1. A Method to Correct Data Corrupted by Overflow Ben Christensen Jay Brady Dec. 5, 2013

2. Topics A brief introduction to overflow. How do we identify a signal which has been corrupted by overflow? How can we correct a corrupted signal? What kind of signals can be corrected? New moving average based unwrap method. Conclusion.

3. Overflow Overflow is a problem that can occur in Digital Signal Processing (DSP). Overflow occurs when a value is represented in binary using an insufficient number of bits. Systems are usually designed to avoid overflow, but this is not always possible.

4. Overflow The value of ‘5’ using two’s compliment is 0101 this requires 4 bits -- 1 ‘sign’ bit plus 3 ‘value’ bits If you try to represent a number using fewer bits, you may lose information and possibly change the sign bit. 0101 has a decimal value of 5 101 has a decimal value of -3!

5. Overflow Example: 9-bit Sine wave represented with 8 bits

6. Overflow Identification For continuous signals, look for sharp jumps or discontinuities. If the signal is noisy, you need to look at the distribution. ◦ The distribution will be limited to values within the representable range

7. Overflow Identification Example:

8. Correcting Signals with Overflow If continuous and noise-free, a simple ‘unwrap’ function can be used. But how can we correct the signal if noise is present? ◦ A more sophisticated ‘unwrap’ function can be used.

9. Assumptions The signal must be slowly-varying. The signal must be finely sampled. The noise cannot exceed the total bit- range. ◦ Our simulations were done using fairly low- frequency sinusoids.

10. Signal Model Signals are sinusoidal with added Gaussian noise.

11. Signal Model If the signal overflows, it has an added ‘overflow factor.’ m – number of times the signal has overflown (positive or negative). b – number of bits.

12. Moving Average Unwrap If can be subtracted from the original signal can be restored. The trick is finding m and n (i.e. how much correction and where?)

13. Correcting the signal Regions of overflow are identified using a moving average estimator ◦ The difference between the moving average and signal must be under the overflow detection threshold for some number of samples “c” The value of the function before and after these regions are used to determine the correction factor needed ( -2 b, 0, or 2 b ) The areas of overflow are removed, and the correction terms added

14. Using moving averages to locate overflow regions

15. 0-bit noise (no noise)

16. 7-bit noise

17. 8-bit noise

18. Conclusions Continuous signals with added noise that suffer from overflow can be corrected for given: The sampling frequency is much greater than frequencies that make up the signal The noise is limited below the bin limits

19. Questions?