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`DSP-pP routine computes magnitude
`
`Amarnath Palacherla
`Minochip Teclwwlngg, ClwndW, AZ
`The 320C10 DSP rrP pmgram in Listing I computes
`the magnitude of a complex number, X+jY. Fiirding
`the sum of che squarcs of the real (X) and imaginary
`(Y) parts of a complex number is no problem with a
`320C10 because the IC performs a multiplication in
`one instruction cycle (122 nsec at 32.8 MHz).
`But finding the square root of the sum of the squares
`is not trivial. If you use sommon iterative methods,
`such as the Newton'Raphson rnethod, the B20Cl0 wiU
`take 350 or more clock cycles for this calculation (4p
`psec).
`The method embodied in the listing takes only about
`30 clock cycles (3.? psec) and uses the following ap-
`proximation:
`' I - \fE.o
`MAX (X,y) + KrMIN (x,y).
`the MAX(X,Y) function examines the real and imagi-
`nary parts of a complex number and selects the part
`having the largest absolutp value. MIN(X,Y) similarly
`looks for the absolute minimum part of each complex
`number in your sample set. You can choose any vdue
`for the eonstsnt K between 0.25 and 0.31. Recom.
`mended values for K are 0.26?304 for exact extimated
`mean, and 0.300585 for minimum variance.
`Fig I graphs the pertentage error between the ac-
`tual value of the complex numbefs magnitude and the
`dgorithm's estimate for phase angles in the filst oc-
`tant. Note that the maximum error is about ll,% for
`K=0.26?304, but the mean ernor is 0. The second rou-
`tine in Listing t modiftes the algorithm slightly to
`lessen the estimate's error for phase angles greater
`lhan 41". In pseudocode, the modified algorithm is
`
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`xl = MAX f A3S(x), ABS(y) I ;
`yl = MIN I ABS(x), AAS0) I ;
`IF ( xl == yl ) THEN Z - SQRT(Z).xI ; RETURN ;
`z=xl+K.yl;
`IF I yl > xlrTAN({l) ] THEN z -- 2.1.09865; RETURN
`
`This algorithm yields a manimum ertor of less thar
`67o, Qf cource, with this modification, the rnean ener
`6t
`is no longer zero.
`
`References
`l. Onoe, Morio, "Fast Amptitude Approximation Yielding
`Either Exact Illean or Minimum Deviation for Quatrature
`Pairs," hoceedings of the IEEE, July l9?2.
`2. Knuth. D E, The At1' Of Computcr Prcgmmming: Vol
`ume Il, Addison-Wesley Publishing Co, Reading, MA.
`
`To Vote For This Design, Circle No ?50
`
`LISTING I-COMPLEX.NUMBER MAGNITUDE ROUTINE
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`Petitioner Sirius XM Radio Inc. - Ex. 1005, p. 1
`
`

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`Petitioner Sirius XM Radio Inc. - Ex. 1005, p. 2
`
`