lecture11-f08

lecture11-f08 - Administrative EE264 Digital Signal...

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EE264 Digital Signal Processing Lecture 11 Quantization in Digital Filter Implementations - II October 22, 2008 Ronald W. Schafer Department of Electrical Engineering Stanford University STANFORD UNIVERSITY, EE264 Administrative • HW 5 due on Tuesday, Nov. 4 by 5pm in EE264 drawer on 2 nd floor Packard. • Mid-term exam: Weds., Oct. 29 in class. – Building 200 - Room 203 – Covers material through Lecture 9 and HW 4. – Open textbook and one 8.5x11 sheet of notes (both sides • READ: Chapter 6 of DTSP and start Chapter 7. • Review Session: Thursdays 4:15 - 5pm Gates B03 (recording available online) • Office Hours: – RWS: Mon./Weds 10-11, and 12:15-12:45 – Raunaq: Mon. 5-7pm, Tues. 9-11am – Rahim: Friday 4-6pm • Grader: Pegah Afshar, Ramin Miri STANFORD UNIVERSITY, EE264 Overview of Lecture • Quantization in FIR implementations – Coefficient quantization – Roundoff before and after accumulation • Linear noise model – Scaling to avoid overflow • Quantization in FIR implementations – Coefficient quantization – Roundoff before and after accumulation • Linear noise model – Scaling to avoid overflow – Limit cycles STANFORD UNIVERSITY, EE264 Fixed-Point Implementation Issues • We need to represent coefficient and signal values by integers in a fixed range. • Quantization errors in coefficients imply shifts of poles and zeros (even instability). • For a given word-length, the quantization error is fixed in size. Therefore, signal values should be maintained as large as possible to maximize SNR. • If signal values get too large, additions can overflow (or clip), thereby creating large errors. • Thus, fixed-point implementations require careful attention to scaling the signal values.
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STANFORD UNIVERSITY, EE264 FIR Digital Filter y [ n ] = h [ k ] x [ n k ] k = 0 M = ((( h [0] x [ n ] + h [1] x [ n 1]) + ) + h [ M ] x [ n M ]) To implement this filter, we must do multiply followed by accumulation (MAC). accumulation ± ² ³ ³ ³ ³ ³ ³ ´ ³ ³ ³ ³ ³ ³ STANFORD UNIVERSITY, EE264 Roundoff Noise in FIR Filters • Using the MAC instruction with quantized coefficients and quantized input, we can compute • This sum of products can be computed with 32-bit precision; i.e., with no quantization of partial sums. • The result is usually quantized to 15 bits + sign for storage or D-to-A conversion. • The resulting noise power in the output is therefore y [ n ] = h [ k ] x [ n k ] k = 0 M ˆ y [ n ] = Q 15 y [ n ] {} = y [ n ] + e [ n ] σ e 2 2 /12 = 2 30 STANFORD UNIVERSITY, EE264 Remember the Linear Noise Model • Error is uncorrelated with the input. • Error is uniformly distributed over the interval • Error is stationary white noise, (i.e. flat spectrum) π ω Δ = = Φ | | , 12 ) ( 2 2 e j ee e ( Δ /2) < e [ n ] ( Δ /2). STANFORD UNIVERSITY, EE264 x [ n ] h [ M ] h [ M 1] h [2] h [0] h ˆ y [ n ] e [ n ] y [ n ] FIR Digital Filter with Quantized Output ˆ y [ n ] = Q 15 y [ n ] = Q 15 h [ k ] x [ n k ] k = 0 M = y [ n ] + e [ n ] Q 15 x [ n ] h [ M ] h [ M h [2] h [0] h y [ n ] ˆ y [ n ] In this case, noise is added directly to the output.
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lecture11-f08 - Administrative EE264 Digital Signal...

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