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For many real-time applications it is useful to be able to either process or display an array of time-domain or frequency-domain vectors in compact numerical form. One way of implementing this with an eye toward rapid code execution is to convert the vectors to 8-bit log-power form. The procedure consists of two steps -- first converting each vector into unsigned magnitude and then converting this magnitude to log form. STEP 1 -- QUICK COMPUTATION OF MAGNITUDE Although power can be computed by squaring and summing the vector components, the doubling in significant bits required leads to a loss in dynamic range, as well as to doubling the number of shifts required for normalization. On the other hand, using the magnitude requires computing a square root, a time-consuming process. The relationship $MAGNITUDE\; =\; SQRT(A2+\; B2)$ can be approximated to within a dB using the relationships: MAGNITUDE = |A| + 1/2 |B| for A > B MAGNITUDE = |B| + 1/2 |A| for A < B thereby affording considerable computational savings by bypassing the square root operation. The approximation is best when either A or B is zero, and worst when A=B, where SQRT(2) is taken as 1.5. STEP 2 -- QUICK COMPUTATION OF LOG POWER Log power can be obtained either by normalizing a number or by utilizing its normalized floating-point representation. For an array scaled to fractions less than unity (other scalings are also possible) the basic relationship is: if x = $SQRT(A2+\; B2)$ < 1, then log (x) = 8 x (31 + exp(x) ) + man(x). where exp(x) refers to the unbiased signed exponent of x, while man(x) refers to the uppermost bits of the mantissa, with the most significant bit dropped (which is always 1). The factor of 8 represents a shift of 3 places, while the bias of 31 is used to remove the negative exponent for the fractional form. Thus, the upper 5 bits of the 8-bit word are used for the exponent, while the 3 lower bits are used to interpolate between successive 6-dB steps in the exponent (6-dB, not 3-dB, because the numbers represent magnitude, not power). 0 dB = 255 eeeee mmm = 11111 111 (255 used since 256 is out of range)) - 6 dB = 248 eeeee mmm = 11110 000 -12 dB = 240 eeeee mmm = 11100 000 . . -192 dB = 0 eeeee mmm = 00000 000 The division of 6 dB into 8 parts (3 bits) gives slightly better than 1 dB resolution. The maximum dynamic range is 192 dB. Notice that this operation is essentially one of masking and shifting to remove the MSB of the mantissa (the DEC floating-point format automatically drops this redundant bit, referring to it as a "phantom" bit). For display purposes, using a bipolar DAC, the MSB can be used as a negative d.c. scope trigger by using only 4 (instead of 5) bits for the 6-dB divisions 0 dB = 127 eeeee mmm = 01111 111 (127 used to leave 128 free for triggering bipolar DAC) - 6 dB = 120 eeeee mmm = 01111 000 -12 dB = 112 eeeee mmm = 01110 000 . . -96 dB = 0 eeeee mmm = 00000 000 One result of this is a reduction in maximum possible dynamic range from 192 to 96 dB. Other possibilities for display purposes are to use only 2 bits for the mantissa and 5 bits for the exponent or to use a DAC with more bits (10 or 12). For a DAC with a +/-10-volt output, the scaling is close to 10 dB/volt (96 dB over 10 volts). A simultaneous display of up to 8 channels can be accomplished by using a DAC with 12 bits and off-setting each consecutive channel by about 1/2 volt. Sequential or stacked displays can be generated from the same output by setting the trigger of the first channel to minus full scale and the other channels to minus half scale and adjusting the scope d.c. trigger level to trigger on the desired display, while adjusting the time base to sweep more slowly for the sequential-channel display and faster for the stacked-channel display.