A Method to Correct Data Corrupted by Overflow

Ben ChristensenJay Brady

Dec. 5, 2013

TopicsA 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.

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.

OverflowThe value of ‘5’ using two’s compliment is 0101this 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!

OverflowExample:

9-bit Sine wave represented with 8 bits

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

Overflow IdentificationExample:

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.

AssumptionsThe 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.

Signal Model

Signals are sinusoidal with added Gaussian noise.

Signal ModelIf the signal overflows, it has an

added ‘overflow factor.’

m – number of times the signal has overflown (positive or negative).

b – number of bits.

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?)

Correcting the signalRegions 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 ( -2b, 0, or 2b)

The areas of overflow are removed, and the correction terms added

Using moving averages to locate overflow regions

0-bit noise (no noise)

7-bit noise

8-bit noise

ConclusionsContinuous 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

Questions?