thi cae the Nyquit rate i eceeded in order to provide guardband between aliae and to take into account the non-ideal frequency repone of filter. The practice of eceeding the Nyquit rate i referred to a overampling [8].
9.6 Quantization A dicued in ection (.3), the quantization proce take a ignal that i dicrete in time and continuou in amplitude and convert it into a ignal that i dicrete in both time and amplitude. In order for the ignal to be tored digitally, the rounded-off value of the ignal amplitude at dicrete point in time mut be converted into binary number. In thi chapter uniform quantization will be dicued in which each quantization region ha the ame quantization width. A dicued in ection (.3), the number of quantization region determine the number of bit required to repreent each dicrete value in amplitude. The number of bit per ample (n), or quantization bit, required in uniform quantization i calculated a follow: n = LOG (number of quantization region) (.17) Conider a digitizer that i et to ample at 0,000 ample/ec and require 4 bit/ample for quantization. Since the ignal i being ampled at a certain rate with a certain number of quantization bit, the output of the digitizer mut be outputting data at a certain rate. The encoding rate i the number of bit per econd that i required to digitally repreent the ignal. Quantitatively, the encoding rate i the product of the ampling frequency and the number of quantization bit [3]. Encoding Rate = (f )*(n) (.18) In thi eample the encoding rate i equal to 80,000 bit per econd. Increaing the ampling frequency puhe the aliae away from the original ignal pectral content and allow analyi of higher frequency ignal, but it alo increae the amount of required memory to tore the ignal digitally. The ame tradeoff applie to increae in
10 the number of quantization bit: a more accurate repreentation of the ignal can be achieved, but the amount of required memory to tore the ignal alo increae. The dynamic range of the ignal i the range of the ignal amplitude [3]. In the eample in ection (.3) the dynamic range wa 8 volt becaue the ignal amplitude ranged from 4 volt to 4 volt. Dynamic Range = (Signal_Amplitude ) (Signal_Amplitude min ) (.19) The width of the quantization region (for uniform quantization) can be defined by the following relationhip: Dynamic Range Quantizati on Region Width = (.0) n where n = number of bit per ample [3]. The quantization region width i inverely proportional to the reolution of the quantization. It hould alo be noted that while a decreae in the quantization region width increae reolution, it alo increae the number of bit required to digitally repreent the ignal. Since quantization involve rounding-off to the center of the quantization region, the proce can caue error to be preent between the actual amplitude value and recorded amplitude value..7 Uniform Quantization Error The reaon for chooing the center of the quantization region to be deignated a the rounded-off value i een by evaluating the quantization error (alo called quantization noie) aociated with the rounding-off proce. Quantization error i the difference in the ignal actual amplitude and the amplitude aumed by the proce of quantization. If the top of the quantization region i choen a the round-off value,
11 then the minimum quantization error i zero and the imum quantization error i equal to the quantization region width. The ame error i aociated with chooing the bottom of the quantization region a the round-off value. If the center of the quantization region i choen to be the round-off value, then the minimum error i till zero but the imum error i ± one-half the quantization region width [3]. Error in chooing the center of the quantization region i defined by: X min = 0 (.1) 1 X = ± (Quantization Width) (.) Dynamic Range X = ± (.3) n+ 1 From equation (.3) it i evident that the accuracy i increaed by increaing n (the number of quantization bit). Since quantization error i equally likely to be any value ranging from the minimum quantization error to the imum quantization error, the error i aid to be probabilitic. Quantization error can be repreented a a uniformly ditributed random variable ince it ha an equal probability to lie at any point in the range of poible quantization error value (minimum quantization error to imum quantization error). The probability denity function of the uniformly ditributed random variable for quantization error i defined a the following [3]: 1 ) = X 0 f (,, X other X X (.4) Since the quantization error i a uniformly ditributed random variable centered at zero, the average value, or mean, of the quantization error i zero.
1 µ = mean = 0 (.5) The variance of the quantization error i the amount that the error varie about the mean. The higher the variance i, the le predictable the random variable i. Variance i given by the following relationhip [6]: σ = ( µ ) f ( ) d (.6) For random variable with a mean of zero, the variance i equal to the average normalized power. Thi i demontrated a follow [3]: σ = ( 0) f ( ) d = f ( ) d (.7) The variance in the above epreion i equal to the average value of X, which i the average normalized power. For uniform quantization, the mean (µ ) of the quantization error i zero [3]. Alo taking into account the fact that the probability denity function i zero for -value greater than X or le than X min, the variance of the quantization error equal to σ X X = f ( ) d = d = X X 1 X X 3 (.8).8 Summary In order to convert an analog ignal to a digital format, ampling and quantization mut be performed on the ignal. Although quantization introduce error, thi error can be reduced by increaing the number of quantization bit. Once a ignal ha been ampled and quantized, it can be tored in a digital format and manipulated and analyzed by DSP (digital ignal proceing) algorithm. Chapter 3 dicue the ue of applying
13 the dicrete-time Fourier erie (DTFS) to digital ignal in order to gain inight into the frequency content of the original ignal.