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Unformatted text preview: MODULE 13 QUANTIZATION EFFECTS • Filter Coefficient Quantization • Input Signal Quantization • Arithmetic Roundoff • Arithmetic Overflow • ZeroInput Limit Cycles Page 13.1 INDEX Quantization of FixedPoint Numbers SignMagnitude and Two's Complement Representation Rounding and Truncation Quantizer Characteristics Table of Quantization Error Ranges FIR Coefficient Quantization FIR Coefficient Sensitivity Analysis IIR Coefficient Quantization IIR Coefficient Sensitivity Analysis Input Signal Quantization Additive Noise Model Uniform Quantization Model Input and Output Quantization Noise Statistics Controlling the MSE Arithmetic Roundoff Arithmetic Roundoff Error Model Arithmetic Overflow Scaling to Prevent Overflow l 1Scaling Maximum Scaling ZeroInput Limit Cycles Bound on Limit Cycle Amplitude Limit Cycle: SecondOrder Section MAIN INDEX Page 13.2 13. QUANTIZATION EFFECTS index READ : Sections 6.6  6.9 of Oppenheim & Schafer. Work as many related problems as possible. • Here we are concerned with the effects of quantization inherent in the digital processing of signals. Mostly, the error that occurs in linear filtering is of interest. • These can be divided into three main categories: filter coefficient quantization input signal quantization arithmetic roundoff (in calculations) Page 13.3 Quantization of FixedPoint Numbers index • The analysis of floating point quantization is more involved , and won’t be considered here. Also, uses the fixed point results. • Quantizer : Q [·] x ( n ) x ( n ) • Quantization is a loss of information , hence the process introduces an error : ε ( n )= Q [ x ( n ) ] x ( n ) = x ( n )  x ( n ) • This error depends on the numerical representation scheme used to store numbers digitally. Two will be examined here: signmagnitude two's complement • The error also depends on the type of quantization that is used: rounding truncation Page 13.4 SignMagnitude Representation index • Here it is assumed that a quantized number x is represented as a sequence of bits indicating the magnitude of x , and a sign bit indicating the polarity of x . Symbolically separated by a binary point: . ······ binary point sign bit 64 44 44 44 74 44 44 44 8 x • The sign bit takes logical value ≥ < ˆ if ; ˆ if ; 1 x x Page 13.5 Two's Complement Representation index • A quantized number x stored in two's complement form is: the same as x in signmagnitude form if x ≥ if x < 0, then 2   x  is stored • For negative x : x =  ∑ = B k 1 γ ( k ) 2k the two's complement representation can be accomplished via: (1) Start with: 0 . γ (1) γ (2) γ (3) · · · γ ( B ) (2) Complement: 1 ....
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 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Signal Processing, Finite impulse response, Infinite impulse response, IIR Coefficient Quantization

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