Quick and Easy Binary to dB Conversion

George Weistroffer, Jeremy Cooper, and Jerry Tucker

Electrical and Computer Engineering

Virginia Commonwealth University

Richmond, VA.

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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.

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Motivation

Analog

Mux

n-bit

ADC

Level detector

Level detector

Level detector

V1

V2

Vk

w, n-bit word

Binary output samples of analog input voltages.

V1,V2,…Vk

dB

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The problemConsider 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

1 2 0 2( ... ... )n n iw w w w w

10 10 1020log 20log ( ) 20log (2 1)2 1

nn

wG w

Changes in real time Fixed – depends only on

number of bits in w.

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1020log (2 1)ncG

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

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Computing 20log10(w)

• The difficulty is the evaluation of the 20log10(w) term.

• The input w changes for each sample.

• 20log10(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.

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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 wmin = (1wm-1wm-2…wm-r0…0)2

and wmax = (1wm-1wm-2…wm-r 1…1)2

• Then wmin ≤ w ≤ wmax

Note that when m ≤ r then wmin = w = wmax.• w wmin = (1wm-1wm-2…wm-r0…0)2

1 2 0 2(1 ... )m mw w w w

r-bitsBit m

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The approximation of GRecall that G = 20log10(w)-20log10(2n-1)

Therefore, Gmin = 20log10(wmin)-20log10(2n-1)

Using G Gmin

and letting R = (wm-1wm-2…wm-r)2

it is easy to show that G can be approximated using three terms.

where 10

10

10

( ) 20log (2) 6.02

( , ) 20log 1 2

( ) 20log (2 1)

m

rR

nC

G m m m

G R r R

G n

( ) ( , ) ( )m R CG G m G R r G n

Depends only on m

Depends only on R

A constant

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The approximation error( )

10

1 2max 20log 1 when >

2

m r

rerror m r

max 0 when error m r

( 1 )

10

1 220log 1

2

n r

rworst case error

The worst case error occurs when m=n-1.

Using ( ) ( , ) ( )m R CG G m G R r G n

The error will never exceed this worst case error.

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Refinements to the algorithm• Combine the Gc value with each term in the Gm table.

– This is slightly more efficient than subtracting the Gc 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(GC – Gm) - kGR

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Example: Let w = (1 0101 01011000011)2

m = 15, R = (0101)2 = 5, 2(Gc – Gm(15)) = 12, 2GR(5) = 5

-2G 12 – 5 = 7, G -3.5 dB, Actual value G = -3.517 dB

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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.