电子科技大学：《DSP算法实现技术与架构 VLSI Digital Signal Processing Systems Design and Implementation》课程教学资源（课件讲稿）Chapter 11 缩放噪声 Scaling and Roundoff Noise

电子种越女学 University of Electroe Scioncad TechofChina 986 Chapter 11 Scaling and Roundoff Noise Xiang LING National Key Lab of Science and Technology on Communications

Chapter 11 Scaling and Roundoff Noise Xiang LING National Key Lab of Science and Technology on Communications

11.1 Introduction 96 In a fixed-point digital filter implementation,the overall input-output behavior is non-ideal due to quantization of signals and coefficients. There are two basic types of quantization effects in any implementation. Coefficient quantization Signal rounding Limit-cycle oscillation ■Roundoff noise 2021年2月 2

2021年2月 2 11.1 Introduction  In a fixed-point digital filter implementation, the overall input-output behavior is non-ideal due to quantization of signals and coefficients.  There are two basic types of quantization effects in any implementation.  Coefficient quantization  Signal rounding  Limit-cycle oscillation  Roundoff noise

11.2.1 Scaling Operation /96 Scaling: -A process of readjusting certain internal gain parameters in order to constrain internal signals to a range appropriate to the hardware with the constraint that the transfer function from input to output should not be changed. D(Z) H()=D()+F()G() IN OUT F(Z) G(Z) (a) D(Z) OUT F(Z)/B x' BG(Z (b) 2021年2月 3

2021年2月 3 11.2.1 Scaling Operation  Scaling:  A process of readjusting certain internal gain parameters in order to constrain internal signals to a range appropriate to the hardware with the constraint that the transfer function from input to output should not be changed. H(z)  D(z)  F(z)G(z)

/966 The scaling parameter B can be chosen to meet any specific scaling rule such as h-scaling:B=∑of(l (11.2) 12-scaling:B=δV∑f2(l (11.3) where f(i)is the unit-sample response from input to the node x, ■ And the parameter 6 can be interpreted to represent the value of standard deviations representable in the register at node x if input is unit-variance white noise. 2021年2月 4

2021年2月 4  The scaling parameter β can be chosen to meet any specific scaling rule such as  where f(i) is the unit-sample response from input to the node x,  And the parameter δ can be interpreted to represent the value of standard deviations representable in the register at node x if input is unit-variance white noise

/96 ■If the input is bounded by |u(n)l≤l,then ron=|∑f0un-sofo间 (11.4) ■ Equation (11.4)represents the true bound on the range of x and overflow is completely avoided by /scaling in (11.2),which is the most stringent scaling policy. 2021年2月 5

2021年2月 5  If the input is bounded by |u(n)| ≤1, then  Equation (11.4) represents the true bound on the range of x and overflow is completely avoided by l1 scaling in (11.2), which is the most stringent scaling policy

96 Input can be generally assumed to be white noise.For unit-variance white noise input,variance at node x is given by: -∑f0 (11.5) /scaling is commonly used because most input signals can be assumed to be white noise. (11.5)is a variance (not a strict bound),there is a possibility of overflow. We can increase 6 in (11.3)to prevent possible overflow. But increasing o will decrease SNR (signal-to-noise ratio). Thus,there is a trade-off between overflow and round-off noise. 2021年2月 6

2021年2月 6  Input can be generally assumed to be white noise. For unit-variance white noise input, variance at node x is given by:  l2 scaling is commonly used because most input signals can be assumed to be white noise.  (11.5) is a variance (not a strict bound), there is a possibility of overflow.  We can increase δ in (11.3) to prevent possible overflow.  But increasing δ will decrease SNR (signal-to-noise ratio). Thus, there is a trade-off between overflow and round-off noise

11,2.2 Round-off Noise /986 Product of two Wbit fixed-point fractions is a (2 W1)bit number.This product must eventually be quantized to Wbits by rounding or truncation, which results in round-off noise. 2021年2月 7

2021年2月 7 11.2.2 Round-off Noise  Product of two W-bit fixed-point fractions is a (2W-1) bit number. This product must eventually be quantized to W-bits by rounding or truncation, which results in round-off noise

966 Example: Consider the 1st-order IIR filter. Assume that the input wordlength W=8 bits,and the multiplier coefficient wordlength is also 8 bits. To maintain full precision in the output,we need to increase the output wordlength by 8 bits per iteration. This is clearly infeasible. Thus,the result needs to be rounded or a 15-bits 8-bits truncated to its nearest 8-bit representation. This introduces a round-u(n) D x(n) off noise e(n). 8-bits 2021年2月 8

2021年2月 8  Example:  Consider the 1st-order IIR filter.  Assume that the input wordlength W=8 bits, and the multiplier coefficient wordlength is also 8 bits.  To maintain full precision in the output, we need to increase the output wordlength by 8 bits per iteration. This is clearly infeasible.  Thus, the result needs to be rounded or truncated to its nearest 8-bit representation. This introduces a round￾off noise e(n)

/96 Round-off Noise Mathematical Model:usually modeled as an infinite precision system with an external error input. Rounding is a nonlinear operation.But its effect at the output can be analyzed using linear system theory with the following assumptions about e(n): e(n)is uniformly distributed white noise; e(n)is a wide-sense stationary random process,i.e.,mean and co-variance are independent of time index n; x(n) e(n)is uncorrelated to all other signals such as input and other noise signals. e(n):round-off error 2021年2月 9

2021年2月 9  Round-off Noise Mathematical Model: usually modeled as an infinite precision system with an external error input.  Rounding is a nonlinear operation. But its effect at the output can be analyzed using linear system theory with the following assumptions about e(n):  e(n) is uniformly distributed white noise;  e(n) is a wide-sense stationary random process, i.e., mean and co-variance are independent of time index n;  e(n) is uncorrelated to all other signals such as input and other noise signals

/966 Let the wordlength of the output be W-bits,then the round-off error e(n)can be given by 2W-2 sgn 2-(m-1) 2-(m-1) W-1 W-1 se(n) (11.6) 2 2 2 22m1 2-(m-1) The error is assumed to be uniformly distributed over the interval in (11.6),the corresponding probability distribution is shown below,where A is the length of the interval,i.e.,A=2-(W-1) Pe() 片 2021年2月 10

2021年2月 10  Let the wordlength of the output be W-bits, then the round-off error e(n) can be given by  The error is assumed to be uniformly distributed over the interval in (11.6), the corresponding probability distribution is shown below, where Δ is the length of the interval, i.e., Δ=2-(W-1) sgn 2W-2 W-1 W-1