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Linear to log conversion

Posted By Krishna Sankar On November 20, 2008 @ 6:40 am In DSP | 10 Comments

In signal processing blocks like power estimation used in digital communication, it may be required to represent the estimate in log scale. This post explains a simple linear to log conversion scheme proposed in the DSP Guru column on DSP Trick: Quick-and-Dirty Logarithms. ^[1]The scheme makes implementation of a linear to log conversion simple and small in a digital hardware like FPGA.

Consider an integer $N$ . The floating point representation is,

$N=M2^E$

where,

$E$ is the exponent and

$M$ is the mantissa.

Assume that $M$ is normalized, i.e $1\le M < 2$ .

Taking logarithm to the base 2,

$\log_2N=\log_2M+E$ .

In digital hardware implementations, finding the exponent $E$ is simple. Its just noting the index of the first bit which is 1 starting from MSB side.

For example consider an input number $N=12$ .

Expressed in binary on 8 bit bus, $N_b=00001100$ .

The value of $E$ in this example is 3.

Now, the part which remains to be computed is the mantissa $M$ . In this example,

$M=\frac{12}{2^E} = 1.5$ .

Given that $M$ lies in the range $1\le M < 2$ . this can be computed using a Look Up Table. The LUT can store $\log_2$ values of input between 1 to 2. The precision requirement determines the number of elements in the LUT. Let us assume that we want to have a precision of $\frac{1}{2^k}=0.0625$ , where $k=4$ . The look up table values will be as follows:

k=4
index. j	Linear = 1+j/2^k	LUT = log_2(Linear)
1	1.06250	0.0874628
2	1.12500	0.1699250
3	1.18750	0.2479275
4	1.25000	0.3219281
5	1.31250	0.3923174
6	1.37500	0.4594316
7	1.43750	0.5235620
8	1.50000	0.5849625
9	1.56250	0.6438562
10	1.62500	0.7004397
11	1.68750	0.7548875
12	1.75000	0.8073549
13	1.81250	0.8579810
14	1.87500	0.9068906
15	1.93750	0.9541963
16	2.00000	1.0000000

Table: Look up table values for logarithm computation

From the above look up table, we can see that mantissa of $M=\frac{12}{2^E} = 1.5$ corresponds to index of $j=8$ . It is inituitive to note that the array index $j$ can be found out by the simple formula,

$j=\left(\frac{N}{2^E}-1\right)2^k$ . To handle cases where this number can $j$ can be a fraction, the result is floored to the nearest integer, i.e.

$j=\lfloor\left(\frac{N}{2^E}-1\right)2^k\rfloor$

So the value of $N$ in $\log_2$ base is,

$N_{\log_2} = LUT(j) + E$ .

Once we have the number in $\log_2$ base, conversion to any other base is simple.

$\log_b(N) = \frac{\log_c(N)}{\log_c(b)}$ .

So the number $N$ in $\log_{10}$ base is,

$N_{\log_{10}} = \frac{N_{\log_2}}{\log_2(10)}$ .

Simulation Model

Simple Matlab/Octave script for computing the logarithm via the LUT based approach is provided. Click here to download Matlab/Octave script for performing linear to log conversion using LUT based approach ^[2]

linear to log using LUT

Figure: Linear to log conversion using LUT

Reference

DSP Trick: Quick-and-Dirty Logarithms – Ray Andraka, June 2000 ^[1]

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URL to article: http://www.dsplog.com/2008/11/20/linear-to-log-conversion/

URLs in this post:

[1] DSP Trick: Quick-and-Dirty Logarithms. : http://www.dspguru.com/comp.dsp/tricks/alg/quicklog.htm

[2] Matlab/Octave script for performing linear to log conversion using LUT based approach: http://www.dsplog.com/db-install/wp-content/uploads/2008/11/script_linear_to_log_conversion.m

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Simulation Model

Reference

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