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Finite word lengths introduce quantization error in fixed-point systems. Truncation quantization causes a larger maximum error and a negative bias compared to rounding, but is easier to implement in hardware. Similarly, wraparound overflow is typically worse than saturation, but also requires more hardware.

The fractional B -bit two's complement number representation evenly distributes 2 B quantization levels between -1 and 1 2 B 1 . The spacing between quantization levels is then 2 2 B 2 B 1 B Any signal value falling between two levels is assigned to one of the two levels.

X Q Q x is our notation for quantization. e Q x x is then the quantization error.

One method of quantization is rounding , which assigns the signal value to the nearest level. The maximum error is thus B 2 2 B .

Another common scheme, which is often easier to implement in hardware, is truncation . Q x assigns x to the next lowest level.

The worst-case error with truncation is 2 B 1 , which is twice as large as with rounding. Also, the error is always negative, so on average it may have a non-zeromean (i.e., a bias component).

Overflow is the other problem. There are two common types: two's complement (or wraparound ) overflow, or saturation overflow.

Wraparound

Saturation

Obviously, overflow errors are bad because they are typically large ; two's complement (or wraparound) overflow introduces more error than saturation, but is easierto implement in hardware. It also has the advantage that if the sum of several numbers is between -1 1 , the final answer will be correct even if intermediate sums overflow! However, wraparound overflow leaves IIR systemssusceptible to zero-input large-scale limit cycles, as discussed in another module. As usual, there are many tradeoffs to evaluate, andno one right answer for all applications.

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Source:  OpenStax, Digital signal processing: a user's guide. OpenStax CNX. Aug 29, 2006 Download for free at http://cnx.org/content/col10372/1.2
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