1. Quick and Easy Binary to dB Conversion George Weistroffer, Jeremy Cooper, and Jerry Tucker Electrical and Computer Engineering Virginia Commonwealth University Richmond, VA.

2. 2 New algorithm for converting binary integers to decibels (dB) Advantages over known alternative methods –Faster No real-time floating point operations Only real-time arithmetic is one integer subtraction –Requires fewer resources (i.e. less memory) –Simple to implement Limitations –Reduced precision High precision is not typically needed for dB’s Precision can be increased at the cost of more memory.

3. 3 Motivation Analog Mux n-bit ADC Level detector V1V1 V2V2 VkVk  w, n-bit word Binary output samples of analog input voltages. V 1,V 2,…V k dB

4. 4 The problem Consider an unsigned binary integer w expressed as Typically w is the output of an n-bit ADC and is changing in real-time. Let G be the comparison in dB of w to the maximum value of w. Then G Changes in real time Fixed – depends only on number of bits in w.

5. 5 For any particular value of n, the G c term is a constant and does not have to be computed in real time.

6. 6 Computing 20log10(w) The difficulty is the evaluation of the 20log 10 (w) term. The input w changes for each sample. 20log 10 (w) must continually be recalculated. A fast method is needed for this calculation. Such a method can be developed by taking advantage of the fact that dB values typically do not need to be represented with high precision. Therefore, it is possible to use an approximation for w.

7. 7 The approximation of w Then assuming w ≠ 0, w can be expressed as where bit m with m < n is the left-most (most significant) “1” of w. To approximate w, use only the first r-bits to the right of the most significant “1” of w. Let w min = (1w m-1 w m-2 …w m-r 0…0) 2 and w max = (1w m-1 w m-2 …w m-r 1…1) 2 Then w min ≤ w ≤ w max Note that when m ≤ r then w min = w = w max. w  w min = (1w m-1 w m-2 …w m-r 0…0) 2 r-bits Bit m

8. 8 The approximation of G Recall that G = 20log 10 (w)-20log 10 (2 n -1) Therefore, G min = 20log 10 (w min )-20log 10 (2 n -1) Using G  G min and letting R = (w m-1 w m-2 …w m-r ) 2 it is easy to show that G can be approximated using three terms. where Depends only on m Depends only on R A constant

9. 9 The approximation error The worst case error occurs when m=n-1. Using The error will never exceed this worst case error.

10. 10

11. 11

12. 12 Refinements to the algorithm Combine the G c value with each term in the G m table. –This is slightly more efficient than subtracting the G c value each time a dB value is computed. Scale table values to avoid fractional values. –Let k be the scale factor. –If resolution is ½ dB let k = 2, if resolution is ¼ dB let k = 4, etc. Represent the negation of G as an unsigned integer. –G is always negative. Therefore, evaluate –kG  k(G C – G m ) - kG R

13. 13

14. 14 Example: Let w = (1 0101 01011000011) 2 m = 15, R = (0101) 2 = 5, 2(G c – G m (15)) = 12, 2G R (5) = 5 -2G  12 – 5 = 7, G  -3.5 dB, Actual value G = -3.517 dB

15. 15

16. 16 Conclusion A technique that is fast, easy to implement, and memory efficient has been developed for converting a binary integer to dB’s. It takes advantage of the fact that in practice decibels do not need to be calculate to high precision. It uses two small lookup tables to find the dB values. The method has been illustrated for the specific case where w is a 16-bit unsigned integer, and where four bits following the most significant “1” of w are used to approximate w. –Requires two 16 byte tables. –The dB value obtained is always within ½ dB of the actual value. The algorithm has been implemented and verified using both 8051 assembly language and VHDL.