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 roundoff 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